.. 5:626... . 4 t 5‘! l! x v. ... L..!P.,I1....ufimuwfih_ V , . .. DI Al! ‘ ll} 2. .- al‘; . our: ‘1 .v I». 3.}. I 5.3me fl. 5.)}!!! 3'" 4!: t .. . hohuhflh . ,3. Ir .H . . “Hf ufismww‘rhm . Is: 1... ... : i gageuhix .. . Hun“: I .- . .. It. 3.. . rilcstli; :4: 1....“ Rita“ itsaflfl. {NIH-’0 . L9. .122 ‘1 V ‘1 .1 in .fiWI. h , X. . ... - if” I: 11.1. If): .6}!!! .l 5!.» . .1 it x i. 7 it. w . .nwuum 94. 2%? .nfiznnnnnunfi » . x, .l.‘ italicuwjnfi .fllflhnfiflfl urn. Y‘i-lbfia' « 5 MM“ c an tate 2M5 Unisersity This is to certify that the dissertation entitled QUANTITATIVE STRUCTURE DETERMINATION OF NANOSTRUCTURED MATERIALS USING THE ATOMIC PAIR DISTRIBUTION FUNCTION ANALYSIS presented by AHMAD SALAH MASADEH has been accepted towards fulfillment of the requirements for the Ph.D. degree in PHYSICS 'Major Professor's finature 10/l?/o? Date MSU is an affirmative-action, equal-opportunity employer PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 5/08 K;/Prolecc&Pres/ClRC/DaIeDue indd QUANTITATIVE STRUCTURE DETERMINATION OF NANOSTRUCTURED MATERIALS USING THE ATOMIC PAIR DISTRIBUTION FUNCTION ANALYSIS B y Alunad Sal-ah Masadeh A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Physics 2008 ABSTRACT QUANTITATIVE STRUCTURE DETERMINATION OF NANOSTRUCTURED MATERIALS USING THE ATOMIC PAIR DISTRIBUTION FUNCTION ANALYSIS By Ahmad S. hilasadeh The employed experimental method in this PhD. (.lissertation research is the atomic pair distribution function (PDF) technique specializing in high real space resolution local structure determination. The PDF is obtained via Fourier transform from powder total scattering data including the important local structural information in the diffuse scattering intensities underneath, and in-between, the Bragg peaks. Having long been used to study liquids and amorphous materials, the PDF technique has been recently successfully applied to highly crystalline materials owing to the advances in modern X-ray and neutron sources and computing power. The conventional XRD experiments probe for the presence of periodic structure which are reflected in the Bragg peaks. Local structural deviations or disorder mainly affect the diffuse scattering background. In order to have information about both long-range order and local structure disorder, a technique that takes both Bragg and diffuse scattering need to be used, such as the atomic pair distribution function (PDF) technique. This PhD. work introduces a PDF based methodology to quantitatively study nanostructure materials in general. The introduced methodology have been applied to a size-dependent structural study on CdSe nanoparticles (NPs). Quantita- tive structural information about structure, crystallinity level, core size, NP size, and inhomogeneous internal strain in the studied NPs have been obtained. This method is generally applicable to the characterization of the nano-scale solid, many of which may exhibit complex disorder and strain. The introduced methodology have been also applied on technologically important system, ultra-small CdSe NPs. To my parents To my wife Eiman and my daughter Jana To my family, friends, and all those who once helped me To the memory of my beloved brother Mustafa Yousef Al-Hajdarwish iii Acknowledgments It’s such a thrill to be in this moment. This moment wouldn’t have been possible without all the loving and beloved people around me. This moment wouldn’t have any meaning without them. Each one of them is so special on the way to this moment. This moment belongs to them all. No words can express my greatest appreciation and respect to S. J. L. Billingc, my Ph.D. adviser. The past four years, I spent in his group, has been invaluable and extremely beneficial. I saw unlimited patience, unsparing guidance, and unfailing trust. Above all, he always strives to help me succeed. I am full of gratitude to my thesis committee members, W. G. Lynch, S. D. Mahanti, M. Donahue and S. Tessmer for their devoted supervision. My highest respect to each of them. My PhD. research benefits greatly from collaborations with M. Kanatzidis, M. Jansen, S. Bruehne and S. Rosenthal. Special thanks to each of them. It has been a great pleasure working with them. I have fortunately received magnanimous help from D. Robinson and D. Wermeille during experiments, to whom I am very very thankful. I also like to thank J. McBride, C. Malliakas and A. Karkamkar for help with sample characterizations. Deepest appreciation goes to X. Qiu who has helped me with a great deal in almost all aspects when I just started my research. May thanks to P. Juhas and H. J. Kim for the valuable assistance. My most sincere salute to E. Bozin who was my teacher when I joined the group and gave his lavish encouragements along with his amusing accompany. It has been great fun working with Gianluca Paglia, Mouath Shatnawi, Moneeb Shatnawi, and He Lin. They are all great teachers and friends. I would like to thank D. Simmons and C. Cords for taking care of all the aspects of my graduate study and research. This moment is especially for my family. My father, my mother, my wife, all my family, I should have spent all the time with you all. This moment belongs to you all. iv Contents List of Tables ................................. viii List of Figures ................................ x 1 Introduction ................................ 1 1.1 The birth of nanoscience ......................... 1 1.2 Importance of nanostructure determination ............... 2 1.3 Classification of nanostructures ..................... I 1.4 Nanostructure types in three dimensions ................ 4 1.4.1 Mesoporous materials ....................... 6 1.4.2 A-‘letallic nanoparticle ....................... 7 1.4.3 Semiconductor nanoparticles ................... 7 1.5 The effect of nanoscale dimensions on properties ........... 9 1.6 Experimental methods for nanostructure characterization ...... 11 1.6.1 Transmission Electron Microscopy (TEM) ........... 12 1.6.2 Extended X-ray Absorption Fine Structure (EXAFS) ..... 13 1.6.3 Nuclear Magnetic Resonance (NMR) .............. 15 1.6.4 X-ray diffraction (XRD) ..................... 16 1.6.5 Small-angle scattering X—rays (SAXS) .............. 17 1.6.6 Scanning probe microscope (SPM) ............... 18 1.6.7 Total scattering technique .................... 19 1.7 N anostructure determination problem ................. 20 1.8 Outline of this thesis ........................... 23 2 The Atomic Pair Distribution Function (PDF) Method ...... 25 2.1 Introduction ................................ 25 2.2 The atomic pair distribution function (PDF) technique ........ 26 2.3 PDF method: past and present ..................... 29 2.4 Description of the PDF experiment ................... 31 2.5 The rapid acquisition PDF (RAPDF) experiment ........... 32 2.5.1 The image plate (IP) and PDF method ............. 33 2.5.2 Description of the experiment .................. 34 2.5.3 Real space refinement ....................... 36 2.5.4 Early results: standard nickel .................. 39 2.6 The PDF for nanostructured materials ................. 41 2.7 Extracting information from the PDF .................. 42 2.7.1 Peak position ........................... 44 2.7.2 Peak width ............................ 44 2.7.3 Peak intensity ........................... 44 2.8 Conclusions ................................ 45 3 Real Structure of Na3BiO4 ....................... 47 3.1 Introduction ................................ 48 3.2 Experimental methods .......................... 49 3.2.1 Sample syntheses ......................... 49 3.2.2 High resolution X-ray powder diffraction ............ 52 3.2.3 PDF data collection ....................... 54 3.2.4 High Resolution Transmission Electron Microscopy ...... 56 3.3 Structural analyses ............................ 56 3.3.1 X—ray analysis and structure solution .............. 56 3.3.2 Pair distribution function analysis ................ 58 3.3.3 High Resolution Transmission Electron IVIicroscopy ...... 61 3.4 Conclusion ................................. 64 4 Quantitative Structure Determination of CdSe N anoparticles 66 4.1 Introduction ................................ 67 4.1.1 Previous work ........................... 70 4.1.2 Stacking Faults in Close-packed Structures ........... 73 4.2 Experimental details ........................... 75 4.2.1 Sample preparation ........................ 75 4.2.2 The atomic PDF method ..................... 76 4.2.3 High-energy x-ray diffraction experiments ........... 78 4.3 Results and discussion .......................... 82 4.3.1 Nanoparticle structure ...................... 82 4.3.2 Nanoparticle size ......................... 92 4.3.3 Internal strain .......................... 95 4.4 conclusion ................................. 97 5 Structure of ultra-small CdSe nanoparticles using PDF analysis 99 5.1 Introduction ................................ 99 5.2 Experimental details ........................... 100 5.2.1 Sample preparation ........................ 100 5.2.2 The atomic PDF method ..................... 101 5.2.3 High-energy x—ray diffraction experiments ........... 103 5.3 Results and discussion .......................... 107 5.3.1 Nanoparticle structure ...................... 108 5.3.2 Nanoparticle size ......................... 117 5.3.3 Internal strain .......................... 120 5.4 Conclusion ................................. 122 6 Concluding Remarks ........................... 124 6.1 Summary of the thesis .......................... 124 6.1.1 Combining complementary probes ................ 126 vi 6.1.2 Applying the PDF-based—Methodology in new name—systems . 127 6.2 Future work ................................ 128 6.3 Acknowledging funding agencies ..................... 128 RAPDF Data Collection ........................ 129 A.1 Sample preparation ............................ 129 A2 Requirements of the experiment ..................... 130 A21 Extended Q range ........................ 130 A22 Achieving good counting statistics ................ 133 A23 Achieving good signal to noise ratio ............... 133 A.3 Setup of the RAPDF experiment .................... 134 A4 Diffraction intensity ............................ 141 A5 Collecting the RAPDF data ....................... 141 A51 Sample position .......................... 145 A52 Varying the setup ......................... 145 A53 Exposure time ........................... 146 A54 Measuring the background .................... 146 RAPDF Raw Data Processing .................... 148 B1 Startup with FIT2D ........................... 149 B2 RAPDF data reduction .......................... 150 B.2.l Data homogeneity ........................ 150 B.2.2 Overcxposed spots ........................ 151 B.2.3 Objects in the beam path .................... 154 B.2.4 Usage of masks .......................... 154 8.3 Calibration ................................ 155 B.3.1 Calibration using an external standard ............. 156 B.3.2 Standard-free calibration ..................... 158 B4 RAPDF data integration ........................ 159 B.4.1 Debye—Scherrer rings geometry ................. 159 B.4.2 Integration parameters ...................... 160 8.4.3 Integration Process ........................ 160 B.4.4 Normalization by the monitor counts .............. 161 RAPDF Data Analysis ......................... 164 CI Data Analysis .............................. 164 C2 Additional corrections to the RAPDF data ............... 166 C21 Oblique incident angle ...................... 167 C22 Compton intensities at high Q and fluorescence ........ 167 C23 Energy dependence of the IP response ............. 168 C3 Estimate of standard deviations ..................... 168 Image plate characteristics as an X-ray detector .......... 171 DJ Use of the image plate as an X-ray detector .............. 171 D2 Principle of X-ray image plate detection ................ 171 D3 Characteristics of image plate detectors ................. 172 vii D4 The IP and PDF method ......................... E The PDF of a single spherical particle ................ Bibliography ................................. viii 176 List of Tables 1.1 3.1 3.2 4.1 4.2 4.3 4.4 5.1 5.2 5.3 C1 .5}. 5.5 Examples of reduced-dimensionality systems .............. 5 The average crystallographic structure of ,i3—Na3BiO4 from synchrotron powder data in comparison with a—Na3BiO4 . ............. 53 The resulted refined parameters for average structure using Rietveld refinement .................................. 58 The results of the refined paran‘ieters from PDF analysis ....... 62 CdSe nanoparticle diameter as determined using various methods. . 76 The refined residual (Rm) values obtained from PDF analysis assuming the wurtzite and zinc—blend structure models with space group P63mc and F 43m, respectively. In both models isotropic atomic displacement factors (Uiso) are used. ......................... 84 The refined parameters values obtained from PDF analysis assuming the wurtzite structure , space group P63mc, with different stacking fault densities (SFDs). .......................... 90 The first PDF peak position (FPP) and width (F PW) for different CdSe nanoparticle sizes and the bulk. ................. 97 CdSe nanoparticle diameter as determined using various methods. . 101 The refined parameters values obtained from PDF analysis. The wurtzite structure (Wur), space group P63mc, with no stacking faults present in the structure. ............................. 112 The refined parameters values obtained from PDF analysis. The wurtzite structure (Wur). space group P63mc, with different stacking fault den— sities present in the structure. ..................... 114 The refined parameters values obtained from PDF analysis. The wurtzite structure (Wur), space group P63mc, with different stacking fault den— sities present in the structure. Rw = 0.25. ............... 117 The first PDF peak position (FPP) and width (FPW’) for different CdSe nanoparticle sizes and the bulk. ................. 120 ix B.1 Comparison of the image plate formats Mar3rl50 and Mar2300. . . . . 149 List of Figures 1.1 1.2 1.3 1 .4 1.5 1.6 2.1 2.2 2.3 2.4 2.5 3.1 3.2 3.3 3.4 3.5 h‘lesoscopic regime between molecules, namiparticle and bulk matter . Example of three-dimensional nanostructure material types . . . . . . Schematic comparison of the band structure and electronic states of semiconductor nanocrystals and bulk semiconductor. ......... Calculated surface-to-volume atom ratio for a spherical gold nanopar- ticles as a function of particles size. ................... Schematic diagram showing the principle of a STM ........... Diffraction pattern of CdSe. bulk vs 2 nm nanoparticle. ........ Calculating the radial distribution function from two dimensional square lattice. .................................. T we dimensional contour plot of the \i powder diffraction raw data from the Mar345 image plate detector .................. Experimental F (Q) C (r), and the crystallogrzmhic model fit of the. Ni powder ................................... Calculating the pair distribution function from cupper (Cu). ..... Using sphere envelope function 136, the pair distribution functions are calculated for two nanoparticles with different sizes. (top) 20.0 and (bottom) 40.0 A. used to ........................ Cell used for the electrocrystallization of ,13-Na3BiO4 . (1) wires leading to the potentiostat, (2) das inlet, (3) connectors, (4) Pt electrodes, (5) furnace, (6) nickel crucible ........................ STM images of ,B—Na3BiO4 ....................... High resolution X—ray powder diffraction of Na3BiO4 at ambient con- ditions .................................... The experimental reduced structure function F (Q) of Na3BiO4 and the corresponding PDF. ......................... Average crystal structure of ,d-Na3BiO4 at ambient conditions. xi 11 19 21 28 40 42 43 3.6 3.7 3.8 4.2 4.3 4.5 4.6 4.7 4.8 4.9 The experimental C(r) (solid dots) and the calculated PDF from re- fined structural model (solid line) of fi-NagBiO4 .The difference curve is shown offset below. (a) without refining the occupancy, (b) with refining the occupancy, (c) manually setting the occupancy of Bi atom at (0,0.0) site to be 0.0. ......................... The experimental C(r) (solid dots) and the calculated PDF from the refined structural model (solid line) of ,B-NagBiOLjThe difference curve shown offset below. (a) without refining the occupancy, (b) manually setting the occupancy of Bi atom at (0,0,0) site to be 0.0. ...... HRTEM on domain crystals of Na3BiO4 ................ Comparison of the optical absorption thresholds of CdS nanocrystals of 3 run average diameter: experimental data, tight-binding calculation, and effective-mass calculation ...................... Powder X—ray diffraction spectra of CdSe nanocrystallites compared with the bulk wurtzite peak positions ................. Relative positions of the three layers A, B and C for a close-packed structure. ABAB... close—packed hexagonal arrangement (left) and ABCABC'... face-centered cubic arrangement (right). ......... TEM image of CdSe nanocrystal prepared using the method described in the text. CdSe obtained by 1200 seconds (left) and 15 seconds (right) nucleation. The line-bar is 10 nm in size in both images. ....... (a) Room temperature UV-vis absorption and (b) photoluminescence spectra from the sample of CdSe nanocrystals. (O) CdSeI, (A) CdSeII, (Cl) CdSeIII ................................. Two dimensional XRD raw data collected using image plate detector from (a) CdSe bulk and (b) nanoparticle CdSeIII samples. ...... (a) The experimental reduced structure function F (Q) of CdSe nanopar- ticle with different diameters and (b) the corresponding PDF, C(r), obtained by Fourier transformation of the data in (a) with Qmag; :- 25.0 A—l, from top to bottom: bulk, CdSeIII, CdSeII and CdSeI. Fragments from the (a) wurtzite structure, space group (P63mc) and (b) zinc-blende structure, space group (F 43m). ............ (Color online) The experimental PDF, C(r), with Qmam = 19.0 A" 1 (blue solid dots) and the calculated PDF from refined structural model (red solid line), with the difference curve offset below (black solid line). PDF data are fitted using (a) wurtzite structure model, space group P63mc and (b) zinc-blende model with space group F 43m. In both models isotropic atomic displacement factors (Uiso) are used. xii 60 62 63 69 72 74 76 77 79 81 4.10 (Color online) The experimental PDF, C(r), with Q7nag; = 19.0 A"1 (blue 4.11 4.12 4.13 5.1 5.2 5.3 5.4 solid dots) and the calculated PDF from refined structural model (red solid line), with the difference curve offset below (black solid line). PDF data are fitted using wurtzite structure model (a) with no stack- ing fault and (b) with 33% stacking fault density for bulk and 50% for all nanoparticle sizes. In both cases anisotropic atomic displacement factors (UaniSO) are used ........................ 86 The the enlargement in the the ADPs along the z-direction for Se site U33, as a function of the stacking fault density. ............ 88 (Color online) The experimental PDF, C(r), shown as solid dots. Sphere envelope function (Eq. 4.5) is used to transform the calculated PDF of bulk CdSe, using wurtzite structure containing 50% stacking fault density, to give a best fit replication of the PDF of CdSe nanoparticles (red solid line). The inset shows on an expanded scale for the high- 1‘ region of experimental (7(7) on the top of simulated PDF data for different diameters of CdSe nanoparticles (solid line). (a) CdSeIII, (b) CdSeII, (c) CdSeI. Dashed lines are guides for the eye. ........ 94 (a) The first PDF peak. (0) bulk, (o) CdSeIII, (Cl) CdSeII and (A) CdSeI fitted with one Gaussian (~--.) Dashed line represents the posi- tion of first PDF peak in the bulk data. (b)(A) The first PDF peak width vs nanoparticle size, obtained from one Gaussian fit. Dashed line represents the width of first PDF peak in the bulk data. (c) Strain in Cd-Se bond (A‘r/r)(%) vs nanoparticle size. (I) Bond values obtained from the local structure fitting and (0) obtained from one Gaussian fit to the first PDF peak. Dotted curves are guides for the eye. ..... 96 Z-STEM image of CdSe nanocrystal prepared using the method de— scribed in the text. The associated band edge absorption centered at 417 nm. The line-bar is 4 nm in size. .................. 102 (Color online) UV—Vis spectra of the CdSe nanoparticles used in this study. The ultra-small nanoparticles that had absorption maximum at 402 red-shifted to 417 nm after washing. ................ 103 Two dimensional XRD raw data collected using image plate detector from (a) CdSe bulk and (b) nanoparticle CdSe552 samples. ..... 105 (a) The experimental reduced structure function F (Q) of CdSe nanopar- ticles with different diameters and (b) the corresponding PDFs, C(r), obtained by Fourier transformation of the data in (a) with Qmam = 25.0 A—l. From top to bottom: bulk, CdSe586, CdSe552, CdSe470 and CdSe417. ............................... 106 xiii 5.5 5.6 5.7 C1 CD A.1 A.2 A.3 A.4 Fragments from the (a) wurtzite structure, space group (P637720) with (ABAB ...) layer sequence and (b) zinc-blende structure, space group (F 43m) with (ABCABC' ...) sequence. ................. 109 (Color online) The experimental PDF, 0 (r), with Qmm- = 19.0 AT1(blue solid dots) and the calculated PDF from refined structural model (red solid line), with the difference curve offset below (black solid line). PDF data are fitted using wurtzite structure model (a) with no stack— ing fault and (b) with 33% stacking fault density for bulk and 50% for all nanoparticle sizes. In both cases anisotropic atomic displacement factors (Uamjso) are used ........................ 111 (Color online) The experimental PDF, C(r), with QmaJ: = 19.0 AT1(blue solid dots) and the calculated PDF from refined structural model (red solid line), with the difference curve offset below (black solid line). PDF data are fitted using wurtzite structure model with 50% stacking fault density (a) one phase fit CdSe (Wur) (b) two phase fit, Cd (cubic) and CdSe (W'ur). ............................. 116 The experimental PDF, 6' (1'), shown as solid dots. The simulated PDF of CdSe nanoparticles (red solid line), where sphere envelope function (Eq. 4.5, Ch. 4) is used to transform the calculated PDF of the refined wurtzite bulk structure, to give a best fit replication of the PDF of CdSe nanoparticles. The inset shows on an expanded scale for the high—r region of experimental C(r) on the top of simulated PDF data for different diameter of CdSe nanoparticles (solid line). (a) CdSe586, (b) CdSe552, (c) CdSe470, (d) CdSe417. ................ 119 (a) The first PDF peak, (0) bulk, (o) CdSe586, (Cl) CdSe552, (A) CdSe470 and (I) CdSe417 fitted with one Gaussian (, ~ ). Dashed line represents the position of first PDF peak in the bulk data. (b)(A) The first PDF peak width vs nanoparticle size, obtained from one Gaussian fit. Dashed line represents the width of first PDF peak in the bulk data. (c) Strain in Cd-Se bond (Ar/r)(%) vs nanoparticle size. (I) Bond values obtained from the local structure fitting and (0) obtained from one Gaussian fit to the first PDF peak ................. 121 Sample holders used in rapid acquisition PDF (RA-PDF) experiment 130 Experimental F (Q) of Ni powder data using RAPDF technique with two sample geometries, flat plate vs. capillary ............. 131 Experimental (7(7) of Ni powder data using RAPDF technique with two sample geometries, flat plate vs. capillary geometry ....... 132 Schematic diagram of the rapid acquisition PDF (RA—PDF) experi- ment layout ................................ 135 xiv A5 A6 A7 A8 B.l B.2 C.1 C.2 D.1 D.2 E.1 (a) Air scattering intensity from the second flight path region, with and without balloon vs the typical background. (b) Improving instrument resolution by optimizing the experiment setup factors .......... 140 Two dimensional contour plot of Ni data from the Mar345 Image Plate Detector .................................. 142 Two dimensional contour plot of fci-HoMan data from the ZVIar345 Image Plate detector, spotty vs. not spotty data ............ 143 Experimental C(r) and F (Q) of fci-HoMan processed from RAPDF data, spotty vs. not spotty data ..................... 144 Two dimensional contour plot of Til 5210.581) raw data from the Mar345 Image Plate Detector. .......................... 153 The effect of the overexposed spots on the integrated intensity. . . . 163 Three panels figure, I (Q), F (Q), and 0(1) of an analyzed Ni data. . 165 Estimated relative standard deviations of the IP response as a function of photon number absorbed in the area .................. 169 Energy level scheme of detection media BaFBrzEuQ+ used in the image plate phosphor layer ........................... 173 The dependence of the IP response on the incident photon energy . . 176 A spherical particle with radius R. An atom a distance 'r' from the center of the particle can have a shell of radius 1“ that is only partially embedded within the particle. ...................... 180 XV Chapter 1 Introduction 1.1 The birth of nanoscience .\Vlichael Faraday, one of the greatest experimental physicists. became famous for his works on electro—magnetism. What is less known is that he made the first experiments with nanoparticles (gold colloids) and thus initiated the fields of nanoscience and nanotechnology [1] In December 1959, R. Feynman gave a very interesting talk entitled “There’s Plenty of Room at the Bottom.”1 Feynman’s ~ 7,000 words talk was an invitation I. to explore a new field of science ‘ nanoscience and nanotechnology” before “nano” terminologies appeared on the horizon. “What I want to talk about,” Feynman said, “is the problem of manipulating and controlling things on a small scale. . . . What I have demonstrated is that there is room-that. you can decrease the size of things in a practical way. I now want to show that there is plenty of room. I will not now discuss how we are going to do it, but only what is possible in principle....We are not doing it simply because we haven’t yet gotten around to it.” Now, nearly 50 years later, the nanotechnology revolution is in full swing. However, as I will discuss, the basic problem of determining 3D structure at the nanoscale is not fully solved. 1R. Feynman, There’s plenty of room at the bottom, Eng. Sci. 23, 22 (1960) reprinted in J. Micromech Systems 1, 60 (1992). NUMBER OF ATOMS RADIUS (nm) 1 MOLECULES 10 1 102 “ NANOPARTICLES 104 10 10‘5 v l BULCK Figure 1.1: l\:‘Iesoscoi)ic regime between molecule, nanoparticle and bulk matter 1.2 Importance of nanostructure determination Nanostructured materials (nanocrystalline materials) are materials possessing grain sizes on the order of a billionth of a meter. The prefix “nano” means one billionth. One nanometer (abbreviated as 1 nm) is l / 1,000,000,000 of a meter. To get a sense of the nano scale, a human hair measures 50,000 nanometers across, a bacteria cell measures a few hundred nanometers cross. The smallest things that can be seen with the unaided human eye are 10,000 nanometers cross. Nanoscale materials, the foundation of nanoscience and nanotechnology, have be- come one of the most popular research topics in a very short period of time. Nanos- tructured materials, or nanomaterials, reside in the so—called mesoscopic regime be- tween isolated atoms or molecules and bulk matter (see Fig. 1.1). These materials have unique physical and chemical properties that are distinctly different from bulk materials. Their size—dependent properties, their sensitivity to surface phenomena, and how they spatially arranged present a significant challenge to our fundamental understanding of how these materials should behave. Intense interest in nanotechnology and nanostructured materials is fuelled by the tremendous economical, technological and scientific impact anticipated in several ar- eas [2]: (1) The exponential growth of the capacity and speed of semiconducting chips, the key component that virtually enabled all modern technology, is rapidly approaching their limit of art and demands new technology and new materials sci- ence on the nanoscale. (2) Novel nanoscale materials and devices hold great promise in energy, enviromnental, biomedical, and health sciences for more efficient use of energy sources, effective treatment of environment hazards, rapid and accurate de— tection and diagnosis of human diseases, and improved treatment of such diseases. (3) When a material is reduced to the dimension of the nanoscale, only tens of the dimension of a hydrogen atom, its properties can be drastically different from those of either the bulk material that we can see and touch even though the composition is essentially the same, or the atoms or molecules that make up the materials. Therefor, nanoscale materials prove to be a very fertile ground for great scientific discoveries and explorations [3, 4]. The research and development in nanomaterials involve three aspects: assembly and synthesis of the nanomaterials, characterization of their structures and properties, and exploring or implementation of their applications. N anomaterial characterization is very important key aspect, since it is a bridge between synthesis and applications. Nanomaterials are made up of a small number of atoms with charge and spins, and all carry some properties/ functions. How atoms are assembled together refers to the atomic and magnetic structure of materials. The nanomaterial structure and prop- erty/function relationship proves to be very important, as well as very interesting. For example, in the bulk case, graphite (dark, soft, cheap) and diamond (transparent, hard, precious) are dramatically different materials. However, they are both made up of the same kind of carbon atoms and the only difference is their atomic crys— tal structures. “(hat is more, in the nanomaterials case, the buckyball (C60) and nanotube are made up of the same carbon atoms too, but they have different proper- ties/ functions. In a word, the structure property relationship is fundamental and its in-depth understanding is vital to manipulate the material function. Thus, precise knowledge of the structure is of paramount importance. 1.3 Classification of nanostructures Nanostructured materials are structures with at least one characteristic dimension measured in nanometers. All conventional materials like metals, semiconductors, glass, ceramic or polymers can in principle be obtained with a nanoscale dimension. A reduction in the spatial dimension, or confinement of particles in a particular crystallographic direction within a structure generally leads to changes in physical properties of the system in that direction. Hence one classification of nanostructured materials and systems essentially de— pends on the number of dimensions which lie within the nanometer range, as shown in Table 1.1: (a) systems confined in three dimensions, (b) in two dimensions, (c) in one dimension. Generally there are different approaches for a classification of nano- materials depending on dimensions, phase composition or manufacturing process [2] 1.4 N anostructure types in three dimensions In general, nanoparticles may have a random arrangement of the constituent atoms or molecules (e.g., an amorphous or glassy material) or the individual atomic or molecular units may be ordered into a regular, periodic crystalline structure which may not necessarily be the same as that which is observed in a much larger system. Example of nanostructured materials is shown in Fig. 1.2. (a) Three dimensions confinement Fullerenes Colloidal particles N anoporous silicon Activated carbons Nitride and carbide precipitates in high-strength low-alloy steels Semiconductor particles in a glass matrix for non-linear optical components Semiconductor quantum dots (self-assembled and colloidal) Quasi-crystals (b) Two dimensions confinement Carbon nanotubes and nanofilaments Metal and magnetic nanowires Oxide and carbide nanorods Semiconductor quantum wires (c) One dimension confinement Nanolaminated or compositionally modulated materials Grain boundary films Clay platelets Semiconductor quantum wells and SlpoI‘lattlt‘PS Magnetic multilayers and spin valve structures Silicon inversion layers in field effect transistors Surface—e1igineered materials for increased wear resistance or corrosion resistance Table 1.1: Examples of reduced—dimerisionality systems. Adapted from [2] The work presented in this thesis is focusing on discrete nanoparticles, an example is shown in Fig. 1.2(C). as well as nanostructured bulk materials (bulk crystals with short-range structure orders) an example is shown in Fig. 1.2(A). In these nanostruc- tured bulk materials, domains of local structural order are exist where correlations extend over nanometer length—scales without destroying the average lattice [5]. There is a growing realization that this behavior is not rare but widespread, and can be ex— tremely useful [6, 7]. WI . as. ‘1 a ,.x\1k\1\7i" Figure 1.2: Example of three-dimensional nanostructure materials. (A) Nanostruc- tured bulk materials. (B) Intercalated mesoporous materials. (C) Discrete nanopar- ticles. In each case, ball-and-stick renditions of possible structures are shown on the top, and transmission electron microscopy (TEM) images of examples are shown on the bottom. Adapted from [7] The TEM image in Figure C is from [8] 1.4.1 Mesoporous materials Mesoporous materials, an example is shown in Fig. 1.2(B), are materials with pores (holes). Simply they are bulk materials that contain porosity with nanometer scale dimensions, between 1 and 100 nm [9]. What distinguishes nanoporus materials, and gives them their great importance in applications, is the high surface to volume ratio with high surface area and large porosity which enables them to discriminate and interact with molecules and clusters. They have a wide range of uses including catalysis, chemical separation, waste remediation, hosts for hydrogen storage and passivation of reactive species. Just about every bulk material is a candidate for being manufactured in mesoporous form, but most progress has been made in covalent network systems such as alumino-silicates [10], carbon [11], silicon [12] and silica materials containing functionalized organic monolayers [13, 14, 15]. The internal pores of these materials can also be functionalized with molecules to modify their reactive properties. These kind of materials have not been studied here in this these, detailed study can be found in [16, 17] 1.4.2 Metallic nanoparticle Synthesis of metallic nanoparticles, such as gold, can be tracked back as early as Faraday’s times [1] Gold nanoparticles have been taken as a model system for the— oretical and experimental investigations of the properties of nanoparticle materials, such as size effect on color, size induced quantum effect, light absorption and many more [18, 19, 20]. Experimentally, gold is an ideal because it. has no impurity and no oxidation. Theoretically, gold is easily described using quantum theory because all of it is inner shells are filled orbitals ant its outer-most shell only has one electron. The agreement between theory and experiment is confirmed early work done by Leff et al. [21]. The reduction of metal ions in solution is the most. popular and economic technique for preparing metallic nanoparticles [22]. Platinum nanoparticles can also be produced by the reduction of platinum ions in the presence of capping materials [23, 2'1]. Pileni’s group has extensively carried out studies of metallic nanoparticles. They have focused on the collective properties of self-assembly of magnetic nanoparticles [25]. Ag nanoparticles coated with a silica shell can also be self-assembled [26]. The silver (Ag) nanoparticles are dominated by tetrahedra and their assembly is affected by their shape [27]. Metallic nanocrystals with specific size and shape were synthesized by an aerosol technique [28, 29]. 1.4.3 Semiconductor nanoparticles In semiconductor terminology three-dimensional confinement systems are often called quasi-zero dimensional, as the structure does not permit free particle motion in any "J electron states Conduction band hole states I [ Valence band / @ Quantum dot Bulk semiconductor Figure 1.3: Schematic comparison of the band structure and electronic states of semiconductor nanocrystals and bulk semiconductor. Adapted from [4] dimension [30]. The band structure of semicomluctor nanocrystals is quite different from that. of the bulk material (Fig. 1.3). Nanocrystals have discrete excited electronic states and an increased band gap in comparison to the bulk semiconductor materials. The smaller nanoparticle size is the larger the band gap width. The difference in the band gap width, can be easily differentiated by optical absorption spectroscopy. The band gap is tunable via controlling nanocrystal size, providing an effective way in adjusting the electronic structure in addition to controlling of particle chemistry [30]. The most typical example is that of silicon nanocrystals. The dynamics and spec- troscopy of silicon nanocrystals that emit at visible wavelengths were analyzed [31]. Due to the oxidation on the surface of silicon nanocrystals, intensive research focuses on compound VI-II and V -III types sciniconductor nano(':rystals. In 1988, Brus’ group reported a process for synthesis of pure and stable organic capped CdSe nanocrys- tals using an inverse mieelle (IM) method [32]. Up to now, CdSe is still the most intensively studied semiconductor nanocrystal system [33, 34, 35]. Synthesis of CdSe nanocrystals has been developed rapidly so that there is now enormous control of size, shape [36, 37]. The self-organization of CdSe nanocrys- tallites into 3D semiconductor quantum dot (QD) superlattices (colloidal crystals) has been demonstrated in early work of ‘.\-'Iurray et al. [38]. The size and spacing of the dots within the superlattice is controlled with near atomic precision. This con- trol is a result of synthetic advances that produce CdSe nanocrystallites exhibiting monodisperse within the limit of atomic roughness. The methodology is not limited to semiconductor QDs, but provides general procedures for the preparation and char- acterization of ordered structures of nanocrystallites from a variety of materials [38] The semiconductor nanoparticles. such as CdSe, have a wide range of applications, for example, CdSe quantum dots have been used for laser diodes [39], nanosensing [40], and biomedical imaging [33]. They also have been used as a model system for inves- tigating a wide range of nanoscale electronic, optical, optoelectronic, and chemical processes. [41] Assemblies of nanoparticles offer an entirely new and hardly explored frontier for applications [42]. 1.5 The effect of nanoscale dimensions on prop- erties Interesting changes in ratio of surface to volume atoms have been observed in systems of reduced dimension, there is a considerable proportion of atoms that are in contact with the surface comparing to the total number of atoms. If an atom is located at a surface then it is clear that the number of nearest-neighbor atoms are reduced, giving rise to differences in bonding (leading to the well—known phenomenon of surface recon- struction) and electronic structure. For instance, in a 5 nm particle, approximately 30%-50% of the atoms are influenced by the surface, compared with approximately a few percent for a 100 nm particle. Such structural differences in reduced-dimensional systems would be expected to lead to very different properties from bulk systems [43]. In nanoscale materials, the surface—effects are comparable in some cases to the chemical composition effects that influence the chemical, electronic, magnetic, and optical behaviors. A first-order approximation of a particle shape is spherical. The ratio between the number of surface atoms (M) to the volume atoms (NU) is N“ : 38f (11) IV} V} 7' I) v where [is and p1,» are the densities of surface atoms and volume atoms, respectively, and -r is the nanoparticle radius. Figure 1.4 shows a plot of the NS/Nv as a function of nanoparticle diameter calculated for gold (Au). The unique chemical and physical properties of nanoparticles are determined not. only by the large portion of surface atoms, but also by the atomic structure of the particle surface. The former is determined by the size of the particles, and the latter relies on the particle shape. One typical example is that the melting temperature of nanoparticles strongly depends on the crystal size and is substantially lower than the bulk melting temperature. The melting point of Au nanoparticles of core size 2.5 nm is 40% lower than that of the bulk gold [44]. Similar behavior have been observed for sodium clusters [45] and CdS nanoparticles [46]. The other interesting property is the quantum confinement effect in the small size metallic and semiconductor nanoparti— cles [47, 37]. The shift of the electron energy levels as a function of particle size gives rise to emission of photons with unique wavelengths. 10 s; ...... :_ ....... : ....... r. ....... : ....... j o o o o g - Q. : : : . ‘0 . . . . 30° -------- .Q ----- : ------- . ------- : ------ - o » ' ' ' ' + 00- ...... 1....» ....... 1 ....... 1 ........ 00 . . . . e - - y - - ¢ 0 O 0 I 30' """" ‘.' """" : """" .' °°°° ' 301. ...... 3. ...... Q ...... E ....... 3 ........ co : :\. : : 3%”; ....... i ....... ...... ‘. *H. ...... - 55°- 5 3 : 9‘ 4 3.. ...... L ....... - ....... a. ....... ° ....... .. o . : : : . r . O 2 4 6 8 10 Particle size (nm) Figure 1.4: Calculated s11rface-to—volume atom ratio for a spherical gold nanoparticles as a. function of particles size. Adapted from [i] 1.6 Experimental methods for nanostructure char- acterization In the past few years many experimental probes l'iave been employed in investigation of nanostructure at different length scales. Some are inherently local, such as trans- mission electron microscopy (TEM) and scanning probe microscopies. Others are bulk average probes that are sensitive to local structure, such as the atomic pair dis- tribution function (PDF) method, nuclear magnetic resonance (N MR), and extended x—ray absorption fine structure (EXAFS) analysis. Such probes can yield direct or indirect information about atomic arrangement. PDF is an example of the former and NMR the latter. The atomic pair distribution function (PDF) method [48, 6] has been employed ll in this thesis, as a local probe, to investigate the local structure of nanomaterials at different length scales. Before we describe the PDF method in more detail, let us first briefly review a few other techniques exist for probing nanostructured materials. 1.6.1 Transmission Electron Microscopy (TEM) The TEM allows both the imaging of individual crystallites and the developi‘nent of a statistical description of the size and the shape of the particles in a sample. High resolution TEM (HRTEM) imaging allows determination of individual crystallite n'iorphology and internal structure. The TEM technique operates on the same basic principles as the light microscope but uses electrons instead of light. What you can see with a light microscope is limited by the wavelength of light. T EMS use electrons as the ” light source” and their much shorter wavelength makes it possible to get a. resolution a thousand times better than with a light microscope [49, 50]. Two major mechanisms provide contrast in TEM imaging. Lattice image contrast and Z contrast provide important and complimentary information. Lattice imaging highlights the crystalline core of particles with planes oriented perpendicular to the electron beam but is less sensitive to misoriented or disordered portions of the struc- ture. When electrons are diffracted by different portions of a periodic sample they have a definite phase relation. When refocused in the image plane, an interference pattern characteristic of the lattice is generated. Specific crystal reflections can be selected by the choice of objective apertures. The aperture limits the angular range of reflections contributing to the final image. In addition to being diffracted by the crystal lattice, electrons are also diffusely scattered from individual atoms. This dif- fuse scattering is proportional to the atomic number (Z) of the element and is referred to as Z contrast or Z scattering. It provides contrast in both the ordered and the disordered or misoriented regions [51]. The HRTEM as a high spatial resolution probe has been the primary technique used to characterize nanostructured materials [52]. Using HRTEM you can see ob- jects to the order of a few angstrom (10’10 m). For example, you can study small details in the nanomaterials down to near atomic levels. The possibility for high mag- nifications has made the HRTEM a valuable tool in nanomaterials research. HRTEM allows us to analyze individual nanocrystals and obtain an average size distribution. Unfortunately, it can be very difficult to determine the precise shape and size of the nanocrystal because of nanocrystal movement under the electron beam and poor con- trast near the surface of the nanocrystal. One of the difficulties with HRTEM is that image formation relies on phase-contrast and it is limited to a surface or thin sections of a sample. 1.6.2 Extended X-ray Absorption Fine Structure (EXAFS) The EXAF‘S is one of the most powerful characterization methods to investigate local atomic structure of nanostructured materials. The EXAFS can select a specific type of atom as a probe atom and measure the structural environments near the probe atom. The EXAF S can therefore give structural information about the bond lengths and the species of the atoms around the probe atom. The EXAF S technique has become a useful structure analysis of nanostructured materials [53, 54, 55, 56, 57]. The EXAFS measurements are usually carried out at synchrotron facilities, since the technique requires the incident X-ray energy to be scanned. The measured quan— tity in one EXAFS experiment is the sample absorption coefficient as a function of the incident X-ray energy usually ranging between 20 and 800 eV above the K or L absorption edge of one specific constituent element [58, 59]. Either the transmitted photons or the fluorescence intensities can be measured for this purpose. How the lo- cal structure information is contained in the absorption coefficient can be understood qualitatively as the following. The incident photons excite either the K or L shell core electrons into the continuum as a consequence of the photoelectric effect. when 13 the photon energy is closely above the absorption energy edge. The excited electron will have kinetic energy, that is the difference between the photon energy and the absorption threshold of the probed atomic shell. When the photoelectron energy (2 20 eV) is large compared with its interaction energy with the surrounding atoms (~ 3 eV), simple plane wave approximations can be made on those photoelectrons with the surrormding atoms treated as perturbations [59, 60]. The cross section of the plmtoelectric effect is then proportional to the wave function overlap between the pho- toelectron plane wave and the atomic Ii or L core level states. While the atomic core levels are little affected by the neighboring atomic environments, the photoelectrons are scattered by the surrounding atoms. Therefore, the final state of the photoelec- tron is modified by the backscattered electrons, and the backscattered electron waves can either add or subtract depending on their relative phases. As the wavelength of the photoelectron simply follows the de Broglie relation A = L], their relative phases directly depend on the distance between the prode atom and its neighbors, i.e. half of the travelling distance of the backseattered electron. Oscillations in the sample absorption with incident photon energy are then expected, as the photoelectron en- ergy, thus the wavelength A, and the relative phases, are varied. To properly extract the local structure information from EXAF S data, many other complications have to be taken into considerations, such as backscattering phase shift, finite life time of electron states, and multiple electron scattering events. The basic theory of the EXAFS method have been covered in other places, such as [59, 60, 61, 62]. EXAF S method is a powerful tool for studying nanostructured systems [54, 55, 56]. Chen et al. [63, 64] used EXAFS to probe the origin of the unique functions of titanium dioxide. Using EXAF S combined with some other methods, Gilbert et al. [65] observed a compressive strain compared to the bulk in ZnS nanocrystals. Using Fe K—edge Chen et al. [66] studied the structures of Fe203 nanoparticles with different sizes. The nature of the surface defects and their correlation with the unique properties of lrl the nanoparticles were discussed. 1.6.3 Nuclear Magnetic Resonance (NMR) Within the present context, the NMR teclmique probes the chemical environment of the specific nucleus under resonance [67]. During an NMR measurement, the energy of different nuclear spin states are shifted by an amount proportional to their net spin by the external static magnetic. field. Transitions between the different spin states can be initiated by a secondary oscillating magnetic. field by either changing its frequency or the amplitude of the applied static magnetic field. The zero static field NMR method is called nuclear quadrupole resonance (NQR). An important quantity coming from NMR measuren'ients is called chemical shift, which is defined as the difference between the resonance frequency of the nucleus and a standard [68]. One origin of the chemical shift is from the electrons around the nucleus. The applied magnetic field also causes the electrons to circulate about. the direction of the applied magnetic field. This circulation causes a small n'iagnetic field at the nucleus which opposes the externally applied field (in some cases, it enhances the external field). This shifts the nuclear resonance frequency, which is proportional to the magnetic field strength at the nucleus. Therefore, the magnitude of the chemical shift carries information about the surrounding electrons, thus the local chemical environment, such as valence, bonding, geometry. etc.. As the probed nucleus is required to have a net spin, isotope substitution is sometimes necessary. Using H-l NMR Ladizhansky et al. [69] have studied CdS nanoparticles precip- itated from aqueous solution. Zelakiewicz et al. [70] have reported in recent work an investigation of metal-ligand interactions in Au nanoparticles with 13C NMR. Tomaselli et a1. [34] have reported the results of P-31 NMR measurements on tri- octylphosphine oxide (TOPO) passivated InP quantum dots. With decreasing parti- cle size the In P—31 core resonz—rnce reveals an increasing up field chemical shift related to the overall size dependence of the hip electronic structure [34]. 1.6.4 X-ray diffraction (XRD) XRD is a powerful tool for investigating the atomic structure of nanoclusters. This is possible if nronodispersive nanocrystals can be prepared and separated in a reasonable quantity. If the particles are oriented randomly so that the entire assembly can be treated as a ” polycrystalline” specimen composed of nanocrystals with identical structure but random orientations so that their scattering from each can be treated independently, the diffraction intensity I(Q) Equ. 1.3 as a function of the diffraction vector length Q, 4 7r Q = —- sinf), (1.2) A where 6 is the diffraction half—angle, A is the wavelength of the X-ray photons, N N I(Q) = Z:fifjs'f’lerijl/Qrija (1.3) 'i j where r,- j is the distance between atom i and atom j in the cluster of a total N atoms, and fz- and f, are their X—ray scattering factors, respectively. To make a quantitative comparison of the calculated intensity profile with the XRD spectrum, the contribution from the background created from the substrate used to support the nanocrystals rrrust be subtracted. The structure of nanocrystals can be refined by quantitative comparison of the theoretically calculated diffraction spectra for different nanocrystal models [71] Palosz et al. [72] have shown that the conventional tools developed for elaboration of powder diffraction data are not directly applicable to nanocrystals [72]. They proposed an alternate evaluation of diffraction data of nanoparticles, based on the so—called “apparent lattice parameter” [72]. 16 1.6.5 Small-angle scattering X-rays (SAXS) SAXS is another powerful tool in characterizing nanostructured rrraterials. Strong diffraction peaks result from constructive interference of X-rays scattering from or- dered arrays of atoms and molecules. A lot of inforn'ration can be obtained from the angular distribution of scattered intensity at low angles. Fluctuations in electron density over lengths on the order on 10 run or larger can be sufficient to produce an ap- preciable scattered X—ray intensity at angles 26’ less than 5 degrees. These variations can be in density, from differences in composition, or from both, and do not need to be periodic. The amount and angular distribution of scattered intensity provides in— formation such as the size of very small particles or their surface area per unit volume, regardless of weather the sample or 1_)articles are crystalline or amorphous [73]. SAXS is observed from almost all kinds of materials, and it is widely used in structural studies of non-crystalline [7.1] materials at relatively low resolution. The term ”small angle” here refers to the angular range within a few degrees, containing structural information on the order of approximately a nanometer to submicromc— ters [51, 75]. Many applications of SAXS technique are found in structural biology, polymer science, colloid chemistry and nanostructured materials [38, 65] SAXS usually requires a thin section of the material that is scanned across a narrow x—ray beam. The diameter of the x-ray beam defines the lateral resolution of the scanning procedure. It is in the order of 100 micrometer on laboratory x—ray source and in the order of I micrometer (or even below) at a synchrotron source. Ideally, the thickness of the specimen should be the same as the beam diameter, d. Then the scattering volume for each individual measurement will be about d3, making the evaluation of the scattering patterns yield structural information on the nanocomposition, within each (13 -volume separately. Such local information from X- ray diffraction on nanostructured materials [76] can be advantageously combined with local information from other techniques, eg. imaging microscopy or/ and PDF [7]. l7 Using SAXS, EXAF S and HRTEM Ramallo—Lopez et al. [77] have reported re- cently a structural characterization of palladium nanoparticles, with an average di- ameter of 1.2 nm, capped with three different n-alkyl thiol molecules (11 = 12, 16 and 18). Greaves et al. [78] have used the SAXS, EXAFS and Diffuse Optical Spec- troscopy (DOS) to characterize cadmium oxide particles nucleated within cadmium exchanged sodium zeolite Y-treated with sodium hydroxide. On the experimental side of SAXS there are a. few aspects of synchrotron radiation that have improved SAXS studies such as very small beam divergence, high beam flux, and energy tunability. It is crucial to have small beam divergence in order to isolate weak scattering at very small angles from the direct beam which is orders of magnitude stronger. The high flux of the beam allows one to use a smaller beam size, resulting in better isolation of scattering from the direct. beam. The flux level at a synchrotron source is usually several orders of magnitude higher than those from conventional x—ray sources, thus studies of weak scatterers have become much more practical. The high flux beam also made it possible to conduct time-resolved i‘neasuren'lents of small angle scattering [79]. It is possible to conduct anomalous small angle x—ray scattering only when beam energy tunability of synchrotron radiation is used [80] 1.6.6 Scanning probe microscope (SPM) SPM represents a group of techniques, such as scanning tunnelling microscope (STM), atomic force microscope (AF M), magnetic force microscopy and chemical force mi- croscopy, etc., which have been extensively applied to characterize nanostructures [81]. A common characteristic of these techniques is that an atom sharp tip scans across the specimen surface and the images are formed by either measuring the current flowing through the tip or the force acting on the tip (longitudinal or transverse). SPM has impacted greatly the research in many fields because it can be operated in a variety of environmental conditions, in liquid, air, or gas, allowing direct imaging of inor— 18 Figure 1.5: Schematic diagram showing the principle of a scanning probe microscope (STM). ganic surfaces and organic molt-Krules. SPM not. only provides the ”eyes" for imaging nanoscale world, it also provides the ” hands” for manipulation and construction of nanoscale objects. The discovery of SPM is a landmark contribution to nanotechnol- ogy. STM is based on the vacuum tunnelling effect. in quantum mechanics [82]. The wave function of the electrons in a solid extends into the vacuum and decays expo- nentially. This barrier prevents the passage of the electrons belonging to the solid. If a tip is brought sufficiently close to the solid surface, the overlap of the electron wave functions of the tip with that of the solid results in the tunnelling of the electrons from the solid to the tip when a small electric voltage is applied (see Fig 1.5). 1.6.7 Total scattering technique Probing the atomic structure in nano crystalline materials proves very challenging to conventional crystallography. The assumption that Bragg peaks contain all structural information does not hold in the presence of local structural distortions due to finite size effects. This is because the. local deviations result in diffuse. scattering iii-between and underneath the Bragg peaks in reciprocal space, the diffraction intensity I(Q) 19 Eq. 1.3 . Neglecting those diffuse scattering intensities results in loss of the local structure information. The PDF method [418], that includes both Bragg (where it is present.) and diffuse scattering intensities, have been employed in this thesis to investigate the local struc- ture of studied nanomaterial at different length scales. Using the PDF method quan- titative structural information about structure, crystallinity level, core size, nanopar- ticle size, and inhomogeneous internal strain in the studied nanoparticles could be obtained. This method is generally applicable to the characterization of the nano- scale. solid, many of which may exhibit complex disorder and strain. 1.7 Nanostructure determination problem The atomic arrangement. inside new materials is prerequisite knowledge to help in understanding the new material’s properties. The nanostructure problem can be simply stated as “the need to determine atomic arrangements inside nanostructured materials, quantitatively and with high precision” [7]. In the case of crystalline materials there already exist robust and quantitative solution methods to this problem, as long as the average atomic positions are con- cerned. This is not the case for nanostructured materials, for example, nanoparticles, mesoporous materials and bulk nanostructured (see Fig. 1.2). What usually appears from the x-ray diffraction measurements when a sample of nanoparticles is loaded is not a well defined x-ray diffraction pattern as merges from well crystalline mate- rials Fig. 1.6 (top), but a broad and continuous intensity distribution, as shown in Fig. 1.6 (bottom), which is not suitable to be analyzed by a crystallographic structure solution. Hence, the need for a robust solution method for nanostructure determina- tion, quantitative and precise, is still a dream. A number of powerful probes exist. for studying local and nanoscale structures, shown below, but in general we have no widely applicable solution to the nanostructm'e problem. ‘20 ¢ ,_ . e: . n n- . ,\ . 3 “2 . o “F \, . ._ N_ . e: - e: l o 4 4 8 12 16 20 24 28 0 or") * I I ' l ' I ' l ' N CdSe nanoparticle 2nm 7 &- . q_ . A a - 3 P 3,:- 4 o as_oe_osv l l A. l A L l l 12 16 20 24 28 o (3“) O .p co Figure 1.6: Diffraction pattern of CdSe bulk (top) vs '2 nm nanoparticle (bottom). The two samples were measured at the same experiment a1 conditions. Q is the diffrac- tion vector length defined in Equ. 1.2 and the y—axis label (a u) refers to arbitrary unites. Many techniques exist for probing nanostructured materials. Some are inherently local, such as transmission electron microscopy (TEM) and scanning probe micro- scopies. Others are bulk average probes that are sensitive to local structure such as the atomic pair distribution function (PDF) method or extended x-ray absorption fine structure analysis (EXAFS). Usually the first thing done when a material’s properties need to be characterized is XRD measurements. For nanoparticle samples, the XRD measurements don’t give as accurate structural information as in the crystalline material case. Palosz et at. [72] have shown that the conventional tools developed for elaboration of powder diffraction data are not. directly applicable to nanocrystals. [72]. The application of the Scherrer formula in order to determine nanoparticle diameter is based on the assumption of a perfect crystal, limited in size. Nanoparticles with finite size are often terminated by a variety of different hid-planes, and have a large fraction of their atoms on the nanoparticle surface with a high degree of disorder. This can be seen from the broad and continuous intensity distribution of the nanoparticle (see Fig. 1.6). These features are inconsistent with the basic assumptions of the Scherrer formula. For that reason, the characteristic diameter of nanoparticles cannot be reliably estimated from the XRD peak-width-based analysis. Electron microscopy techniques such as high-resolution transmission electron mi- croscopy (HRTEM) have been the primary technique used to characterize nanocrystal size and shape. HRTEM allows us to analyze individual nanocrystals and obtain an average size distribution [36]. Unfortunately, it can be wry difficult to determine the precise shape and size of the nanocrystal because of nanocrystal movement under the electron beam and poor contrast near the surface of the nanocrystal. Additionally, phase-contrast imaging relies on lattice fringes, which arise from the periodic structure of the crystal lattice. This periodicity is broken at. the surface and nearby lattice defects, complicating image interpretation. Z-contrast scanning transmission electron microscopy (ZSTEM), however, can pro- vide highly detailed images of the nanocrystal surface, 3D information, and mass con- trast simultaneously, which can all be directly discerned from the image. Z-STEM uses an incoherent imaging process, that yields images that are directly interpretable to the structure of the object being observed. With this technique, combined with a spherical aberration (Cs) -corrected STEM, the resulting highly—spatially resolved images are able to show subtle details about the shape and faccting of nanocrystals with near sub—angstrom precision [8]. The principal difficulty with the application of these, methods to solving the nanos- tructure problem is that, in general, any one technique does not contain sufficient in- formation to constrain a unique structural solution. A coherent strategy is required for combining input from multiple experimental methods and theory in a self—consistent global optimization scheme - something we refer to as “complex modelling” [7]. 1.8 Outline of this thesis Investigation of the structure of nanoscale materials at different length scales, em- ploying the PDF technique, is the the central theme of this thesis work, covering both technical developments and scientific applications. This thesis work is illustrated as part of the solution to the (nanostructure problem) [7]. This thesis is organized in five chapters: Chapter 2 provides a brief introduction into the atomic pair distribution function (PDF) technique as powerful tool, used in this thesis work, to quantitatively study nanostructured materials. Also in this chapter, I will describe recent development in the PDF technique as nanoscale prob and derive the PDF of single spherical particle. Chapter 3 focuses on the high resolution PDF analysis of x—ray powder diffraction data. on Na3BiO4 nanostructured bulk material, where a short range chemical order 23 have been observed. A real structure of Na3BiO4 will be discussed. Chapter 4 introduces a PDF based methodology to qualitatively study nanostruc- tured materials. The size-dependent structure of CdSe nanoparticles, with diameters ranging from 2 to 4 nm, has been studied using the introduced PDF methodology. Chapter 5 focuses on applying the introducing a PDF based I‘nctl’iodology to study on ultra-small CdSe nanoparticles. Chapter 6 summarizes the thesis and discusses some future work. Appendixes A, B, and C are technical description of the RAPDF method, in detail, as a non-specific-user-friendly technique. In this manual, I will describe recent development of a new way to collect PDF data: rapid acquisition pair distribution function (RAPDF) method, where the coupling of high energy X-ray with an area detector leads to three to four orders of magnitude decrease of data collection time. Appendix D has an information about image plate characters as an X-ray detector. Appendix C has information about the PDF of a single spherical particle. The materials in this appendix have been adapted from [83] The work presented in Appendixes A, B, C and D is a collaboration work with Dr. Xiangyun Qiu. 21.1 Chapter 2 The Atomic Pair Distribution Function (PDF) Method 2. 1 Introduction Modern synthetic new materials are subjected to different levels of distortion, and one growing trend is that their structures become more and more disordered and complex. On the other hand low-dimensional systems such as nanoparticle are subjected to structural distortions due to size effect and surface tension [84]. Currently, it. is of great interest to characterize those nanocrystalline material structures effectively and routinely. However, the long-range structural coherence length at nanoscales challenges conventional crystallographic analysis as has been reviewed by Billinge et al. [6, '7]. The atomic pair distribution function (PDF) technique is a promising candidate as evidenced by its successful application to solve structures with different levels of complexities [48, 6] and lower-dirnensionality [65, 85, 83, 86, 87, 88]. In this chapter, I will briefly introduce the PDF technique as powerful tool, used in this thesis work, to quantitatively study narmstructured materials. Also, I will describe a recent development in the PDF technique as a nanoscale probe and show the derivation of the PDF of a single spherical particle However. conventional PDF measurements 25 are very slow and generally take more than eight hours, even at a third generation synchrotron or intense spallation neutron sources. This, coupled with the novelty of the approach and the somewhat intensive data analysis requirements, has prevented widespread application of the technique in areas such as nano-materials. This chapter will describe a recent development -— the rapid acquisition PDF (RA-PDF) method where the data collection time is reduced by three to four orders of magnitude [89]. A new program PDFgetX2 [90], as a user friendly program, is used in this thesis work to obtain the PDF from X-ray powder diffraction. 2.2 The atomic pair distribution function (PDF) technique As can be understood from it. is name, the atomic pair distribution function (PDF), C(r), tells the probability of finding atomic pairs separated by the real space distance 7'. It should be noticed that the PDF of interest is radially averaged and is a one dimensional function. In order to calculate the C(z) for a given a structure, we first sit on one atom i, then look out for neighbors. A peak is assigned for every atom j found at the position corresponding to the inter-atomic distance Tija and zero everywhere else, as demonstrated in Fig. 2.1. Then we repeat the same procedure over all atoms, and average all obtained curves to get the total C(r) of the structure. The mathematical definition of the PDF is bibr _ Goff) = $22 [Kw—$00 — 7'ij)[ — 47rrp0 i j = 47r'r[p(r) — m], (2.1) where rij is the distance between atoms i and j; the value bi is the scattering length for atom i; (b) is the average scattering length over all atoms; and p0 is the number 26 density. The p(7') here is the pair density function [48]. The PDF can be obtained experimentally by Fourier transforming the measured scattering intensities. The measured scattering intensities can be expressed as +30 bib- sin(QI'-.-) 1 '00 bxb: sin(Q7',- ) _ .7 7'] .-.*.__ ‘J U (2.2) from the Debye scattering equation [92]. The PDF can be directly obtained via a sine Fourier transform: C(r) = % fox Q[S (Q) — 1] sin Q7“ (IQ. The key difference between the PDF method and conventional average structure analysis of powder diffraction data is how to treat the diffuse scattering intensities in- between and underneath the Bragg peaks. The Bragg peaks come from the long range ordered structure, while the diffuse scattering intensities come from local/ temporal deviations from the long range ordered structure. The PDF technique takes into account both, as evident from the equations above, while conventional average struc- ture analysis only looks at the Bragg peaks. Therefore, the PDF method is capable of probing both the local and average structure. This combination of local and average studies from the same scattering data represents a clear advantage for a wide range of applications. For example, with the presence of Bragg peaks in the case of crystalline materials, the PDF technique and conventional analysis are complementary while the former provides additional information about the local structure. However, in the case of nano-crystalline and non—crystalline materials, with few or no Bragg peaks, only the PDF technique is applicable. Local structural information, by intuition, is contained in the low-r PDF peaks. Thus, the real space resolution is an important experimental factor to resolve closely neighboring peaks. PDF peaks are first broadened by the atomic thermal motions. Second, the finite measurement range also broadens the peaks coming from termina- tion effects [93]. High real space resolution usually requires extended Q range (2 25.0 A—l) data to be collected, and low temperatures if it is possible. Other than identi- 27 (a) B l l M 0‘ l A l n l n I n 1 l 1 3 (b) “ " G 0 G 0 0 G 0 D C G 0 $5 . can 0 2A Figure 2.1: Schematic of how the total pair distribution function is built. up by finding atoms at a distance, r, from an atom at the origin. Circles indicate distances where neighbours to the central atom exist and these correspond to peaks in the radial distribution function shown on the right. The number of atoms on the circle is the multiplicity (scale) of that peak. The scattering power of each atom in the pair also contributes to this scale, as per equation (2.1). The total pair distribution function is obtained by repeating this process systematically by placing each atom at the origin. Adapted from [91] fyiug low-r peak positions (existing atomic pair distances), extracting the peak width and area reveals further details about the local distortions and number of neighbor- ing atoms as well. The average structural information is usually recovered from the PDF structure model refinements using the non-linear least square regression fitting program PDF FIT [91.] The agreement between the average structure and experi- mental G (1') provides a. good check for the possible existence of local disorder, and hypothetical distorted structure models can then be fit to inmrove the agreement. Knowledge of the detailed local and average structures often is the basis of many theoretical calculations of material properties. For example. the bond valence sum can be computed to investigate the strength of chemical bondings. Electronic band structure calculations can be carried out if necessary to check whether the obtained structure gives macroscopic properties consistent. with other types of measurements. For more details about the PDF technique, please turn to a recent book by Egami and Billinge [48]. An overview of the technique and some applications have been referred to in Billinge et al. [6]. 2.3 PDF method: past and present The PDF method has been used as early as 1931 by Debye [95] to study liquid mercury, and has mostly focused on glass and amorphous materials [96, 97, 98, 99]. Only recently, the PDF analysis has been successfully applied to crystalline and nano- crystalline materials [6, 100], owing to advances in modern x-ray and neutron sources and much improved computing power [48]. The PDF technique has been applied on perfect crystalline materials as well, such as Si and Ni, to not only obtain the average structures in quantitative agreement with conventional crystallographic methods [101], but also to give additional information about the correlated motions between neighboring atoms [102, 103]. Proffen et al. have successfully applied the PDF method to probe the chemical short range order in ‘29 C113Au [104]. Complementary to the average structure analysis, Billinge et. al. and Bozin et al. have applied the PDF technique extensively to study the local disorder and inhomogeneities in the highly crystalline colossal magneto-resistive manganites and high temperature superconducting cuprates [105, 106, 107, 108, 109]. Recently many applications have proved the PDF technique as a pmverful local structural probe of nanostructured materials [‘18, 6, 65, 85. 86, 87, 88]. Nanoparticles with finite size are often terminatt-rd by a variety of different hkl- planes, and have a large fraction of their atoms on the nanoparticle surface which can have a high degree of disorder. However, difficulties are. experienced when standard methods are applied for small nanoparticles. In this domain the presumption of a. peri- odic solid, which is the basis of a crystallographic analysis, breaks down. Quantitative determinations of the nanoparticle structure require methods that. go beyond crys- tallography. This was noted early on in a seminal study by Bawendi et al.[71] where they used the Debye equation, which is not based on a crystallographic assumption, to simulate semi-quantitatively the scattering from some CdSe nanoparticles. How- ever, despite the importance of knowing the nanoparticle structure quantitatively with high accuracy, this work has not been followed up with application of modern local structural methods [48, 6] until recently [65, 85, 83]. In this thesis we return to the archetypal CdSe nanoparticles to investigate the extent of information about size—dependent structure of nanoparticles from the atomic pair distribution function (PDF) method. This is a local structural technique that yields quantitative structural information on the nanoscale from x-ray and neutron powder diffraction data. [6]. Re- cent developments in both data collection [89, 110] and modeling [94, 111] make this a potentially powerful tool in the study of nanoparticles. Additional extensions to the modelling are necessary for nanoparticles, and some of these have been successfully demonstrated [65, 85, 86, 87, 88]. The history of the PDF method is briefly reviewed by work presented by Billinge [112] Recent results, principally from the Billinge group, are also reviewed. 2.4 Description of the PDF experiment In this section I will briefly review modern PDF experiments. As we are trying to resolve inter—atomic distances to one tenth of an Angstrom resolution, this sets the relevant wavelength of the probing wave to use. From the De—Broglie wavelength equation A = if with A of 0.2 A, the corresponding energy scales are around 60.0 keV for X-rays (E = Li—(OffiaWd/lla 2.04 eV for neutrons (E = f2l51%(meV)), and 3.76 keV for electrons (E = fiQl/fl). Electrons have rarely been the choice for bulk studies due to their short penetration depth (strong scattering). High energy X-rays and neutrons have much better penetrating power suitable for bulk studies, which also means their interactions with matter are rather weak. Thus most quantitative experiments need to be performed at either high—flux synchrotron source or intense neutron spallation sources, and the experiments are slow because of the weak inter- actions. X-rays and neutrons are complementary probes for PDF studies, providing different contrasts for the different elemental species in the structure. For example, X—rays are scattered by the electrons, thus heavy atoms with more electrons are better “seen” by X-rays [48]. In advanced synchrotron sources the electromagnetic radiation that is generated due to the acceleration of cycling electrons travelling at relativistic speeds, gives a wide energy spectrum of X—rays, from which a desired single energy (mono-wavelength) can be selected. Wigglers and insertion devices are used in situations to enhance the brightness at certain photon energies. lV’Ionochromaters are used to select the desired wavelength X-rays. High energy X—rays, e. g. 2 100 keV, can be routinely achieved at third generation synchrotron sources such as the Advanced Photon Source (APS) at Argonne National Laboratory (AN L). Multiple slits are used to confine the beam to dimensions in consideration of either sample sizes or other reasons such as experimen- 31 tal resolution, background, etc.. With monochromatic X—rays, the scattering angle 26 is scanned step by step by moving a point solid state detector (SSD). Again from the formula Q = if sin 6, the scattering intensity I versus Q data are collected. For this purpose, the menu driven program PDFgetX [113] is used. Conventional PDF X—ray measurements with a SSD normally take more than 8 hours, though the data collection time at a single 26 point usually takes less than 10 seconds. As the wide Q range required by PDF analysis needs easily more than 2000 points, it is the step by step scanning that. significantly changes the experimental data collection time scale. Considerable flux increase is expected when the fourth generation free electron laser X-ray source becomes available. However, the data col- lection time then will possibly be limited by the time required to move and stabilize the detector. Attempts with an area detector such as an image plate have been real- ized with the Debye-Scherrer geometry [114]. The counting rate is greatly improved. However, the limiting factor becomes the time to take out the image plate, scan it, and then put it back in. Also, signal-background ratio issues were never statistically resolved. The developed RA—PDF technique [89], that is used for the work presented in this thesis, makes use of a planar 2D image plate used in a transmission geometry mode. 2.5 The rapid acquisition PDF (RAPDF) experi- ment The rapid acquisition pair distribution function (RAPDF) technique is a recent de- velopment technique [89], where the data collection time is reduced by three to four orders of magnitude comparing with the conventional PDF measurements which that are very slow and generally take more than eight hours, even at a third generation synchrotron or intense spallation neutron sources. These RAPDF technique devel- opments open the way for more widespread applications of the PDF technique to study the structure of nanocrystalline materials. They also Open up the possibility for experimentally demanding experiments to be carried out such as time—resolved studies of local structure. A technical manual that describes the RAPDF teclmique is provided in the Ap— pendixes (A, B and C ). Descriptions about the RAPDF experiment including the ex- perimental setup diagram and data collection protocol is presented in Appendix (A). Data integration including the corrections that need to be applied to the image plate data along with the procedures of how the corrections were made with software FIT2D [115] is presented in Appendix (B). Data reduction using PDFgetX2 soft- ware package [90] including some explanations about the corrections that need to be applied is presented in Appendix (C). The Imaged plate characters as an X-ray detector, including usage of the image plate as an X-ray detector and principle of X—ray image plate detection are cover in details in Appendix 0. The RAPDF data described in these Appendixes were collected with an image plate detector MAR3450 at APS 6ID-D MUCAT. All raw two dimension data were integrated and converted to intensity versus 26’ format using the software F IT2D [115]. The data were corrected using the standard methods [48, 6] to obtain the total scat- tering function, S (Q), and the PDF, C(r). This data correction was done using PDFgetX2 software package [90]. The work presented in Appendixes A, B, C and D was done by collaboration with Dr. Xiangyun Qiu. 2.5.1 The image plate (IP) and PDF method Recent developments have shown the utility of 2D detector technology in scattering studies of liquids. A recent report by Crichton et al. [116], has made use of integrated two-dimensional IP data for in-situ studies of scattering from liquid GeSeg. This study, and others more recently, demonstrate the feasibility of using IPs for diffuse 33 scattering measurements though the measurements have been limited in real-space resolution, with Qmaa: _<_ 13.0 A‘1 [117, 118], making them less suitable for the study of crystalline and nanocrystalline materials. Image plates have also been successfully used to study diffuse scattering from single crystals [119]. A Debye-Scherrer camera utilizing IPs has also been tested in our group and shows promise for lower energy X-ray sources such as laboratory and second generation synchrotron sources [114]. Successful application of IP technology to the measurement of quantitatively re- liable high real-space resolution PDFs requires that a number of issues be resolved. For example, it is necessary to correct for coutan'iination of the signal from Compton and fluorescence intensities and for angle and energy dependencies of the IP detection efficiency [120, 121]. These corrections have been covered in details in Appendix (C) 2.5.2 Description of the experiment The. diffraction experiment was performed at 6ID—D MUCAT beamline at the Ad- vanced Photon Source (APS) at Argonne National Laboratory. Diffraction data were collected using the recently developed rapid acquisition pair distribution function (RA—PDF) technique [89] that benefits from 2D data collection. Usually finely powdered samples are packed in a flat plate with thickness of 1.0 mm sealed between kapton tapes (irradiated volume 0.25 mm3, beam size 0.5x0.5 mm?) Data usually are collected with a specific selected x-ray energy, which is usually larger than 60.0 keV, to avoid the K-edges for elements with high atomic number (Z). An image plate camera (Mar345) with a diameter of 345 mm was mounted orthogonal to the beam path with a sample to detector distance of 198.90 mm. The image plate was exposed for (N) seconds and this was repeated (T) times for a total data collection time of (Nx T) seconds. This approach avoids detector saturation whilst allowing sufficient statistics to be obtained. To reduce the background, lead shielding was placed before the sample with a. small opening for the incident beam. 31 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2600 2400 2200 Rows 1 600 1400 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 Columns l I I l I I 1 l l l 1 —: 1 , 1 [ : l l 1 a l J 300 1000 3000 30000 Intensity Figure 2.2: (color) Two dimensional contour plot from the Mar345 Image Plate Detec— tor. The data are from nickel powder measured at room temperature with 97.572 keV incident X-rays. The concentric circles are where Debye-Scherrer cones intersect the area detector. To protect the image for the direct beam, a beam stop is aligned with the direct beam and loaded few millimeters (mm) away from detector surface. The beam stop is a solid tantalum cylinder (diameter 3.1 mm), with an indentation machined to a depth of approximately 2 mm to accept the beam. With a beam stop to sample distance of approximately 150 mm, the scattering angle blocked by the beam stop is less than 1 degree, dependent upon sample—to—detector distance. This gives a Qmm limited to approximately 1 A‘lWith our beam stop to sample distance of approximately 150 mm, the scattering angle blocked by the beam stop is less than 1 degree. This gives the energy dependent sz‘n limited to approximately 0.6 A_1 with 80.725 keV X- rays. In many crystalline materials with small unit cells this is not a problem. When Bragg-peaks are lost at low-Q due to this limit. a weak, long-wavelength oscillation results in C(r), which is not fatal but, ideally, is to be avoided. Ni sample is usually what is measured first, after all the background and standard runs have been performed. Ni was purchased from Alfa Aesar (99.9%, 300 mesh) and was used as received. Fine powders of all the samples were measured in flat plate transmission geometry, with thickness of 1.0 mm packed between kapton foils. The beam size on the sample as defined by the final slits before the goniometer was 0.5 mm x 0.5 mm. Lead shielding before the goniometer, with a small opening for the incident beam, was used to reduce background. All raw data were integrated using the software F it2D and converted to intensity versus 20 (the angle between incident and scattered X-rays). An example of the data from nickel measured at 97.572 keV is shown in Fig. 2.2. The integrated data were then transferred to a home-written program, PDFgetX2 [90], to obtain the PDF. 2.5.3 Real space refinement As the experimental PDF data is obtained, then the next step would be extracting many structural information through a modeling process. Determining of the struc- 36 ture is a prerequisite to further theoretical understanding of the materials properties. Conventional structure determination depends on the intensity and position of Bragg peaks. The most common method for such analysis is the Rietveld method [122]. A least-squares refinement between the calculated and observed intensities is per- formed until the best match with the measured profile is obtained. The calculated intensities are obtained based on the crystal structure, thermal factors, diffraction op— tics, instrumental factors, lattice parameters and other specimen characteristics [123]. Considering only the Bragg peaks assumes perfect long range periodicity of the crys- tal. Such a presumption prevents the studying of non-periodic structures, or aperiodic modifications to otherwise crystallographic materials. However many important ma- terials are disordered and many of these materials owe their important properties to these deviations from the average structure. These deviations result in the occur- rence of diffuse scattering, which contains information about two-body interactions and which is disregarded as a background in the Rietveld method. This shows the need for performing a similar refinement to the PDF since it considers both kinds of scattering. Full profile refinement of the PDF can be carried out using the program PDFfit2 [124]. In this method the model is defined in a small unit cell with atom positions specified in terms of fractional coordinates. The PDF is calculated from the model structure as follows (Figure 2.4), ( 1) assume that we have a sample that consists of N atoms at position ’I'n with respect to some origin. Place the origin of our space randomly at any atom. (2) Systematically find every other atom in the sample and measure its distance from the origin. (3) Each time we find an atom we place a unit of intensity at that position, Ti, on the axis of the radial distribution C(r). When we have cycled over all of the atoms in the sample we move the origin to another atom and repeat the process, adding the intensity to the G (r) function. We multiply the unit of intensity for each atom-pair by bibj/ < b >2 where b,- is the 37 scattering length of the ith atom. 1 bib' 7. 00(1') 2 7:: [(5)43200' - 700] — 177pr i j = wipe) —/)ol: (2.3) where r, j is the distance between atoms 2' and j; the value b,- is the scattering length for atom i; (b) is the average scattering length over all atoms; and [)0 is the number densitythe sum goes over all the pairs of atoms 12 and j within the crystal. (1)), is the average scattering power of the sample. T he goodness of the fit is determined by the p‘dl‘alllOlUI’ Ru, which is computed according to the relation: 21,21 ""'("i)lG()bs("i) _ (’cr'zlcf’ill2 R i : 2.1 ” ZN ,( 002 c) l ) 2:1” ’1 obs 1‘ Here it) is the weighting factor, w(r',j) = 1 / a? where a is the estimated standard deviation on the ith data-point at position 72,-. The sum goes over all measured data points 1‘, in the experimental PDF. Performing the sum for all the atoms in the sample (1023) is impractical, but the sample is made of many equivalent cells which are periodically repeated in the space so that we can limit the calculation to one unit cell. The refined parameters are the same as those used in Rietveld. The main difference from Rietveld is that it allows for different 'r scaling refinement, which enables one to study the local structure for different r-ranges [48]. The ability to refine the local structure yields information about disordered and short-range atomic correlations. Because of the similarity between the Rietveld methods and the PDF modeling, a quantitative comparison between the resulting structures of both refinements may be made. This is an important first step in revealing the existence of local distortions beyond the average structure. It should be noted that the pr for the PDFfit2 is similar to that used in Rietveld analysis but the functions being fit are significantly 38 different. Hence, direct comparison of pr from PDF and Rietveld analysis should not be made. pr values are useful measures of the goodness-of-fit when comparing how different models fit to the same PDF data. For well-crystallized samples, PDF pr values greater than 10%, are not uncommon. Obtaining a. pr value of less than 20%, for nanocrystalline structures is excellent. 2.5.4 Early results: standard nickel In RA-PDF technique, the first OXPQI'llIlClltal data is collected from a Ni sample in order to comment on the data quality as well as instrumental resolution. The Ni data usually shows excellent counting statistics over the entire Q range, reflecting the significant advantage obtained by extracting 1D data sets by integration of a 2D area detector. Image plate exposure time for a Ni sample is 12 seconds. The exposure time is chosen carefully for each sample to maximize the dynamic range of the IP without saturating the phosphor on the Bragg peaks. Data. collection was repeated a number of times until acceptable statistical errors were observed at high-Q in the reduced structure functions, as shown in Figures 2.3(top). The PDFs, C(r), show either superior or acceptable qualities. The example shown in Fig. 2.3, is from standard Ni powder with Qmax of 28.5 A71. The Ni PDF. C(r), in Fig. 2.3 (bottom) appears to have minimal systematic errors which appear as the small ripples before the first PDF peak at r = 2.4 A. These result from imperfect data corrections and their small amplitude is a good indication of the high quality of the data. A structural model (space group Fm3m) was readily refined, and gave excellent agreement with the data as shown in Fig. 2.3(bottom). The lattice parameter (3.5346(2) A) and an isotropic thermal displacement parameter (U = 0.005184(6) A2) were refined. The lattice parameters reproduce the expected values given in previously published data [101]. In spite of the simplicity of the Ni crystal structure, the exceptional quality of both the experimental Ni PDF and re- 39 Fat“) 18‘24'30'36' '12 4 8 12 16 20 24 28 Q at") . ; E ‘ t , ~ I ( ' A : : <3( - i 3 : z i: :‘i ‘t "I I I I l J_ 1 ._ I I 10 12 14 16 rm) Figure 2.3: (top) The experimental reduced structure function F(Q) = Q* (8(Q) — I) of Ni powder. (bottom) The experimental C(r) (solid dots) and the calculated PDF from refined structural model (solid line). The difference curve is shown offset below. 40 finement indicates that the necessary data corrections of image plate data with high Q range (28.5 A‘l in this case) can be carried out properly and with an acceptable level of accuracy. 2.6 The PDF for nanostructured materials The pair distribution function (PDF) reveals directly in real space the inter-atomic distances in a material. Recent applications have proved the PDF technique as a powerful local structural probe of nanostructured materials [48, 6], as well as its traditional use to study liquids and glasses [125, 126]. In the ideal case of infinite perfectly ordered crystalline structure the calculated PDF goes over large distance range approaching 1:00, as an example shown in Fig. 2.4 Not the case for finite size crystals. For example, in the case of spherical nanoparticles, the calculate PDF inten— sity will fall-off due to the finite size. Then corresponding calculated PDF will look like the PDF of the bulk material, that shown in Fig. 2.4, that has been attenuated by an envelope function given by the PDF of a homogeneous sphere, as follows [127, 83] G (r, d)... = G (r) f (a). (2.5) where C(r) is given in Eq. 5.1, and f ('r, d) is a sphere envelope function given by f ('r, a) = (1— 311 + % (3)3] @(d — r), (26) where d is the diameter of the homogeneous sphere, and @(r) is the Heaviside step function, which is equal to 0 for negative 2: and 1 for positive. Examples of the calculated PDF for spherical nanoparticles, 20 and 40 A in size, are shown in Fig. 2.5. Detailed information about the PDF of a single spherical particle and the derivation of Eq. 4.5 can be found in Appendix E. 41 Z I 4 l . - .' - . . - . . j j I. ,_ O. - oo- - (D- - C? 1 «t- on: v o N- O > 3 . . I n I n I . I . I . I 2 4 6 8101214161820 r00 Figure 2.11: Calculating the pair distribution function from cupper (Cu). 2.7 Extracting information from the PDF The PDF for a perfectly ordered crystalline structure is the sum of well-defined delta functions, the positions of which give the separations of pairs of atoms in the structure. In real materials atoms are displaced from their perfect position due to the thermal motion and/or static displacements of the atoms. This gives rise to a distribution of atom-atom distances, which causes the PDF peaks to be broadened. Three basic properties of the PDF peaks may reveal great information about the structure: The position of the peak yields information about the atom-pair distances, the width of the peak reveals information about the disorder of the atoms involved in the pair and the integrated intensity imder the peak gives the coordination number of the origin atom. These three important properties of the PDF peaks can be extracted 42 i l, (.1 it. . , law u ~ O 4,!“ g _ in» [J g L L L L A J L -o.1 ' 10 Figure 2.5: Using sphere envelope function E6, the pair distribution functions are calculated for two nanoparticles with different sizes. (top) 20.0 and (bottom) 40.0 A. used to 43 easily by fitting Gaussian peaks convoluted with a sine function that accounts for the termination effects. 2.7 .1 Peak position The peak position yields bond-lengths directly, which can be very useful in under- standing the local atomic structure. For example while investigating the structure of the semiconductor CdSe nanoparticles Masadeh et al [84] studied the evolution of the bond lengths with nanoparticle size. Based on the dependence of the peak position on nanoparticle size they concluded that local atomic-level strain exists and it is size dependent. 2.7.2 Peak width The peak width may be obtained by fitting a. Gaussian function. Since the number of neighbors is constant the integrated area under the peak is invariant and the peak height, extracted directly from the data, gives the inverse peak width. This often gives a more accurate determination of the evolution of peak width with some experimental parameter such as temperature or composition. Many kinds of information may be obtained by studying the peak width such: (1) The width as a function of temperature yields information about the Debye temperature of a bond [106]. (2) The width as a function of atomic separation yields information about correlated atomic dynamics [102]. (3) The width as a function of doping gives information about doping induced disorder [128] (4) The width as a function of nanoparticle size gives information about the level of internal disorder [84] 2.7 .3 Peak intensity The integrated intensity under the PDF peak yields information about the number of atoms at a specific distance i.e. coordination number). This type of analysis is 44 widely used in studies of glasses [129] and in partially crystalline samples [130]. For example; while studying the structure of nanoporus carbon; Petkov et al [131] used the fall-of the first peak peak intensity with temperature as evidence of disorder in the sample heated to 800°C compared to the one heated to 1000°C. The fall-off PDF peak intensity as a function of 7‘ used for estimating the nanoparticles size in nanoscale materials [84]. 2.8 Conclusions The PDF technique goes beyond crystal]ography yielding to a new structural infor- mation. The RA-PDF method features high energy X-rays coupled with an area detector. Use of high energy X-rays provides enough Q space range for quantitative PDF analysis. Use of an area detector offers three to four orders of magnitude reduc- tion of data collection time. The new data analysis program PDFgetX2 implements necessary corrections to account for the characteristics of the IP detector used. we have shown with convincing examples that high quality medium-high resolution PDFs can be obtained from wide range of materials. The RA—PDF method and easy to use program PDFgetX2 expect to significantly lower the barrier for wide-spread PDF ap- plications to a broad scope of crystalline and nano-crystalline materials. For example, more samples can be measured, more detailed phase diagrams can be mapped out with quicker data collection. The PDF technique can now be introduced to new types of experiments. Time resolution of the RA-PDF method is around 2 minutes, which is limited by the read- out and erasing time. A large fraction of time resolved experiments can benefit from high real space resolution PDF local structure analysis simply by moving the detector closer to the sample and using high X-ray energies. The overall experimental setup is also significantly simplified with no moving parts during data collection. Materials under extreme conditions such as high pressure, high temperature become :15 possible for PDF studies. Recent developments in both data collection[89, 110] and modeling[94, 111] make this a potentially powerful tool in the study of nanoparticles. Additional extensions to the modelling are necessary for nanoparticles, and some of these have been successfully demonstrated[65. 85, 86, 87, 88]. As the high energy sources, such as APS, becoming more accessible and the data collection time get more faster (with the usage of RAPDF technique) and user friendly software [124] joined with technique manual (presented in the Appendix), this will make the PDF as “nonspecific user friendly” technique and useful tool for structural analysis in the next decades. The user friendly software [124] with technique manual are. available for public. 46 Chapter 3 Real Structure of NagBiO4 The work presented in this chapter was done in collaboration with S. Vensky, L. Kicnlc, R. E. Dtnnebier', S. J. L. Billinge and M. Jansen. This work has appeared in a publication in 2005 [132 . The real structure of a new crystalline high temperature phase, metastable at room temperature, in the system of sodium-bismuth-oxygen, ,B-Na3BiO4, was de— terminate using high resolution X-ray powder diffraction (HRXRD), atomic pair distribution function (PDF) analysis and high resolution transmission electron mi- croscopy (HRTEM) [132]. The ,B-Na3BiO4 phase was synthesized by anodic oxida— tion of bismouth(III)-oxide in a sodium hydroxide melt. The sample synthesis was done by Dr. Jansen Group as well as the HRXRD and HRTEM analysis [132]. Based on the HRXRD analysis [132], the average crystal structure of fi-Na3BiO4 at am- bient conditions (R3m, a = 3.32141(9) A, c = 16.4852(5) A is structurally related to B-NaFeOg with metal layers almost statistically occupied by a NazBi ratio 3:1. Analysis of the long-range order on the bulk material by Rietveld refinement revealed to approximately NazBi ration of 2:1 and 4:1 in the consecutive metal layers, while a detailed PDF analysis of the local order, which is presented here in this chapter, revealed the existence of almost pure sodium layers and mixed NazBi layers of ratio 1:1. Complen‘ientary study using HRTEM exhibited a. complex domain structure with short-range ordered among the one layer. The HRTEM results came out in a good agreement with the PDF results, as a support for existence of almost pure sodium layers locally. Much of this work has appeared in print in 2005 as collaboration with Dr. Jansen group [132]. 3. 1 Introduction The rock salt arrangemtmt is among the fundamental building principles in three- dimensional space. Besides the vast families of chemically different AB compounds, it is realized in salts containing complex anionic and / or cationic constituents, even including extended cluster ions. Examples are calcite CaCO3, [133, 134, 135] sodium nitrate NaNO3, [135, 13G] sodium ozonide NaO3, [137] calcium carbide CaC2, [138] sodium azide N aNg, [139] and fulleride compounds of the [l\=‘I(NH3)6]C50.6 N H3 type, M 2 (Cd, Co, Mn, Ni, Zn) [140]. Substitution variants, with either the cationic or anionic sublattices occupied by different species in an ordered manner, represent another class of rock salt. derivatives. Here various ternary alkali metal oxides of general formula types ABOg, A2B03, A3804, A6B05,... (A = alkali metal, B 2 metal or nomnetal) need to be included. Some of the latter show order-disorder transitions within their cationic sublattices, and are reluctant to fully order, during the synthesis along the solid state route. In the past, this phenomenon has caused some confusion with respect to the correct indexing of the powder patterns of e.g. Li5SnO3, [141, 142] LigMnO3, [143] or NagRu03 [144, 145]. The room temperature modification of N a3BiO4, referred to as a-Na3BiO4, hereafter, is a fully ordered rock salt substitution variant with monoclinic symmetry [145]. Here we report on a heav- ily disordered high temperature modification of Na3BiO4, i.e. fi-Na3BiO4, grown electrochemically from a NaOH/Bi203 melt. The oxidation state of +V for bismuth in oxides is generally rare. However, it has been realized in a number of alkali bis- muthates: ABiO3 and A3BiO4 (with A = Li, Na, K), Li5BiO5 and Li7Bi06 [146]. 48 Out of these, the only one accessible through electrocrystallization from a melt, be- sides solid state routes, was KBiO3. [1117] 3.2 Experimental methods 3.2. 1 Sample syntheses The sample syntheses was done by Dr.Jansen group as follow. Crystalline material of Vii—Na3BiO4was obtained by electrocrystallization from alkali hydroxide melts con- taining bismuth(III) oxide Bi2O3. The components of the melt, 1 g Bi203 (Riedel- de Haen, 10305), 12 g NaOH (Merck, 106498), 3.4 g LiOH (Merck, 105691), and 0.4 g ZnO (Chempur, 008417), were used without pre—treatment. Fig. 3.1 shows a schematic drawing of the electrolysis cell used. A nickel crucible containing the com- ponents of the melts was placed into a closed glass reaction vessel and heated during three hours starting from a temperature of T = 200 CC up to a temperature slightly above the electrolysis temperature (330-350 0C), allowing the melt to equilibrate. 2110 levels the amount of water in the melt. After one hour, the temperature was decreased to the electrolysis temperature, the platinum electrodes were inserted, and the reaction vessel was closed. A platinum wire (9: 1 mm) was used as the cathode, and a second platinum wire (9 = 0.3 mm) as the anode. A constant current density of 1 mA/cm‘2 was applied for 18—42 h using a VMP multipotentiostat (Bio-Logic, France). fi-Na3BiO4 crystallized as dark red, shiny crystals at the platinum anode. The material was washed with bidestillated water and acetone, and was stored under an argon atmosphere. Crystals of ,B-NagBiO4 exhibits an undulated surface (Fig. 3.2) giving hint towards a distorted crystal structure. Images of the crystals were taken by means of scanning electron microscopy (ESEM XL30 TMP, Philips). Investigations of the stoichiometry of fl-Na3BiO4 by chemical analysis using ICP-OES technique were conducted with an optical emission spectrometer ARL 3580 B. The oxygen content. 'JI U wires leading Figure 3.1: Cell used for the electrocrystallization of d—Xa3BiO4 . (1) (5) furnace. ((3) to the potentiostat, (2) das inlet, ( 3) connectors. (41) Pt electrodes. nickel crucible. Adapted from [132]. Figure 3.2: STM images of ,r3—Na3BiO4 . Adapted from [132]. 51 was measured by use of a hot extraction analysator TC-436. Crystalline material of fl—Na3BiO4 was obtained by solid state reaction [145]. A thoroughly ground mixture of Na202 (Aldrich, 223417) and Bi203 (Riedel-de Haen, 10305) in the ratio 3:1 was placed in a corundum boat and reacted for 12 hat T: 600 CC, in a flow of oxygen. a—Na3BiO4 was obtained as a bright yellow powder. 3.2.2 High resolution X-ray powder diffraction The High resolution X—ray powder diffraction data of fi—Na3BiO4 (Fig. 3.3) were collected at ambient conditions, by Dr..]ansen group, in transmission geometry with x1000 320 160 Int 3 l 80 5 g . ”fl" ’ . (a) 0 .. _..- -1» 1 ~ .: (b) I l I I I I I I r I I 1 I 3 6 9 12 tth (deg.) Figure 3.3: High resolution X-ray powder diffraction of Na3BiO4 at ambient condi- tions as a function of the diffraction angle 26’ (diamonds symbol), the best Rietveld fit profile in space group R3m (a), the difference curve between observed and calcu- lated profile (b), and the reflection markers (vertical bars). The used wavelength is /\ = 0.18528(2) A . Several regions representing the decompositions products sodium hydroxide and bismuth oxide were excluded from the refinement. Adapted from [132] the sample sealed in a 0.5 mm lithiumborate glass (Hilgenberg glass No. 50) capillary (1'-.\la3 8104 [1’15] 1’3-NagBiO4 Formula Na3BiO4 Na0_75Bi0.25O Temperature (K) 295 295 Space group (No) P2/c (13) R3171 (166) a (A ) 5.237(1) 3.32141(9) b (A ) 669(6) 3.32141(9) c (A ) 565(0) 16.4852(5) a O 90.0 90.0 ,8 0 109.8 90.0 c/ O 90.0 120.0 v (A3) 208.8(19) 157.50(1) Table 3.1: The average crystallographic structure of ,B-Na3BiO4 from synchrotron powder data in comparison with o-Na3BiO4[145]. This table adapted from [132]. at beamline X17B1 of the National Synchrotron Light Source at Brookhaven National Laboratory. X-rays of an energy of 67.0 keV were selected by a silicon(220)-Laue— Bragg—monochromator and analyzed by a sagittally bent silicon crystal [148, 149, 150, 151]. The exact wavelength was determined as A = 0.18528(2) Ausing the NIST SRM 660 La86 standard. Data were taken in steps of 0.0010 29 from LOO-15.00O for 16 h. The samples were spun during measurement for better particle statistics. The powder pattern exhibits several peaks of small amounts of sodium hydroxide and bismuth oxide. The extinctions found in the powder pattern indicated R3, R3, R32, R3111, and R3111 as the most probable space group. The latter was confirmed by Rietveld refine- ments. The peak profiles and precise lattice parameters were determined by LeBail- type fits [152] using the program GSAS [122]. The background exhibited various humbs caused by strong diffuse scattering and was modeled manually using GUFI. The peak-profile was described by a pseudo-Voigt function in combination with a special function that accounts for the asymmetry due to axial divergence [153, 154]. Rietveld refinements [155] (Fig. 3.3) were performed using the program package GSAS. Starting parameters for the atomic positions of ,B-sodium bisniuthate were taken from the, structurally related a—sodium ferrate NaF (‘02. Starting values for the CI! to peak profile, background, and lattice parameters were taken from the corresponding LeBail-fit. No additional phases were included in the refinement, but several excluded region containing reflections of sodium hydroxide and bismuth oxide were defined. Structural variations causing diffuse scattering were not included in the refinement. The Rietveld refinement results are presented in Table 3.2. 3.2.3 PDF data collection The diffraction experiment was performed at 6ID-D [NCAT beamline at the advance photon soiu'ce (APS) at Argonne National Laboratory. Data acquisition at 300 K employed the recently developed rapid acquisition PDF (RA-PDF) technique [89] with the X-ray energy of 87.97 keV. Data were collected using an image plate camera (.\Iar345), with a usable diameter of 345 mm, mounted orthogonal to the beam path with sample to detector distance of 159.88 mm. Lead shielding before the goniometer, with a small opening for the incident beam, was used to reduce the background. All raw data were integrated using the software F it2D [115] and converted to intensity versus 29 (the angle between incident and the scattered X—rays). The integrated data were normalized with respect to the average monitor count, then transferred to the program PDFgetX2 [90] to carry out data reduction to obtain S (Q) and the PDF, G (r), which are shown in Fig. 3.4 (a) and (b) respectively. Transformation of the F (Q) = Q(S (Q) -— 1) to Qmax of 25.0 A"1 was found to be optimal. There are basically two considerations. The first is to have sufficient Qmag; to avoid large germinate effects; second is to reasonably minimize the introduced noise level as signal to noise ratio decreases with Q. We found that Qma 1; of 25.0 A-1 has significant lower noise level without losing useful structure information, i.e. no significant change of PDF peaks. l—JIIIILI, I I l I n I 1 l l ooooooo :z“:::3%‘h on o o o o 0.02:...“ 6 81012141613 r01) Figure 3.4: The experimental reduced structure function F (Q) = Q * (S(Q) — 1) of Na3BiO4 with Qmax cut at 25.0 A‘1(a) and (b) the corresponding PDF. 55 3.2.4 High Resolution Transmission Electron Microscopy For HRT EM investigations n‘ricrocrystalline samples of N a3BiO4 were crushed under dry argon atmosphere in a glove box. Copper grids were covered with the powder, leaving the crystallites in random orientations. These sample carriers were fixed in a side—entry, double-tilt holder (maximum tilt: :1: 25 O in two directions). An argon bag was used to transfer the sample holder to the microscope. High Resolution Transmis— sion Electron Microscopy (HRTEM) and Selected Area Electron Diffraction (SAED) were performed in a Philips CM3OST (300 kV) which is equipped with a LaB6, cath- ode. These presentcd HRTEM measurements were performed by Dr. Jansen Group. 3.3 Structural analyses 3.3.1 X-ray analysis and structure solution ,fl-NagBiO4 crystallizes in trigonal symmetry 3.5 in the oNaFe()2—type structure, [156] which may be derived from the rock-salt aristotype (cubic closest-packed oxygen anion arrangement with all octahedral voids occupied by cations) with alternating cation layers along [111] ([001] in hexagonal metric). In contrast to the aNaFeOz-type, where the cation layers are occupied alternately by sodium and iron, in ,B-NagBiO4 all cation layers are mixed with an average Na : Bi ratio of close to 3: 1. Two types of cation layers exist, which are stacked alternatively: one is enriched by sodium up to a Na : Bi ratio of 3.85 : 1 (see table 3.2), while the other is enriched by bismuth to a Na : Bi ratio of 2.40 : 1. The resulting average ratio of 3: 1 is verified by chemical analysis. Both, the Na enriched cation position Na(l)/Bi(l) and the Bi enriched position Na(2)/Bi(2) are coordinated by oxygen anions in an undistorted octahedral coordina- tion sphere with shorter distances (cation oxygen distances 2.273 A for Na(2)/Bi(2); 2.451 A for Na(1)/Bi(1)) for the position enriched by the smaller cation (Bi5+ in contrast to Na+). The coordination sphere of the oxygen anion can best be described .. 4;: ......0L 4’ m “L's; "“4. Figure 3.5: A average crystal structure of B—NagBiO4 at ambient conditions. Oxygen atoms are shown in white. The atoms positions named “Na” represent a mixed occupancy of 80% sodoum and 20% bismuth, While the atoms positions named “Bi" represent a mixed occupancy of 70% sodoum and 30% bismuth atoms. Adopted from [132]. 57 atom x y z soprm Uvad Bi(1) 0.0 0.0 0.0 0.206(1) 00113(3) Na(l) 0.0 0.0 0.0 0.794(1) 0.0113(3) Bi(2) 0.0 0.0 0.5 0.2294(1) 0.0133(3) Na(2) 0.0 0.0 0.5 0.700(1) 0.0133(3) o 0.0 0.0 0.2407(6) 1.0 037(2) Table 3.2: The results of the refined parameters (Rietveld). Positional parameters and temperature factors for the ,8~Na3BiO4 at ambient conditions. Standard uncertainties are given in parentheses [132]. by a slightly distorted octahedral sphere coordinated by sodium and bismuth. An ordering with pure sodium or bismuth layers was not found by Rietveld refinement. Because of the apparent diffuse scattering, the local order at the atomic level was studied by means of pair distribution function analysis. 3.3.2 Pair distribution function analysis Structural informatirm was extracted from the PDFs using a full-profile real-space local- structure refinement method [101] analogous to Rietveld refinement [15 5.] We used an updated version [124] of the program PDFfit [94] to fit. the experimental PDFs. Starting from a given structure model and given a set of parameters to be refined, PDFfit searches for the best structure that is consistent with the experimental PDF data. The residual function (Raw) is used to quantify the agreement of the calculated PDF from model to experimental data: Ru: = N .., 2 Zizl “( zllG 01).rs(73-)2f:(zl(2(” ill _ (3-1) N 22 :1w(ri)C2 obs ) Here the weight w(-ri) is set to unity which is justified because in C(r) the statistical uncertainty on each point is approximately equal [157, 158]. The experimental PDF with Qmax 25.0 A‘1 was refined within the crystallo- graphic model of i3-Na3BiO4 described in Table. 3.3. The constraints of space group R—3m were maintained. Lattice parameters, thermal displacen'ient parameters and some experimental factors are refined. The occupancy of the atoms on each site is fixed according to the values in Table. 3.3. We obtained lattice constants of a = b = 3.344(78) A, and c = 16.481(10) A. Fig. 3.6 (a) shows both the experi- mental and model PDFs. The results are summarized in Table. 3.3 . It is clear from the figure that the fit is quite good (pr = 0.21) [94] in the high-r region above r = 6 A indicating the model agrees with the PDF in this region. However significant deviations between the model and the data exist below r = 6 A. In particular, the two model peaks at 2.45 A and 4.77 A ( Fig. 3.6 (a) and Fig. 3.7(a)) are poorly fit. They are Na/Bi-O and Bi/Na-Bi/Na peaks, respectively, originating from the ONa/B16 octahedra. These peaks can be reduced in amplitude if these correlations have an excess of Na over Bi. We therefore tried relaxing the constraint of Bi occupancy on the (0,0,0) and (0,0,0.5) sites, while maintaining the sample stoichiometry. “('0 obtained a better value of the weighted-profile R-value, (R1111): 0.18) with Bi (0,0,0) occupancy refining to 0.098(4) and Bi(0,0,0.5) occupancy t0 0.402(4). Fig. 3.6 (b) shows the fits with the refinement results summarized in Table. 3.3. In particular, the fit in the low-r region is improved, but still more intensity needs to be removed from the 2.45 A and 4.77 A peaks. Therefore, we manually set the Bi atoms to have 0.0 occupancy at (0,0,0) and occupancy 0.5 at (0,005) and fixed these values. The resulting model agrees extremely well in the low-r region below 5 A (pr = 0.12, Fig. 3.7 (b)). However, the high-r region above 10 A is fit rather poorly (Ru/1p = 0.30). The results are summarized in Table. 3.3. On the surface, these results are in contradiction. The average structure refined from Rietveld refinement, and refined from the PDF when it is fit over a wider range of r, suggests that Bi is distributed approximately equally over the (0 0 0) and (0 0 0.5) crystallographic sites. However, the local structure refinement is clear that Bi prefers (0 0 0.5). Disagreements between local and average structures are not uncommon [6] 4 8 G (A'Z) o -4 -8 8 0 (21-2) 0 —4 -8 4 8 G (ii-2) LAAAAMA‘vAA AVA MAW” A A “A A V'WV“ V WWV VW Vi * vv'v -8 —4 o I I I I I I I I I I I I I I I I L I I 024681012141618 MA) Figure 3.6: The experimental G (7') (solid dots) and the calculated PDF from refined structural model (solid line) of ,B-Na3BiO4 .The difference curve is shown offset. below. (a) without refining the ocmipancy, (b) with refining the occupancy, (c) manually setting the occupancy of Bi atom at (0.0.0) site to be 0.0. 60 [48] and these. differences are always reconcilable by some averaging of local structural motifs that yield a higher—symmetry average structure. It is less clear how extended defects or local domain formation could be used to reconcile the results found here. The (0 0 0) and (0 0 0.5) sites form sheets of l\a/ Bi sites perpendicular to the c- axis coming from the edge-shared ON a/ Big octahedra. Three (0 0 0)-site atoms in a triangle form one face of the octahedra while the three (0 0 0.5) sites, with the triangle rotated 60 degrees, form the opposite face of the same octahedron. According to the average structure the Bi ions are distributed equally over both faces, whereas the local structure indicates that. one face is preferred. In this case, the PDF may be telling us something different. In the average structure another difference between the (0 0 0) and the (0 0 0.5) sites is that the atoms on the former site form a long (2.45 A) bond with the oxygen at the center, whereas in the latter site form a shorter (2.27 A) bond to the oxygen. What is clear from the PDF is that the Bi ion always forms a short (2.27 A) bond to the oxygen. If this were not true we would see a peak in the experimental PDF at (2.45 A) that is clearly absent. Thus, we understand the PDF and average structure results with Bi distributing itself essentially randomly over the different sites in the octahedral; however, wherever it finds itself, the Bi distorts the octahedron to locally make a short (2.27 A) bond with the oxygen. 3.3.3 High Resolution Transmission Electron Microscopy The arrangement of Na and Bi atoms in the real structure was examined by means of HRTEM, by work done by Jansen Group [132]. Interpretation of HRTEM images gives additional evidence about the partial order and the sizes of the domains and their relative orientations. Feature of the real structure is plainly visualized, the formation of anti-phase boundaries between the micro—domains, see arrows in Fig. 3.8 (top). The formation of anti—phase boundaries is interconnected with the ordering of the metal atoms. 61 0 (3'2) G (3'2) - A K A AA M‘A 'VWVV V' J4] I [44.1 I 4*. -8-6-4-20 2 4 6 8-8-6-4-20 2 4 6 8 3.6 4.2 4.8 5.4 r01) 1.8 2.4 3 Figure 3.7: The experimental C(r) (solid dots) and the calculated PDF from the refined structural model (solid line) of fi-Na3BiO4The difference curve shown offset below. (a) without refining the occupancy, (b) manually setting the occupancy of Bi atom at (000) site to be 0.0. Rmnge (A) 1.8-20 1.8-10 10-20 1.8-6.0 SOfPDF sofPDF SOfPDF SOfPDF Bi(1) 0.098(4) 0.019(5) 0.141(5) 0.00 1 Na(l) 0.902(4) 0.981(5) 0.859(5) 1.00 1 Bi(2) 0.402(4) 0.481(5) 0.359(5) 0.50 1 Na(2) 0.598(4) 0.519(5) 0.641(5) 0.50 1 o 1.0 1.0 1.0 1.0 pr 0.18188 0.17954 0.14660 0.11925 UpDF(Bi) 0.01035(5) 0.01088(8) 0.00910(17) 0.00958(11) UppDF(O) 0.0230(5) 0.0238(7) 0.0351(13) 0.0162(7) Table 3.3: The results of the refined parameters from PDF analysis. Positional pa- rameters and temperature factors for the fi-NagBiO4 at ambient conditions. Standard uncertainties are given in parentheses. ’1 This parameter is set manually, it was not refined. Positional parameters were kept fixed as in Table. 3.2. . Adopted from [132]. Figure 3.8: HRTEM 011 domain crystals of Na3BiO4 63 3.4 Conclusion We have investigated the average and the real structure of ,8—NagBiO4 using high res- olution X-ray powder diffraction, pair distribution function analysis, and high resolu- tion transmission electron microscopy. The tools employed show specific weaknesses and strengths, each. Evaluation of Bragg powder reflections, provides insights into the average structure only, while Fourier transformations of the total scattering, still integrating over the whole sample provide the average local pair interactions. Finally, HRTEM images local defect structures, but do not give the statistical weights with which specific defect patterns occur. Thus, it is not coming as a surprise, that, at a. first glance, the different techniques applied are producing different and even de— viating structural information. However, HRTEM and PDF are in good agreement. Fitting the measured Bragg intensities of the powder pattern result in a rock salt structure with a slight trigonal distortion with respect to the cubic unit cell. This symmetry reduction is caused by a partial ordering of the cations. Along [001]t,.,-g, corresponding to [111]Cub, layers containing random distribution of the Na+ and Bi5+ cations, however, with slightly varying Na : Bi ratios follow each other alternatively. Thus, the average structure is coming very close to the a—NaFe02 type of structure. Fitting the total scattered intensity, is basically confirming the results of the Rietveld refinement. However, there are discrepancies concerning the Na : Bi ratios for the two crystallographically different cationic positions. This can be traced back to a larger amount of local structure information revealed by the total scattering experiment. In addition, the HRTEM results document a complex domain structure within single cristallites. Three different local structures are found. One corresponds to the ideal rock salt structure with short-range order of the cations. The second is based on the ordered crystal structure of a-Na3BiO4 with defined atomic positions for the sodium and bismuth atoms. The third, which represents the majority of the investigated grains, matches what has been found as the average crystal structure of 1'3-NagBiO4 64 by PDF analysis with approximately pure sodium and mixed sodium-bismuth lay- ers. Taking into account the many stacking faults of this highly distorted crystal structure, it is evident that superpositions of these local structures yield the average structure, as found by Rietveld analysis. The results of the structural studies shed light on the way how ,B-NagBiO4 forms. It can be assumed that at the conditions of electrocrystallization a rock salt structure forms with the cations randomly dis- tributed. During cooling, the partially ordered structure of ,B-NagBiO4 is obtained, while the total ordered form of a-Na3BiO4 occurs while annealing or by solid state synthesis. It has been shown that even with the lack of single crystals suitable for single crystal diffraction, a detailed analysis of the real structure of highly disordered mate- rials is possible using complementary high resolution diffraction tecl'iniques 011 single grains and powders. Chapter 4 Quantitative Structure Determination of CdSe Nanoparticles [Witch of this work prem—mted in. this chapter has been. published in Phys. Rev. B. This work is in collaboration with E. Bozin, C. L. Farrow. G. Paglia, P. Jahas, A. Kai'kamlrar, M. C. Kanatzidis and S. J. L. Billinge [84]. In this chapter, I will present a detailed analysis of the structural information available from pair distribution function (PDF) data 011 (2-4 nm) CdSe nanoparti- cles. The PDF method is demonstrated here as a key tool that can yield precise structural information about the nanoparticles such as the atomic structure size of the core, the degree of crystallinity, local bonding, the degree of the internal disorder and the atomic structure of the core region, as a function of the nanoparticle diame- ter. Three CdSe nanoparticle samples with different diameters that exhibit different optical spectra have been studied. The purpose of this study is not only to explain the PDF data of CdSe. nanoparticles through a modeling process, but also to system- atically investigate the sensitivity of the PDF data to subtle structural modifications in nanoparticles relative to bulk material. 66 The size-dependent structure of CdSe nanoparticles, with diameters ranging from 2 to 4 nm, has been studied using the atomic PDF method. The core structure of the measured CdSe nanoparticles can be described in terms of the wurtzite atomic structure with extensive stacking faults. The density of faults in the nanoparticles ~ 50% . The diameter of the core region was extracted directly from the PDF data and is in good agreement with the diameter obtained from standard characterization methods suggesting that there is little surface amorphous region. A compressive strain was measured in the Cd-Se bond length that increases with decreasing particle size being 0.5% with respect to bulk CdSe for the 2 nm diameter particles. This study demonstrates the size-dependent quantitative structural information that can be obtained even from very small nanoparticles using the PDF approach. 4. 1 Introduction Semiconductor nanoparticles are of increasing interest for both applied and funda- mental research. W’urtzite—structured cadmium selenide is an important II-VI semi- conducting compound for optoelectronics [159]. CdSe quantum dots are the most extensively studied quantum nanostructure because of their size-tunable properties, and they have been used as a model system for investigating a wide range of nanoscale electronic, optical, optoelectronic, and chemical processes [41]. CdSe also provided the first example of self-assembled semiconductor nanocrystal superlattices [38]. With a direct band gap of 1.8 eV, CdSe quantum dots have been used for laser diodes [39], nanosensing [40] , and biomedical imaging [33]. In fundamental research, particles with a diameter in the 1-5 nm range are of particular importance since they cover the transition regime between the bulk and molecular domains where quantum size effects play an important role. Significant deviation from bulk properties are expected for particles with diameter below 5 nm, and were observed in many cases [33, 160] as well as in this study. 67 Accurate determination of atomic scale structure, homogeneous and inhomoge- neous strain, structural defects and geometrical particle parameters such as diameter and shape, are important for understanding the fundamental mechanisms and pro- cesses in nanostructured materials. However, difficulties are experienced when stan- dard methods are applied to small nanoparticles. In this domain the presumption of a periodic solid, which is the basis of a crystallographic analysis, breaks down. Quantitative determinations of the nanoparticle structure require methods that go beyond crystallography. This was noted early on in a seminal study by Bawendi et al. [71] where they used the Debye equation, which is not based on a crystallo— graphic assumption, to simulate semi-quantitatively the scattering from some CdSe nanoparticles. However, despite the importance of knowing the nanoparticle structure quantitatively with high accm'acy, this work has not been followed up with applica- tion of modern local structural methods [48, 6] until recently [65, 85, 83, 86, 87, 88]. In this study we return to the archetypal CdSe nanoparticles to investigate the extent of information about size-dependent structure of nanoparticles from the atomic PDF method. This is a local structural technique that yields quantitative structural infor— mation on the nanoscale from x-ray and neutron powder diffraction data [6]. Recent developments in both data collection [89, 110] and modeling [94, 111] make this a potentially powerful tool in the study of nanoparticles. Additional extensions to the modelling are necessary for nanoparticles, and some of these have been successfully demonstrated [65, 85, 161]. The information obtained from PDF methods is complementary to more tradi- tional approaches for studying nanoparticle structure. Often local probes, such as high resolution transmission electron microscopy (HRTEM) are used, but it is diffi- cult to obtain accurate results on the average particle diameter, or on the internal structure. Other indirect approaches, which utilize quantum size effects, such as the position of the optical-absorption peaks, are prone to large systematic errors since 68 Optical absorption $275 3.0 “is (o F 1.5 Energy (eV) Figure 4.1: Comparison of the optical absorption thresholds of CdS nanocrystals of 3 run average diameter: experimental data (full curve), tight-binding calculation (broken Curve), and effective-mass calculation (chain curve). Adapted from [162]. they rely on effective-mass or tight-binding models. Comparison of the optical ab- sorption thresholds of CdS nanocrystals of 3 nm average diameter is shown in Fig. 4.1. Although experiment and theory agree reasonably well for large diameters, the simple theory diverges from the experimental values for the small diameter [37]. Lip- penset al. [163] have shown that tight-binding calculations can yield better agreement for smaller sizes. The measurement of the nanoparticle size can lead to significantly different results when performed by different methods, and there is no consensus as to which is the most reliable [164, 165]. It is also not clear that a single diameter is sufficient to fully 69 specify even a spherical particle since the presence of distinct crystalline core and disordered surface regions have. been postulated [71]. 4. 1.1 Previous work Powder diffraction is a well established method for structural and analytical stud- ies of crystalline materials, but the applicability to such small particles of standard powder diffraction based on crystallographic methods is questionable and likely to be semi-quantitative at best. There have been some reports [166, 165, 71] in the past few years extracting nanoparticle diameter from x-ray diffraction (XRD) using the Scherrer formula, which is a phenomenological approach that considers the finite size broadening of Bragg—peaks [75]. This approach will decrease in accuracy with decreasing particle size, and for particle sizes in the range of a few nanometers the notion of a Bragg peak becomes moot. At this point the Debye formula [92] becomes the more appropriate way to calculate the scattering [71]. The inconsistency between the nanoparticle diameter determined from the standard characterization methods and the diameter obtained by applying the Scherrer formula have been observed by several authors [71, 165, 167]. Applying the Scherrer formula to determined nanoparticle diameter is based on the assumption of a perfect crystal limited in size. Nanoparticles with finite size are often terminated by a variety of different hid-planes, and have a large fraction of their atoms on the nanoparticle surface which can have a high degree of disorder. These features are inconsistent with the basic assumptions of the Scherrer formula. For that reason the characteristic diameter of nanoparticles cannot be reliably estimated from the XRD peak—width-based analysis. The XRD patterns of nanoclusters are subjected to local structure deviations rel- ative to bulk material. Previous studies of CdSe nanoparticle structure have demon- strated the sensitivity of the XRD pattern to the presence of planar disorder and 70 thermal effects due to nano—size effects [37, 71]. The diffraction patterns of CdSe nanoparticles smaller than 2.0 nm have been observed to appear markedly different from those of the larger diameters (see Fig. 4.2), the large attenuation and broad— ening in the Bragg reflections in these small nanoparticles, making the distinction between wurtzite and zinc-blende hard using conventional XRD methods. Murray et al. [37] reported that the combination of X-ray studies and TEM imaging yields a de- scription of the average CdSe nanoparticle structure. Strict classification of the CdSe nanoparticles structure as purely wurtzite or zinc-blend is potentially misleading [37]. Bawendi et al. ['71] reported that CdSe nanoparticles are best fit by a mixture of crys- talline structures intermediate between zinc-blend and wurtzite. Here we apply the PDF method to CdSe nanoparticles and refine quantitative structural parameters to a series of CdSe nanoparticles of different sizes. Strain in nano systems has been observed before in different studies, as well as in this study. Using combined PDF and extended X-ray-absorbtion fine structure (EXAFS) methods, Gilbertet at. [65] observed a compressive strain compared to the bulk in ZnS nanocrystals. Using an electric field-induced resonance method, Chen et al. [168] detected the enhancement of Young’s modulus of ZnO nanowires along the axial direction when the diameters are decreased. Very recently, Quyang et al. [169] developed an analytical model for the size-induced strain and stiffness of a nanocrystal from the perspective of thermodynamics and a continuum medium approach. It was found theoretically that the elastic modulus increases with the inverse of crystal size and vibration frequency is higher than that of the bulk. [169]. Experimentally, the CdQ (Q=S, Se, Te) first-neighbor distances have been studied using both XRD and EXAFS methods [57]. The distances were found smaller than those in the bulk compounds by less than 1.0%. Herron et al. [170] studied CdS nanocrystals and showed a bond contraction of ~ 0.5% compared to the bulk. Carter et al. [171] studied a series of CdSe nanoparticles using the EXAF S method. In the first shell 71 A a g ( > S 5 (b) g (c) a (a) g (a) E m (a) (m l l 1 l l l 26 Figure 4.2: Powder X-ray diffraction spectra of (a) 12, (b) 18, (c) 20, (d) 37, (e) 42, (f) 83. and (g) 115 A diameter CdSe nanocrystallitcs compared with the bulk wurtzite peak positions (11). Adapted from [37]. around both the Se and Cd atoms, they found essentially no change in the first- neighbor distance. Chaure et al. [172] studied the strain in nanocrystalline CdSe thin films, using Raman scattering and observed a peak shift with decrease in particle size, which was attributed to the increase in stress with decreasing particle size [172]. 4.1.2 Stacking Faults in Close-packed Structures The most compact arrangement. of spherical atoms is obtained by stacking hexagonal lattice planes in such a way that each atom in a given layer is in contact with three spheres in the adjoining layer, so that it is situated above the center of the equilateral triangle formed by the centers of these three spheres. There are two ways in which the second layer can be placed on the first. The projection of a layer on the reference plane can take three different positions designated by A, B and C layers, as shown in Fig. 4.3. The only necessary condition for the stacking to be close-packed is that two consecutive layers be, different. The two simples arrangements are periodic, and can therefore give a crystal lattice, are the (Tl().5‘(.’-p(L(.‘k(‘(l hexagonal arrangement ABAB... (the other arrangements BCBC... and CA CA... are structurally identical), and the face-centered cubic arrangement ABCABC... for which to the hexagonal planes are the (111) planes. There are other regular arrangements which are more complex and which give rise to until cells extending over more than three layers — for example ABACAB... or ABCBCAB..., which are found in certain forms of carborundum SiC. The energy difference between these various arrangements are small. They would be zero if only neighboring layers showed an interaction energy. It is only the weak interactions between second neighbors which differentiate the hexagonal lattice from the cubic lattice, since the second layers are identical in one case (ABA) and different in the other (ABC). It is therefore possible for a compact lattice to contain faults, a layer A being substituted for a layer B or C. For example, in the hexagonal series ABABIABLCIBCBCBCBCB ...... Figure 4.3: Relative positions of the three layers A, B and C for a close—packed structure. ABAB... close-packed herragonal arrangement (left) and ABCABC... face- centered cubic arrangement (right). there is one fault because there is one group of three consecutive layers which is of the cubic type (three different layers) while all the others are of the hexagonal type (two identical layers on either side of a different one). In the same way, the cubic arrangement ABCABClABilAl CLAIBCABCABCABC ...... contains two hexagonal triplets, and therefore two faults. These types of imperfec— tions have been observed in many crystals and it is therefore important to understand their effects on the different pattern. Later in this chapter we will simulate the effect of the stacking fault density on real space refinement. 4.2 Experimental details 4.2.1 Sample preparation The samples were synthesized by A. Karkamkar at Kanatzidis Group. The CdSe nanOparticles were synthesized from cadmium acetate, selenium, trioctyl phosphinc and trioctyl phosphine oxide. Sixty four grams of trioctylphosphine oxide (TOPO) containing cadmiumacetate was heated to 3600C under flowing argon. Cold stock solution (38.4 ml) of (Se:trioctylphosphine : 2:100 by mass) was quickly injected into the rapidly stirred, hot TOPO solution. The temperature was lowered to 3000C by the injection. At various time intervals, 5-10 1111 aliquots of the reaction mixture were removed and precipitated in 10 ml of methanol. The color of the sample changed from bright yellow to orange to red to brown with time interval variation from 20 seconds to 1200 seconds. Three nanoparticle sizes, CdSeI (small), CdSeII (medium) and CdSeIII (large), were used for this study, as well as a bulk CdSe sample for reference. The samples were further purified by dissolving and centrifuging in methanol to remove excess TOPO. This process also resulted in a narrower particle size distribu- tion. The transmission electron micr(.)graph (TEM) images (Fig. 4.4) show uniformly sized nanoparticles with no signs of aggregation. The ultraviolet visible (UV-vis) absorption and photoluminescence (PL) spectra of the aliquots were recorded by re— dissolving the nanocrystals in toluene. The spectra are shown in Fig. 4.5. The TEM, PL and UV-vis were done at Kanatzidis Group, The band-gap values obtained for the measured samples can be correlated with the diameter of the nanoparticles based on the table provided in supplementary infor- mation of Peng et al. [36] using the data on exciton peaks measured with UV-visible light absorption, and photoluminescence peaks. The particle sizes were measured by TEM as well. The measured values of particle diameter using these various methods Figure 4.4: TEM image of CdSe nanocrystal prepared using the method described in the text. CdSe obtained by 1200 seconds (left) and 15 seconds (right) nucleation. The line-bar is 10 nm in size in both images. CdSeIII CdScII CdSeI Nucleation time (s) 1200 630 15 Diameter (um) TEM 3 5(2) (.2) 2.0(2) UV—vis 3. 5(4) (.3) S 1.90 PL 3.6(4 ) 29(3) 5 2.1 PDF 3.71( ) 3.1(1) 2.2(2) Table 4.1: CdSe nanoparticle diameter as determined using various methods. are summarized in Table 4.1. 4.2.2 The atomic PDF method The atomic PDF analysis of x—ray and neutron powder diffraction data is a power- ful method for studying the structure of nanostructured materials [6, 48, 173, 174, 175, 176]. Recently, it has been explicitly applied to study the structure of discrete nanoparticles [177, 65, 85, 178, 176]. The PDF method can yield precise structural and size information, provided that special care is applied to the measurement and 76 L A -. d”*‘ @L O ‘1.“ "... V 3" o { «Q. a .0 \~, 0 0 (O. x ~53“, m Q . —( O O A; ‘x 1‘ ‘ 5%} “V x. . ‘ .0 x; L: >~ ‘ o U)- — 0° \uvd-‘Mfi‘-':\<. . .. .‘ ~ .8 O 2. "a.“ <2 \ Fluorescence (a. u.) 420 450 480 510 540 570 600 630 Wavelength (nm) Figure 4.5: (a) Room temperature UV-vis absorption and (b) photoluminescence S pectra from the sample of CdSe nanocrystals. (O) CdSeI, (A) CdSeII, (C?) CdSeIII. t- () the method used for analyzing the. data. The atomic PDF, G (r), is defined as cei=wwwurwn, (in where p('r) is the atomic. pair—density, [)0 is the average atomic number density and 'r is the radial distance [179]. The PDF yields the probability of finding pairs of atoms Separated by a distance 7‘. It is obtained by a sine Fourier transformation of the reciprocal space total scattering structure function 3(Q), according to C (1) = 3/0 Q[S(Q) — 1] sin (21" (1Q, (4.2) 77 where S (Q) is obtained from a diffraction experiment. This approach is widely used for studying liquids, amorphous and crystalline materials, but has recently also been successfully applied to nanocrystalline materials [6]. 4.2.3 High-energy x-ray diffraction experiments X-ray powder diffraction experiments to obtain the PDF were performed at the GIDD beamline at the Advanced Photon Source at Argonne National Laboratory. Diffrac- tion data were collected using the recently developed rapid acquisition pair distribu- tion function (RAPDF) technique [89] that benefits from 2D data collection. Unlike TEM, XRD probes a large number of crystallites that are randomly oriented. The powder samples were. packed in a flat plate with thickness of 1.0 mm sealed be- tween kapton tapes. Data were collected at room temperature with an x-ray energy of 87.005 keV (A = 0.14.248 A) An image plate camera (Mar345) with a diameter of 345 mm was mounted orthogonally to the beam path with a sample to detector (listance of 208.857 mm, as calibrated by using silicon standard sample [89]. The i1‘nage plate was exposed for 10 seconds and this was repeated 5 times for a total ( l ata collection time of 50 seconds. The RAPDF approach avoids detector saturation ‘Kr'hilst allowing sufficient statistics to be obtained. This approach also avoids sam- I)]e degradation in the beam that was observed for the TOPO coated nanoparticles ('1 uring longer exposures, on the scale of hours, that were required using conventional point—detector approaches. To reduce the background scattering, lead shielding was placed before the sample with a small opening for the incident beam. Examples of the raw 2D data are shown in Fig. 4.6. These data were integrated and COnverted to intensity versus 29 using the software F it2D, [115] where 29 is the angle 1)et Vveen the incident and scattered x-ray beam. The integrated data were normalized by t he average monitor counts. The data were corrected and normalized [48] using the p 1'0 gram PDFgetX2 [90] to obtain the total scattering structure function, S (Q), and Figure 4.6: Two dimensional XRD raw data collected using image plate detector from (a) CdSe bulk and (b) nanoparticle CdSeIII samples. 79 the PDF, C(r), which are shown in Figs. 4.7 (a) and (b) respectively. The scattering signal from the surfactant (TOPO) was measured independently and subtracted as a background in the data reduction. In the Fourier transform step to get from 8(Q) to the PDF C(r), the data are truncated at a finite maximum value of the momentum transfer, Q = Qmax. Dif- ferent values of (2771013 may be chosen. Here a Qmar = 25.0 A71 was found to be optimal. Qmar is optimized such as to avoid large termination effects and to rea- sonably minimize the introduced noise level as signal to noise ratio decreases with Q value. Structural information was extracted from the PDFs using a full—profile. rea.l-s1.)ace local—structure refinement. method [101] analogous to Rietveld refinement [155]. We used an updated version [124] of the program PDFfit [94] to fit the experimental PDFs. Starting from a given structure model and given a set of parameters to be refined, PDF fit searches for the best structure that is consistent with the experimental PDF data. The residual function (Rm) is used to quantify the agreement. of the calculated PDF from model to experimental data: Z£1w(rillcobs(ri) _ Gcalcir‘in2 Ru) 1' N _ 21:1 w(rz-.)G?,bs(u) (4.3) Here the weight w(ri) is set to unity which is justified because in C(I) the statistical uncertainty on each point is approximately equal [157, 158]. The structural parameters of the model were unit cell parameters, anisotropic atomic displacement parameters (ADPs) and the fractional coordinate z of Se/ Cd atom. Non structural parameters that were refined were a correction for the fi- nite instrumental resolution, (0Q), low-r correlated motion peak sharpening factor ((5), [102, 103] and scale factor. When estimating the particle size, a new version of the fitting program with particle size effects included as a refinable parameter [180] 80 CdSe" . g 3. b _ ,5, i] CdSebulk ( ) _ '° 5; it . ,\ o #fiJL] J]. {mljnA-Wfl W [ [If Wi'IM\\j/\m\/\’]V/ Wf\fi/\MM . V . ‘ n I . _ CdSelll A ‘7 “r 5 . 07% a . CdSeIl l E‘.‘ I :‘2 I 92 '- o 4 8121620242832 rat) Figure 4.7: (a) The experimental reduced structure f1mction F (Q) of CdSe nanoparti- cle with different diameters and (b) the corresponding PDF, C(r), obtained by Fourier transformation of the data in (a) with Qmal- = 25.0 A‘l, from top to bottom: bulk, CdSeIII, CdSeII and CdSeI. 81 was used. The sample resolution broadening was determined from a refinement to the crystalline CdSe and the silicon standard sample and fixed and the particle diameter refined, as described below. Good agreement between these results was obtained. 4.3 Results and discussion The reduced structure functions for the bulk and nanocrystalline samples are shown plotted over a wide range of Q in Fig 4.7(a). All of the patterns show significant intensity up to the highest values of Q, highlighting the value of measured data over such a wide Q-range. All of the diffraction patterns have peaks in similar positions reflecting the similarity of the basic structures, but as the nanoparticles get smaller the diffraction features become broadened out due to finite size effects [75]. The PDFs are shown in Fig. 4.7(b). What is apparent is that, in real-space, the PDF features at low-r are comparably sharp in all the samples. The finite size effects do not broaden features in real—space. The finite particle size is evident in a fall—off in the intensity of structural features with increasing-r. Later we will use this to extract the average particle size in the material. The structure apparent in the C(r) function comes from the atomic order within the nanoparticle. The value of 7‘ where these ripples disappear indicates the particle core region diameter; or at least the diameter of any coherent structural core of the nanoparticle. By direct observation (Fig. 4.12) we can put a lower limit on the particle diameters to be 3.6, 2.8 and 1.6 nm for CdSeIII, II and 1, respectively, where the ripples can be seen to die out by visual inspection. These numbers will be quantified more accurately later. 4.3.1 Nanoparticle structure Features in the PDF at low-r reflect the internal structure of the nanoparticles. The nanoparticle PDFS have almost the same features as in the bulk in the region below 82 Figure 4.8: Fragments from the (a) wurtzite structure. space group (P63mc) and (b) zinc-blende structure, space group (Fl/13171.). 8.0 A, reflecting the fact that they share a similar atomic structure on average. In the finite nano-size regime, local structural deviations from the average bulk structure are expected. A large number of semiconductor alloys, especially some sulfides and selenides, do not crystallize in the cubic zinc-blende. structure but in the hexagonal wurtzite structure [181]. Both wurtzite and zinc—blende structures are based on the stacking of identical two—dimensional planar units translated with respect to each other, in which each atom is tetrahedrally coordinated with four nearest neighbors. The layer stacking is described as ABABAB... along the [001] axis for wurtzite and as AB CABC... along the [111] axis for zinc—blende. As can be seen in the Fig. 4.8, each cadmium and selenium is tetrahedrally coordinated in both structures. However, the next nearest and more distant coordination sequences are different in the two structures. The largest changes in structure are expected in the smallest nanoparticles. In these small nanoparticles, the proportion of atoms on the surface is large making the notion of a well—ordered crystalline core moot. The fraction of atoms involved in the surface atoms was estimated as 0.6, 0.45 and 0.35 for 2 nm, 3 nm and 4 nm nanoparticle diameters, respectively. This was estimated by taking different spherical 83 Table 4.2: The refined residual (Rm) values obtained from PDF analysis assuming the wurtzite and zinc-blend structure models with space group P63mc and F13m, respectively. In both models isotropic atomic displacement factors (Uispl are used. CdSe-bulk CdSeIII CdSeII CdSeI Wurtzite (Rw) 0.16 0.31 0.28 0.31 Zinc-blende (Rm) 052 0.32 0.30 0.35 cuts from bulk structure, then counting the atom with coordination number 4 as core atom and the one with less than 4 as surface atom. For the smallest particles the small number of atoms in the core makes it. difficult to define a core crystal structure, making the distinction between wurtzite and zinc-blende difficult using the conventional XRD methods as nanoparticle size decreases [37]. The principle difference between these structures is the topology of the 01ch connections, which may also be becoming defective in the small nanoparticles. Two structure models wurtzite (space group P63mc) and zinc—blende (space group F713m), were fit to the PDF data. The. results of the full-profile fitting to the PDF data. are shown Fig. 4.9. In this figure we compare fits to the (a) wurtzite and (b) zinc-blende structure models using isotropic atomic displacement factors (in) in both models. The wurtzite structure gives superior fits for the bulk structure. However, for all the nanoparticle sizes, the fits of wurtzite and zinc-blende are com— parable as evident from the difference curves in Fig. 4.9 and the Ray—values reported in Table 4.2. This indicates that classification of the CdSe nanoparticles structure as purely wurtzite or zinc-blend is misleading [37] and it is better described as being intermediate between the two structures, as has been reported earlier [71]. Introducing anisotropic ADPs (U11 2 U22 7é U33) into the wurtzite model, re- sulted in better fits to the data. The refined parameters are reproduced in Table 4.3 and the fits are shown in Fig. 4.10(a). The values for the nanoparticles are rather close to the values in the bulk wurtzite structure. The model with anisotropic ADPs resulted in lower Rw. There is a general increase in the ADPs with decreasing particle 84 CdSe bulk CdSelll G (3'2) CdSell o 4 8 12 16 20 24 23 Fat) CdSe bulk CdSolll CdSell o 4 a 12 16 20 24 28 r01) Figure 4.0: (Color online) The experimental PDF, C(I'), with Qmm; = 19.0 A_1(blue solid (lots) and the calculated PDF from refined structural model (red solid line). with the difference curve offset below (black solid line). PDF data are fitted using (a) wurtzite structure model, space group P631120 and (b) zinc-blends model with space group F13m. In both models isotropic atomic displzwcment factors (LIZ-50) are used. 85 CdSe bulk CdSelll f 11¢ v o CdSell o 4 8 12 16 20 24 28 r01) CdSe bulk (b) CdSelll (fi‘ (4 v 0 o 4 8 12 16 20 24 28 MR) Figure 4.10: (Color online) The experimental PDF, C(r), with Qmaa: = 19.0 A”1(blue solid dots) and the calculated PDF from refined structural model (red solid line), with the difference curve offset below (black solid line). PDF data are fitted using wurtzite structure model (a) with no stacking fault and (b) with 33% stacking fault density for bulk and 50% for all nanoparticle sizes. In both cases anisotropic atomic displacement factors (Liam-SO) are used 86 size. This reflects inhomogeneous strain accommodation in the nanoparticles as we discuss below. However, the values of the ADPs along the z-direction for Se atoms (U33) are four times larger in the nanoparticles compared with the bulk where they are already unphysically large. The fact that this parameter is large on the Se site and small on the Cd site is not significant, since we can change the origin of the unit cell to place a Cd ion at the (1 / 3,2/ 3.2) position and the enlarged U33 shifts to the Cd site in this case. The unphysically large U33 value on the Sc site is likely to be due. to the presence of faults in the basal plane stacking. For example, similar unphysical enlargements of perpendicular thermal factors in PDF measurements are explained by the presence of turbostratic disorder in layered carbons[131], which is a similar effect to faults in the ABABAB wurtzite stacking. Also, the presence of stacking faults in the nanoparticles has been noted previously [71]. It is noteworthy that this parameter is enlarged in EXAF S analyses of bulk wurtzite structures, probably for the same reason [57, 182, 53]. We suspect that the enlargement in this parameter (U33) is related to the stacking fault density present in bulk and that is increasing in the nanoparticles. To test this idea we simulated PDF data using the wurtzite structure containing different stacking fault densities. The stacking faults were simulated for different densities (0.167, 0.25, 0.333, and 0.5) by creating wurtzite superlattices with different. stacking sequences along the C-axis. The program DISCUS [183] was used to create the stacking fault models and PDFgui [124] was used to generate the corresponding PDFS. The PDFs were simulated with all the ADPS fixed at Uiz’ = 0.0133 A2, the value observed in the experimental bulk data collected at room temperature (see Table 4.3). To see if this results in enlarged U33 values we refined the simulated data con- taining stacking faults using the wurtzite model without any stacking faults. Indeed, the refined Se site U33 increased monotonically with increasing stacking fault density. 87 ~_ . . e: I 7’) o ‘0- . "r ’ ’ * O fih /. ] ‘— . vat . ¢ . .. ’ J * Mo_ . D '3. / Q) [\_ O _ m N :L . -O.1 O 0.1 0.2 0.3 0.4 0.5 0.6 Stacking Fault Density Figure 4.11: The the enlargement in the the ADPS along the z-direction for Se site U33, as a function of the stacking fault density. The results are plotted in Fig. 4.11. Fig. 4.11 can be considered as calibration curve of stacking fault density in the wurtzite structure, based on the enlargement in the ADPs along the z-direction U33. From this we can estimate a stacking fault density of ~ 35% for our bulk CdSe sample, and ~ 50% for each of the nanoparticles. It is then possible to carry out a refinement using a structural model that contains an apprOpriate stacking fault density. The PDF data of bulk CdSe was therefore fit with a wurtzite model with a 33% density, and the nanoparticle PDF fit with a model with 50% of stacking faults. The refinements give excellent fits, as is evident in Fig. 4.10(b). The results are presented in Table 4.3. The enlarged U33 parameter on the Se site is no longer present and it is now possible to refine physically reasonable values for that parameter. As well as resulting in physically reasonable ADPs, the 88 quality of the fits to the data are excellent, though the Rw value is slightly larger in the nanoparticles. 89 90 as as ...AAV Mzs :AV AVNAV MVAVAV as ...m As V285 AA. Vs; AV AanAVmAVAV A V3: AV AAVVAAVMVNAVAV AAVVAAVBAV A3223 AAVVNSAVAV Aafi so AAV. VAAVAAVAV AAV VAV:AVAV AaVANAAVAV A VMVMWAVAVAV AstMuAVAV EEAVAV AAVVNAVAAVAV Assess A2: a: n :2 am A VHMWNAVAV AAV VVAVOAV AV AAVVAEVAV AAV VENAV AV AAVVAEAVAV AAVVNMVSAV AMVVNSAVAV AM..VMV:AV.AV ANparticle diameter decreases, due to poor contrast near the surface of the nanopar- ticle. In the literature, CdSe nanoparticles with a diameter of 2.0 nm have been consid- ered to be an especially stable size with an associated band edge absorption centered at 414 nm [185], that size was observed earlier [37, 180] with an estimated diameter of $2.0 nm. There are some reported difficulties in determining the diameter of such small CdSe nanoparticles. Attempts to characterize the structure changes by TEM and X-ray diffraction techniques [184] were unsuccessful due to the small diameter of the particles relative to the capping material. If we assume the nanoparticle to have spherical shape (a reasonable approximation based on the TEM in Fig. 4.4) cut from the bulk, then the measured PDF will look like the PDF of the bulk material that has been attenuated by an envelope function given by the PDF of a homogeneous sphere, as follows [127] C (r, (1)8 = 0(7) f(-,r(1) , (4.4) where C(r) is given in Eq. 5.1, and f (7', d) is a sphere envelope function given by f (7", d) = [1 — 37’; + $— (93] em — r), (4.5) where d is the diameter of the homogeneous sphere, and (9(1) is the Heaviside step 92 function, which is equal to 0 for negative .‘r. and 1 for positive. More information about the PDF of a single spherical particle and the derivation of Eq. 4.5 can be found in Section E. The approach is as follows. First we refine the bulk CdSe data. using PDF fit. This gives us a measure of the PDF intensity fall—off due to the finite resolution of the measurement [48]. Then the measured value of the finite resolution was kept as an unrefined parameter after that, while all the other structural and non structural pa— rameters were refined. To measure the PDF intensity fall-off due to the finite particle size, the refined PDF is attenuated, during the refinement, by the envelope function (Eq. 4.5) which has one refined parameter, the particle diameter. The fit results are shown in Fig. 4.12 and the resulting values of particle diameter from the PDF refine- ment are recorded in Table 4.1. The insets show the calculated and measured PDFs on an expanded scale. The accuracy of determining the nanoparticle size can be. evaluated directly from this figure. Features in the measured PDFS that correspond to the wurtzite structure are clearly seen disappearing smoothly attenuated by the spherical PDF envelope function. The procedure is least successful in the smallest nanoparticles, where the spherical particle approximation on the model results in fea- tures that extend beyond those in the data. In this case, the spherical approximation may not be working so well. The particle diameters determined from the PDF are consistent with those ob- tained from TEM, UV-vis and photolun'iinescence measurements. In particular, an accurate determination of the average diameter of the smallest particles is possible in the region where UV-vis and photoluminescence measurements lose their sensitiv— ity [165]. In this analysis we have not considered particle size distributions, which are small in these materials. The good agreement between the data and the fits jus- tify this, though some of the differences at high-7‘ may result from this and could contribute an error to the particle size. Several additional fits to the data were per- 93 G (3'2) 20 ‘22 24 26 28 30 32 34 38 38- 0 (r2) 6 (1'2) 0510152025303540 re) Figure 4.12: (Color online) The experimental PDF, C(r). shown as solid dots. Sphere envelope function (Eq. 11.5) is used to transform the calculated PDF of bulk CdSe, using wurtzite structure containing 50% stacking fault density. to give a best fit replication of the PDF of CdSe nanoparticles (red solid line). The inset shows on an expanded scale for the high-1‘ region of experimental C(r) on the top of simulated PDF data for different diameters of CdSe nanoparticles (solid line). (a) CdSeIII, (h) CdSeII, (c) CdSeI. Dashed lines are guides for the eye. 94 formed to test the sphericity of the nanol')artic1es. Attempts were made to fit the PDF with oblate and prolate spheroid nanoparticle form factors. These fits resulted in ellipticities very close to one, and large uncertainties in the refined ellipticity and particle diameters, which suggests that the fits are over-paraineterized. Another se- ries of fits attempted to profile the PDF with a lognormal distribution of spherical nanoparticles. Allowing the mean nanoparticle diameter and lognormal width to vary resulted in nonconvergcnt fits, which implies that the particle sizes are not lognormal distributed. Therefore, there appears to be little evidence for significant ellipticity, nor a significant particle size distribution, as fits assuming undistributcd spherical particles give the best results. We cannot rule out the possibility that the particles come in a variety of sizes that are not well described by a lognormal distribution. The simple fitting of a wurtzite structure with ~ 50% SF D to the data will result in an estimate of the coherent structural core of the nanoparticle that has a structure can be described by a mixture of crystalline structures intermediate between zinc- blend and wurtzite. Comparing the nanoparticle core diameter extracted from PDF analysis with the diameter determined from the standard characterization methods yields information about the existence of a surface amorphous region. The agreement between the core diameter extracted from PDF and that determined from the stan- dard methods (Table 4.1), indicates that within our measurement uncertainties, there is no significant heavily disordered surface region in these nanoparticles, even at the smallest diameter of 2 nm (Fig. 4.12). In contrast with ZnS nanOparticles [65] where the heavily disordered surface region is about 40% of the nanoparticle diameter for a diameter of 3.4 nm, the surface region thickness being around 1.4 nm [65]. 4.3.3 Internal strain The local bonding of the tetrahedral Cd—Se building unit was investigated vs nanopar- ticle diameter. The nearest neighbor peaks at r 2 2.6353(3) A come from covalently g ' I ' I ' I ' r ' I . ..c: g— (b) _ 3 :5 N 5 . fl 0: ————— i — — — 5 = (D Q ._ En CD E ST 1 I l I I J l 1 1 1_‘ “I? 2.4 2.7 3 3.3 3.6 0:3; NP diameter (nm) 0 53 , , I I . I . I . I , A co { o— (C) I _ S | v - . _. E. V? its o— _ h | ...; VJ — '.'.'.' .. "d ,0 0,.- . g 0|.T ! I i 1 l I l4 1 1— 2.4 2.8 m 2.4 2.7 3 3.3 3.6 I‘ (A) NP dlameter (nm) Figure 4.13: (a) The first PDF peak. (a) bulk, (o) CdSeIII, (Cl) CdSeII and (A) CdSeI fitted with one Gaussian ( ). Dashed line represents the position of first PDF peak in the bulk data. (b)(A) The first PDF peak width vs nanoparticle size, obtained from one Gaussian fit. Dashed line represents the width of first PDF peak in the bulk data. (c) Strain in Cd—Se bond (Ar/r)(%) vs nanoparticle size. (I) Bond values obtained from the local structure fitting and (0) obtained from one Gaussian fit to the first PDF peak. Dotted curves are guides for the eye. bonded Cd-Se pairs. The positions and the width of these peaks have been deter- mined by fitting a Gaussian (Fig. 4.13(a)) and the results presented in Table 4.4. The results indicate that there is a significant compressive strain on this near-neighbor bond length, and it is possible to measure it with the PDF with high accuracy. The bond length of Cd—Se pairs shorten as nanoparticle diameter decreases, suggesting the presence of an internal stress in the nanoparticles. The Cd-Se bond lengths extracted 96 CdSe-bulk CdSeIII CdSeII CdSeI PDF FPP (A) 2.6353(3) 2.6281(3) 2.6262(3) 2.6233(3) PDF FPW(A) 0.1985(09) 0.1990(19) 0.2021(25) 0.2032(25) Table 4.4: The first PDF peak position (FPP) and width (FPW) for different CdSe nanoparticle sizes and the bulk. from the PDF structural refinement are also in good agreement with those obtained from the first peak Gaussian fit, as shown in Fig. 4.13(c). Thus we have a model independent and a model dependent estimate of the strain that are in quantitative agreement. The widths of the first PDF peaks have also been extracted vs nanopar- ticle. diameter from the Gaussian fits (Table 4.4). They remain comparably sharp as the nanoparticles get smaller, as shown in Fig. 4.13(b). Apparently there is no size- dependent inhomogeneous strain measurable on the first peak. However, peaks at higher-r do indicate significant broadening (Fig. 4.7(b)) suggesting that there is some relaxation taking place through bond-bending. This is reflected in enlarged thermal factors that are refined in the nanoparticle samples. This is similar to what is ob— served in semiconductor alloys where most of the structural relaxation takes place in relatively lower energy l;)ond-bending distortions [187, 188]. 4.4 conclusion The PDF is used to address the size and structural characterization of a series of CdSe nan0particles prepared by the method mentioned in the text. The core structure of the measured CdSe nanoparticles was found to possess a well-defined atomic arrangement. that can be described in terms of locally disordered wurtzite structure that contains ~ 50% stacking fault density, and quantitative structural parameters are presented. The diameter of the CdSe nanoparticles was extracted from the PDF data and is in good agreement with the diameter obtained from standard characterization methods, indicating that within our measurement uncertainties, there is no significant. heavily 97 disordered surface region in these nanoparticles, even at the smallest diameter of 2 nm. In contrast with ZnS nanoparticles [65] where the heavily disordered surface region is about 40% of the nanoparticle diameter for a diameter of 3.4 nm, the surface region thickness being around 1.4 nm [65]. Compared with the bulk PDF, the nanoparticle PDF peaks are broader in the high—r region due to strain and structural defects in the nanoparticles. The nearest neighbor peaks at 7' = 2.6353(3) A which come from covalently bonded Cd-Se pairs, shorten as 1‘1anopartiele diameter decreases resulting in a size—dependent strain on the Cd-Se bond that reaches 0.5% at the smallest particle size. 98 Chapter 5 Structure of ultra-small CdSe nanoparticles using PDF analysis This work presented in this chapter has been. done by collaboration. with James R. McBride and Sandra J. Rosenthal. Following description is taken from mow/script tn the final stage of p1"ep(z.7'a.t'i.on. 5. 1 Introduction The development presented by Chapter 4, have been applied on technologically impor- tant system, ultra-small CdSe nanoparticles. These ultra-small nanocrystals exhibit broadband emission (420—710 nm) that covers most of the visible spectrum while not suffering from self absorption. This behavior is a direct. result of the extremely narrow size distribution and unusually large Stokes shift (40-50 mm). The intrinsic proper- ties of these ultra-small nanocrystals make them an ideal material for applications in solid state lighting and also the perfect platform to study the molecule-to-nanocrystal transition [189]. The interest in ultra-small CdSe has been increased in the past few years as the combination of their intrinsic preperties can make them a good candidate material for applications in solid state lighting. Exploring the fundamental properties of this 99 interesting material could yield to the development of more economical and environ- mentally friendly materials with similar properties, ultimately leading to the next generation of solid state lighting technologies [189, 190, 191, 192, 193, 194]. In this study we used the atomic pair distribution function (PDF) [48] analysis of - x—ray diffraction data to study the size-dependent structure of small (1.5 - 3.6 mm) CdSe nanoparticles. This approach has been shown to provide valuable quantita- tive information about structure in nanoparticles even where crystallography loses much of its power [84, 65. 85. 86, 87, 88]. PDF is a local structural technique that yields quantitative structural information on the nanoscale from x-ray and neutron powder diffraction data. [6] Recent developments in both data. collection[89, 110] and modeling[9rl, 111] make this a potentially powerful tool in the study of nanoparticles. 5.2 Experimental details 5.2. 1 Sample preparation The following reagents and methods were used for the synthesis of CdSe nanocrys- tals. Tri-n-octylphosphine oxide (TOPO, 90% tech. grade), hexadecylamine (HDA, 90% tech. grade) were purchased from Aldrich and used as received. CdO (99.999% Puratrem), tri-n-butylphosphine (TBP, 97%) and selenium powder (200 mesh) were purchased from Strem and used as received. Dodecylphosphonic acid (DPA) was synthesized via the Abruzov reaction from triethylphosphite and l-dodecylbromide followed by acidification with concentrated HCl and recrystallization from cold ethyl acetate. All other solvents were HPLC grade and purchased from Fisher Scientific unless otherwise noted. 0.256 g of CdO, 0.96 g of DPA, 6 g of HDA and 4 g of TOPO was placed in a 100 ml three—neck flask and allowed to heat to 320°C un- der argon. When the solution cleared, 5 ml of 0.2 M SezTBP solution was injected and the ten’iperature was abruptly dropped to 270°C. The reaction was then allowed 100 Table 5.1: CdSe nanoparticle diameter as determined using various methods. CdSeSSG CdSe552 CdSe470 CdSe4l7 Diameter (nm) Z-STEM 4.0(d) 3.0(d) 2. PDF 36(1) 31(1) 22(1) 1.5(2) to proceed until the desired size of nanocrystal was achieved. To obtain smaller sizes, 2 ml of octadecene was added to the SezTBP injection and a careful post- growth injection of 10—20 ml of butanol was used to stop the reaction quickly. Larger sizes were obtained using an equimolar growth solution consisting of cadmium hex- anoate in octanol and SezTBP, which was added to the reaction solution via syringe pump. Nanocrystal samples were cleaned by flocculation followed by centrifugation in methanol, octanol, and methanol in succession. Four nanoparticle sizes, CdSe417 (ultra-sn'lall), C(,lSe~’l70(Small), CdSe552 (n‘iedium) and CdSe586 (large), were used for this study. as well as a bulk CdSe sample as a reference. The numbers in nanopar- ticle sample name here are referring to the associated band edge absorption center. Z-STEM images of the CdSe nanocrystals from the ultra-small CdSe nanoparticles sample (with particle size diameter ~ 1.5 nm) are shown in Fig. 5.1. Figure 5.2 shows the UV-Vis spectra of the CdSe nanoparticles used in this study. 5.2.2 The atomic PDF method The atomic PDF analysis of x—ray and neutron powder diffraction data is a power- ful method for studying the structure of nanostructured materials [6, 48, 173, 174, 175, 176]. Recently, it has been explicitly applied to study the structure of discrete nanoparticles [84, 65, 85, 178, 176]. The PDF method can yield precise structural and size information, provided that special care is applied to the measurement and to the method used for analyzing the data. The atomic PDF is formally defined as G (7') = 4777‘ [pm - pol (5-1) 101 Figure 5.1: Z-STEM image of CdSe nanocrystal prepared using the method described in the text. The associated band edge absorption centered at 4117 run. The line-bar is 4 nm in size. where p(1') is the atomic pair-density, {)0 is the average atomic number density and r is the radial distance [179]. The PDF yields the probability of finding pairs of atoms separated by a distance 7‘. It is obtained by a sine Fourier transformation of the reciprocal space total scattering structure function S (Q), according to 2 00 G (r) = g / QlS(Q) —1]sinQrdQ. (5.2) 0 where S (Q) is obtained from a diffraction experiment. This approach is widely used for studying liquids, amorphous and crystalline ma- 102 Absorption Peak 0.8 — —- 586 nm —-— 552 nm — 470 nm 0.6— - - - 417nm — 402 nm Optical Density (a.u.) $2 I 0.2 ~ -..—sag. —-....._..4.... 0'0 l l l i l I ‘—‘1 300 350 400 450 500 550 600 650 Wavelength ( nm) Figure 5.2: (Color online) UV-Vis spectra of the CdSe nanoparticles used in this study. The ultra-small nanoparticles that had absorption maximum at 402 red—shifted to 417 nm after washing. terials, but has recently also been successfully applied to nanocrystalline materials [6]. 5.2.3 High-energy x-ray diffraction experiments X—ray powder diffraction experiments to obtain the PDF were performed at the GIDD beamline at the Advanced Photon Source at Argonne National Laboratory. Diffrac- tion data were collected using the recently developed rapid acquisition pair distribu- tion function (RAPDF) technique [89] that benefits from 2D data collection. Unlike TEM, XRD probes a large number of crystallites that are randomly oriented. The powder samples were packed in a flat plate with thickness of 1.0 mm sealed between 103 kapton tapes. Data were collected at room temperature with an x-ray energy of 98.111 keV (A = 0.12637 A). An image plate camera (iVlar345) with a diameter of 3115 mm was mounted orthogonal to the beam path with a sample to detector dis- tance of 229.967 mm, as calibrated by using a silicon standard sample [89]. The image plate was exposed for 90 and 860 seconds for bulk and nanoparticle samples, respectively and this was repeated 4 to 5 times for each sample. This approach avoids detector saturation whilst allowing sufficient statistics to be obtained. This approach also avoids sample degradation in the beam that. was observed for the TOPO coated nanoparticles during longer exposures, on the scale of hours, that were required using conventional point-detector approaches. To reduce the background scattering, lead shielding was placed before the sample with a small opening for the incident beam. Examples of the raw 2D data are shown in Fig. 5.3. These data were integrated and converted to intensity versus 20 using the software Fit2D, [115] where ‘26 is the angle between the incident and scattered x-ray beam. The integrated data were normalized by the average monitor counts. The data were corrected and normalized [48] using the program PDFgetX2 [90] to obtain the total scattering structure function, 8' (Q), and the PDFs, G (r), which are shown in Figs. 5.4 (a) and (b), respectively. The scattering signal from the surfactant (TOPO) was measured independently and subtracted as a background in the data reduction. This subtraction is imperfect in the present case since the samples were made with a surfactant that is a mixture of TOPO and HDA. In the Fourier transform step to get from S (Q) to the PDF C(r), the data are truncated at a finite maximum value of the momentum transfer, Q = Qmam- Different values of Qmax may be chosen. Here a Qmax = 25.0 .551 was found to be optimal. Qmam is optimized so as to avoid large termination effects whilst minimizing the statistical noise level, as the signal to noise ratio decreases with Q value. Structural information was extracted from the PDFs using a full-profile real-space local-structure refinement. method [101] analogous to Rietveld refinement [155]. We 101-1 Figure 5.3: Two dimensional XRD raw data collected using image plate detector from (a) CdSe bulk and (b) nanoparticle CdSe552 samples. 105 G (3'2) 7' CdSesae : A W L r- - v m CdSe552 h. N - I "\ Nut/WWW W 93_ CdSe47O . I 0. <7 CdSe417 - ‘. M N - I 1 l l I I l_ A l I ' CdSebulk 15 18 21 24 O 3 6 9 12 Q (3") CdSebulk 0’) CdSeSBG Cd56552 CdSe47O CdSe417 o 4 8121620242832 r01) Figure 5.41: (a) The experimental reduced structure function F(Q) of CdSe nanopar- ticles with different diameters and (b) the corresponding PDFs.‘ G37"). F ouricr transforn bottom: bulk, CdSe-586, CdSe552. CdSe-3170 and CdSe417. ration of the data in (a) with Qmax = 25.0 A" . 106 used an updated version [124] of the program PDFfit [94] to fit the experimental PDFs. Starting from a given structure model and given a set of parameters to be refined, PDF fit searches for the best structure that is consistent with the experimental PDF data. The residual function (Rw) is used to quantify the agreement of the calculated PDF from model to experimental data: Z§11w — 0mm]? 29:1 w('r'i)03b,.('l'i) Ru, = (5.3) Here the weight w(ll) is set. to unity which is justified because in C(r) the statistical uncertainty on each point is approximately equal [157, 1.58]. The structural parameters of the model were unit cell parameters, anisotropic atomic displacement parameters (ADPs) and the Se/ Cd z- fractional coordinate. Non structural parameters that were refined were a correction for the finite instrumental resolution, (0Q), low-r correlated motion peak sharpening factor (6), [102, 103] and scale factor. When estimating the particle size, a new version of the fitting program with particle size effects included as a refinable parameter [180] was used. The sample resolution broadening was determined from a refinement to the crystalline CdSe and the silicon standard sample and fixed and the particle diameter refined, as described below. Good agreement between these results was obtained. 5.3 Results and discussion The reduced structure functions for the bulk and nanocrystalline samples are shown plotted over a wide range of Q in Fig 5.4(a). All of the patterns Show significant intensity up to the highest values of Q, highlighting the value of measured data over such a wide Q-range. All of the diffraction patterns have peaks in similar positions reflecting the similarity of the basic structures, but as the nanoparticles get smaller the diffraction features become broadened out. due to finite. size effects [75]. 107 The PDFs are shown in Fig. 5.4(b). In real-space the PDF features at low—r are comparably sharp in all the samples. The finite size effects do not broaden features in real-space. The finite particle size is evident in a fall-off in the intensity of structural features with increasing-r. Later we will use this to extract the average particle size in the material. The structure apparent in the C(r) function comes from the atomic order within the nanoparticle. The value of r where these ripples disappear indicates the particle core region diameter; or at least the diameter of any coherent structural core of the nanoparticle. By direct observation (Fig. 5.8) we can put a lower limit on the particle diameters to be 3.5, 3.0, 2.8 and 1.6 nm for CdSe586, 552, 470 and 417, respectively, where the ripples die out by visual inspection. These numbers will be quantified more accurately later. 5.3.1 Nanoparticle structure Features in the PDF at low—1‘ reflect the internal structure of the nanoparticles. The nanoparticle PDFs have almost the same features as in the bulk in the region below 8.0 A, reflecting the fact that they share a similar atomic structure on the average. In the finite nano-size regime, local structural deviations from the average bulk struc- ture are expected and become more significant as the nano—size gets smaller. A large number of semiconductor alloys, especially some sulfides and selenides, do not crystal- lize in the cubic zinc-blende structure but in the hexagonal wurtzite structure [181]. Both wurtzite and zinc-blende structures are based on the stacking of identical two- dimensional planar units translated with respect to each other, in which each atom is tetrahedrally coordinated with four nearest neighbors. The layer stacking is described as ABABAB... along the [001] axis for wurtzite and as ABCABC... along the [111] axis for zinc-blende. As can be seen in the Fig. 5.5, each cadmium and selenium is tetrahedrally coordinated in both structures. However, the next nearest and more distant coordination sequences are different in the two structures. F ignrc 5.5: Fragments from the (a) wurtzite structure, space group (Pfigmc) with (ABAB ...) layer sequence and (b) zinc—blende structure, space group (F 43m) with {ABCABC ..) sequence. The largest changes in structure are expected in the smallest nanoparticles. In these small nanoparticles, the proportion of atoms on the surface is large making the notion of a well—ordered crystalline core moot. The fraction of atoms involved in the surface atoms was estimated as 0.6, 0.45 and 0.35 for 2 nm, 3 nm and 4 nm nanoparticle diameters, respectively. This was estimated by taking different spherical cuts from bulk structure, then counting the atom with coordination number 4 as core atom and the one with less than 4 as a surface atom. For the smallest particles the small number of atoms in the core makes it difficult to define a core crystal structure, making the distinction between wurtzite and zinc—blende difficult using the conventional XRD methods as nanoparticle size decreases [37]. Two structure models wurtzite (space group P63mc) and zinc—blende (space group F 33771), were fit to the PDF data. These two structures differ in the stacking arrange- ment of the CdSe4 tetrahedra as shown in Fig 5.5. In an earlier work of Masadeh et al. [84], the PDF data were fitted using isotropic atomic displacement factors (Uiso) in both structure models, wurtzite (space group P63mc) and zinc—blende (space group F13m). The wurtzite structure gives supe- rior fits for the bulk structure. However, for all the nanoparticle sizes. the fits of 109 wurtzite and zinc-blende are comparable as evident from the difference curves in (Fig. 4.9 Ch. 4) and the Rw-values reported in (Table. 4.2 Ch. 4). This indicates that classification of the CdSe nanoparticles structure as purely wurtzite or zinc-blend is misleading [37] and it is better described as being intermediate between the two structures, as has been reported earlier [71]. Introducing anisotropic ADPs (U11 = U22 7é U33) into the wurtzite model, re- sulted in better fits to the data. The refined parameters are reproduced in Table 5.2 and the fits are shown in Fig. 5.6(a). The values for the nanoparticles are rather close to the values in the bulk wurtzite structure. The model with anisotropic ADPS re— sulted in lower Ru», comparing with the (Uiso) structure mode [84]. There is a general increase in the ADPs with decreasing particle size as have been observed earlier [84]. This reflects inhomogeneous strain accommodation in the nanoparticles as we discuss below. However, the values of the ADPs along the z-direction for Se atoms (U33) are four times larger in the nanoparticles compared with the bulk where they are already unphysically large. The fact that this parameter is large on the Se site and small on the Cd site is not significant, since we can change the origin of the unit cell to place a Cd ion at the (1 /3,2/32) position and the enlarged U33 shifts to the Cd site in this case. The unphysically large U33 value on the Se site is likely to be due to the presence of faults in the basal plane stacking. For example, similar unphysical enlargements of perpendicular thermal factors in PDF measurements are explained by the presence of turbostratic disorder in layered carbons [131], which is a similar effect to faults in the ABABAB wurtzite stacking. Also, the presence of stacking faults in the nanoparticles has been noted previously [71]. It is noteworthy that this parameter is enlarged in EXAF S analyses of bulk wurtzite structures, probably for the same reason [57, 182, 53]. We suspect that the enlargement in this parameter (U33) is related to the stacking fault density present in bulk and that is increasing in the nanoparticles. 110 6 (I2) o 2 4 6 8 no 12 14 16 16 r (A) f I v I - I ' ff I ' I fl I r I ' I ' J ‘° cas. bulk (b) 0 ‘° (nausea - i i 736730 —g4_-18 -12 i i [9 if 0 (r2) O 2 4 6 8 1O 12 14 16 18 r (A) Figure 5.6: (Color online) The experimental PDF, C(r), with Qmag; = 19.0 A_1(blue solid dots) and the calculated PDF from refined structural model (red solid line), with the difference curve offset below (black solid line). PDF data are fitted using wurtzite structure model (a) with no stacking fault and (b) with 33% stacking fault density for bulk and 50% for all nanoparticle sizes. In both cases anisotropic atomic displacement factors (Uam-so) are used 111 AAA..AA AANAA 8AA AAAAA mAAA «AAA AAAAAAAAsAA Am .: AA Amd 3:4“ ACQM oo oo AAAAAAV .AvoAAAeAAA 94 AA ASE. AA AAA AAAWAAM AA AAAAAAAAAAAA AAAAABAAAA AAAAAEAAAA AWAEAAAAA .2: -N am As AAAAAAAAA AAAAAAAAmmAA AAAAAAAAAAAA AAAAWAAAAAA AAAAAAAsAAAA AAAASAAAAA AAA: A..A..AA A AA.A.AAA.AA AAAAAAAAAAAAA AAAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAA AWAAAAAAAAAA ANA; a: u AAAA mm A 5.2 AA AAAAAAAAAAAAA AoAmmmAAAA AAAAENAA AA AAAANAAAA AA AA..AAAAAAA AA AAA; a: AAA AAAAAAWAA AA AAAANWWAAAA AAAAAAAAAA AAAAEAAAA AAAAAAAAAAAA ANAAAAAAAA AA AAA; a: n :3 A5 AAAAAAAAAAAAA AAAAWAAAAAAA AAAAAAABAA AAAANAAAAAA AAAAAAAAA A AAAAAAAAAA A A AA A. AAA? AAA. A. AAAAAAAAAAA. AAAEAANA. 32% A. AAAAAAAAAA W AAAAAAAAAA. A AA W 3&on €396 Assad £9.96 AAAAAEAAAAU AWAAAAAAAsAthso .2335? e5 5 AAAemeAAA $.25; mAAEcAAAm AAA: AAA? 65de 95% Scam .A.A:..$v 832:7. AAAANAAASA OAS. .mAmAAmAAm mam AAAOAA AAoAAAAAAAAo $3.5, mAepeEeAmQ A5958 2:. Amh 22:9 112 To test this idea, Masadeh et al. [84] simulated PDF data using the wurtzite structure containing different stacking fault densities. The stacking faults were simu- lated for different densities by creating wurtzite superlattices with different stacking sequences along the C—axis. They used the program DISCUS [183] to create the stacking fault models and PDFgui [124] to generate the corresponding PDFs. Their results indicates that the refined Se site U33 increased monotonically with increasing stacking fault density. Fig. 4.11 (in Ch. 4) can be considered as calibration curve of stacking fault density in the wurtzite structure, based on the enlargement in the ADPs along the z-direction U33. From this we can estimate a stacking fault density of ~ 35% for our bulk CdSe sample, and ~ 50% for each of the nanoparticles. It is then possible to carry out a. refinement using a structural model that contains an appropriate stacking fault density. The PDF data of bulk CdSe was therefore fit with a wurtzite model with a. 33% density, and the nanoparticle PDF fit with a model with 50% of stacking faults. The refinements give excellent fits, as is evident in Fig. 56(1)). The results are presented in Table 5.3. The enlarged U33 parameter on the Se site is no longer present and it is now possible to refine physically reasonable values for that parameter. As well as resulting in physically reasonable ADPs, the quality of the fits to the data are excellent, though the Rw value is slightly larger in the nanoparticles. 113 AWAVAV AVAAV AAAV AVAAV AAV AAAABVAAAA A .VA..A AA VA AA VA.A. AAVAVAA 8 AEAAV AAAAAEAAAAA AAA/A AAAVAAAA... AV AAVVAAAAAVAAV AAV VA. 85 AV AAVAEAVAV AAV VAAAA. AV .3: .N Am AAVVAVRAV AV AAAVmAmAVAV AAV VAAAAAVAV AAAVANAVAAVAV AAAVAV AVAAAVAV ANAAAV AAAAAA AAVVAAVAAV AV AAVVAAAVAAVAV AAV VAAAAAAV. AV AAAVAAAAAAVAV AAVVAAVAAV AV ANS mmAA n :2 Am AAAVA AAA. AV AA. AVAVAVAAVAV Am VAAAVAVAV AAVVAAVAVAVAV Am VAAAAVAV AsAV AVAVAA AAVA. AAVAAAV AV AAAVAAAAAAAVAV AA. VAA.A.AAV AV AAAVAAAAAVAV AAVA. AAAVAAV. AV AAAAV AA: u AAAA AAo AAAVAAAAVAAV AAVVAAAAAAAV AAA V3.3 AV AAAVAAAVAVA AAAVAVNAAV A A 5 .A AAVVEAA. AAAVAAVAANA. A AVAAVAVA A AAAVAVAAAAA. AAVAAAVAVA. A ..AAAAAAAA AAAAAAV AAAAAAAAAAAm 332:6 of E Vsomoa moAfimcew :23 mcicgm AAEAAEE AAA; .oAAAAVoR AAAAAVAw AVAVAAA? Air/C .VAAAAAVAAAAI AVVAxAAAAB AVAE. ..nAm..».AAA.AAAA mam 25¢ AVAVAAAAAZO was?» mAwAAVAAAAAAAAAA @252 23. Ana 22.5. 1111 Attempts to characterize the structure changes using direct measurements such as TEM technique for such small CdSe nanoparticles [184] were unsuccessful due to the poor contrast. However. in the present study we were successful in exploring the local atomic structure for CdSe nanoparticles, in real space, at different length scales. The PDF fits clearly indicate that the structure can be described in terms of locally distorted wurtzite structure containing ~ 50% stacking fault density (i.e., intermediate between wurtzite and zinc-blende) even for the the smallest particle size (~ 1.5 nm) , as evident from Fig. 5.6 and from the Rm values in Table 5.2. looking to the PDF of the smallest particle one can see an extra structural peak appears around 1‘ = 3.5 A indicates there is structure modification happened in this sample. This peak start appearing the nanoparticles PDF data gradually as nanoparticle size decreases. we suspect that the origin of this extra structural peak may be due to unreacted CdO contaminated in the smaller sample. To check that we fitted the PDF data. of the small sample with two phase. CdO (cubic) and CdSe (Wur). The fir results came out with anomalous ADPs for the oxygen tome in the CdO phase (U130 about 16.23 A2), with slight improvement in the fit quality. Another thing we tried is the same procedure with taking out the oxygen tome from the CdO phase. This approach gave excellent fit quality which could fit the extra structural peak that appears around 1‘ = 3.5 A (see Fig. 5.7). The result for two phase fit, Cd (cubic) and CdSe (\Nur). are summarized in Table 5.4. Interestingly, there is little evidence in our data for a significant surface modified region. This surface region is sometimes thought of as being an amorphous-like re- gion. Amorphous structures appear in the PDF with sharp first neighbor peaks but rapidly diminishing and broadening higher neighbor peaks. Thus, in the presence of a surface amorphous region, we might expect to see extra intensity at the first-peak position when the wurtzite model is scaled to fit. the higher-r features coming just 115 Figure 5.7: (Color online) The experimental PDF, C(r), with Qmam = 19.0 A_1(blue solid dots) and the calculated PDF from refined structural model (red solid line), with the difierence curve offset below (black solid line). PDF data are fitted using wurtzite structure model with 50% stacking fault density (a) one phase fit CdSe (Wur) (b) two phase fit, Cd (cubic) and CdSe (Wur). from the crystalline core. As evident in Fig. 5.6, this is not observed. Furthermore, as we describe below, the diameter of the crystalline core that we refine from the PDF agrees well with other estimates of nanoparticle size, suggesting that there is no surface amorphous region in these nanoparticles. The good agreement in the in- tensity of the first PDF peak also presents a puzzle in the opposite direction since we might expect surface atoms to be under—coordinated, which would result in a de- crease in the intensity of this peak. It is possible that the competing effects of surface amorphous behavior and surface under coordination perfectly balance each other out, and this cannot be ruled out, though it seems unlikely that it would work perfectly at all nanoparticle diameters. This is also not supported by the nanoparticle size 116 Table 5.4: The refined parameters values obtained from PDF analysis. The wurtzite structure (Wur), space group P63mc, with different stacking fault densities present in the structure. Ru, 2 0.25. Structure Phase CdSe Cd Stacking fault density (%) 50.0 50.0 a (A ) 4.2561(4) 5.11447 b (A ) 4.2561(4) 4.73978 c (A ) 6.6813(9) 5.30685 (1 0C 90 90.4028(3) 13 CC 90 90.8775(5) QC 120 88.1593(6) Cd U11 = U22 (A?) 0.4218(2) 0.0076(5) U33 (A?) 0.01069(3) 0.0157(6) Se 011 = 0.22 (A?) 0.0177(5) NA U33 (A?) 0.0045(9) NA Phase Fraction (”/6) 0.892 0.108 NP diameter (mu) 15.0(2) 15.0 determinations described below. 5.3.2 Nanoparticle size The determination of nanoparticle diameter is important since the physical proprieties are size dependent. It is also important. to use complementary techniques to deter- mine particle size as different techniques are more dependent on different aspects of the nanoparticle structure, for example, whether or not the technique is sensitive to any amorphous surface layer on the nanoparticle. More challenges are expected in accurate size determination as nanoparticle diameter decreases, due to poor contrast near the surface of the nanoparticle. In the literature, CdSe nanoparticles with a diameter of 2.0 nm are considered to be an especially stable size with an associated band edge absorption centered at 414 nm [185]. That size was observed earlier [37, 186, 189. 195, 184] with an estimated diameter by (51.6 nm). There are some reported difficulties in accurate size determination of such small CdSe nanoparticles. Attempts to characterize the 117 structure changes by TEM and X-ray diffraction techniques for such small CdSe nanoparticles [184] were unsuccessful due to the small diameter of the particles relative to the capping material. The PDF is a direct bulk average probe of the particle size that is a valuable complement to imaging techniques such as T EM. The particle size effect appears in the PDF data as an intensity fall-off with increasing-7'. To quantify this we assume the coherent core of the nanoparticle to have a spherical shape cut from the bulk, a reasonable approximation based on the Z—STEIV‘I in (Fig. 5.1), then the measured PDF will look like the PDF of the bulk material that has been attenuated by an envelope function given by the PDF of a homogeneous sphere [127]. The procedure is explained in details in earlier work of Masadeh et al. [861]. Results of this fit are shown in Fig. 5.8. The insets show the calculated and mea- sured PDFs on an expanded scale. The accuracy of determining the nanoparticle size can be evaluated directly from this figure. Features in the measured PDFS that correspond to the wurtzite structure are clearly seen disappearing smoothly atten- uated by the spherical PDF envelope function. The procedure is least successful in the smallest nanoparticles, where the spherical particle approximation on the model results in features that extend beyond those in the data. The estimate of particle size refined from the data is expected to be an overestimate in these cases where the structural model gives an inadequate description of the data. The particle diameters determined from the PDF are consistent with those ob- tained from HRTEM, UV-vis measurements, as evident in Tab. 5.1 In particular, an accurate determination of the average diameter of the smallest particles is possible in the region where UV-vis and photoluminescence measurements lose their sensitiv- ity [165]. In this analysis we have not considered particle size distributions, which are narrow in these materials [8]. The good agreement between the data and the fits justify this. 118 (a) , u v v r v v I . '26._28 30 32 34 36 38 4o 42 zo‘-2g 24 26 28 so 32 34 as 38 .' 'gb) . 35 4O 30 25 r01) 10 15 20 25 30 35 4O rm Figure 5.8: The experimental PDF, C(r). shown as solid (lots. The simulated PDF of CdSe nanoparticles (red solid line), where sphere envelope function (Eq. 4.5. Ch. 4) is used to transform the calculated PDF of the refined wurtzite bulk structure. to give a best fit replication of the PDF of CdSe nanoparticles. The inset shows on an expanded scale for the high-r region of experimental C(r) on the top of simulated PDF data for different diameter of CdSe nanoparticles (solid line). (a) CdSe586, (b) CdSe552, (c) CdSe470, (d) CdSe417. 119 Table 5.5: The first PDF peak position (FPP) and width (F PW') for different CdSe nanoparticle sizes and the bulk. CdSe-bulk CdSe586 CdSe552 CdSe470 CdSe417 PDF FPP (A) 2.6349(4) 2.6259(9) 2.622(1 2.6181(1) 2.6103(3) ) PDF FPW(A) 0.1999(09) 0.2001(19) 0.2020(25) 0.2055(25) 0.20703(25) The simple fitting of a wurtzite structure to the data. will result in an estimate of the coherent structural core of the nanoparticle that has a structure resembling wurtzite. Comparing the nanoparticle core diameter extracted from PDF analysis with the diameter determined from the standard characterization methods yields information about the existence of a surface amorphous region. The agreement be- tween the core diameter extracted from PDF and that determined from the TEM (Table 4.1), indicates that within our measurement uncertainties, there is no signif- icant heavily disordered surface region in these nanoparticles. This is less clear for the smallest diameter particles where the PDF, whose estimate of NP size may be an overestimate. gives a slightly smaller size (Fig. 5.8). The small surface amorphous re- gion in the CdSe particles is in contrast with 2118 nanoparticles [65] where the heavily disordered surface region is about 40% of the nanoparticle diameter for a diameter of 3.4 nm, the surface region thickness being around 1.4 nm [65]. 5.3.3 Internal strain The position and width of the nearest neighbor peaks in the PDF at 7‘ = 2.6349(4) A give a direct measure of strain in the NP as a function of size [84]. The positions and the width of these peaks were determined by fitting a Gaussian (Fig. 5.9(a)). and also determined from the full-profile modeling. The results are presented in Table 5.5. The results indicate that there is a significant compressive strain which cane be measured accurately from the PDF. The Cd—Se bond lengths extracted from the PDF structural refinement are also in good agreement with those obtained from the first peak Gaussian fit, as shown in Fig. 5.9(c). Thus we have a model independent 120 02? V02 gal—(b) _ E t 'l 4 ‘ N 80' -------- A £- 0) o. - _ hm E;_Il ' 1 11 ll l l r—J 1.5 2 2.5 3 3.5 NP diameter (nm) I E ‘ I ' l ' l ' l ' [pr— ' < ' (C) 2 I ‘4 - F A I 3 co_ 3 ' c: ?[ g ‘ I '8 ‘ co l *3 °?[ I ‘ I ‘ v CIT ‘ i if I ' 8 . I . I . 1 . I . | .‘ 2,4 2.8 m 1.5 2 2.5 3 3.5 r (A) NP diameter (nm) Figure 5.9: (a) The first PDF peak, (0) bulk. (o) CdSe586, (D) CdSe552. (A) CdSe470 and (I) CdSe417 fitted with one Gaussian ( ) Dashed line represents the position of first PDF peak in the bulk data. (b)(A) The first PDF peak width vs nanoparticle size, obtained from one Gaussian fit. Dashed line represents the width of first PDF peak in the bulk data. (c) Strain in Cd-Sc bond (Ar/r)(%) vs nanoparticle size. (I) Bond values obtained from the local structure fitting and (0) obtained from one Gaussian fit to the first PDF peak. (1) TOPO only was used for sample preparation (Ref. [84]), where for this study TOPO/HDA was used. 121 and a model dependent estimate of the strain that are in quantitative agreement. The strain measured in this study also agrees with that measured from CdSe NP samples made with pure TOPO surfactant in an earlier study [84]. The widths of the first PDF peaks have also been extracted vs nanoparticle diameter from the Gaussian fits (Table 5.5). They remain comparably sharp as the nanoparticles get smaller, as shown in Fig. 5.9(b). There is a slight tendency towards a peak broadening at the smallest particle size though the effect is small. Apparently the nanoparticle structure does not have significant inhomogeneous strain. However, peaks at higher-r do indicate significant broadening (Fig. 5.4(b)). also reflected in the enlarged ADPs in the nanoparticles, suggesting that there is some relaxation taking place through bond-bending. This is similar to what is observed in semiconductor alloys where most of the structural relaxation takes place in relatively lower energy l;)ond-bending distortions [187, 188]. 5.4 Conclusion The PDF is used to address the size and structural characterization of a series of ultra- small CdSe nanoparticles. The core structure of the measured CdSe nanoparticles can be described in terms of the wurtzite atomic structure with extensive stacking faults. The density of faults in the nanoparticles ~ 50% . The PDF reveal structure modifi- cation in the smallest particle, an additional structural peak around 7‘ = 3.5 A start appearing in the nanoparticle PDFs data gradually as nanoparticle size decreases. The diameter of the CdSe nanoparticles was extracted from the PDF data and is in good agreement with the diameter obtained from standard characterization methods, indicating that within our measurement uncertainties, there is no significant heavily disordered surface region in these nanoparticles, even at the smallest diameter of 1.5 nm. Compared to the bulk PDF, the nanoparticle PDF peaks are broader in the high—r region due to strain and structural defects in the nanoparticles. The 122 nearest neighbor peaks at 2.6349(4) A which come from covalently bonded Cd-Se pairs, shorten as nanoparticle diameter decreases resulting in a size-dependent strain on the Cd-Se bond that reaches ~ 1.0% at the smallest particle size. The combination of the intrinsic properties of ultra—small CdSe nanoparticles makes them an ideal material for solid state lighting applications. Study of the funda- mental properties of this material could lead to the development of more economical and environmentally friendly materials with similar properties, eventually leading to the next generation of solid state lighting technologies [189]. 123 Chapter 6 Concluding Remarks 6.1 Summary of the thesis We developed a PDF -based-n'1ethodology for quantitative structure determination of 1raocrystalline materials, employing the atomic pair distribution function (PDF) method [48]. As the demand of nanostructure cliaracterizations is growing, the PDF technique is emerging as a promising tool for this purpose. This is a local structural technique that. yields quantitative structural informa- tion 011 the nanoscale from x-ray and neutron powder diffraction data. [6] Recent developments in both data collection [89, 110] and modeling [94, 111] make this a potentially powerful tool in the study of nanoparticles. Additional extensions to the modelling are necessary for nanoparticles, and some of these have been successfully demonstrated. [65, 85, 161] In this thesis, I present a detailed PDF-based—methodology of extracting the struc- tural information available from PDF data on nanostructured materials. The PDF method is demonstrated here as a key tool that can yield precise structural informa- tion about the nanoparticles such as the atomic structure size of the core, the degree of crystallinity, local bonding, the degree of the internal disorder and the atomic structure of the core region, as a function of the nanoparticle diameter. Three CdSe 124 nanoparticle samples with different diameters that exhibit different optical spectra have been studied. The purpose of this thesis is not only to explain the PDF data of CdSe nanoparticles through a modeling process, but also to systematically investi- gate the sensitivity of the PDF data to subtle structural modifications in nanoparticles relative to bulk material. The developed PDF—lased-methodology have been applied on (2-4 11111) CdSe nanoparticle. The size-dependent structure of CdSe nanoparticles, with diameters ranging from 2.2 to 4 nm, has been studied using the atomic pair distribution func- tion (PDF) method. The core structure of the measured CdSe nanoparticles can be described in terms of locally disordered wurtzite atomic structure with estimated stacking fault density of ~ 50% . The diameter of the core region was extracted di- rectly from the PDF data and is in good agreement. with the diameter obtained from standard characterization methods suggesting that there is little surface amorphous region. A compressive strain was measured in the Cd-Se bond length that increases with decreasing particle size being 0.5% with respect to bulk CdSe for the 2 nm di- ameter particles. This study demonstrates the size—depemlent quantitative structural information that can be obtained even from very small nanoparticles using the PDF approach. The developed methodology have been applied also on technologically important system, ultra-small CdSe nanoparticles (1.5 nm). These ultra-small nanocrystals exhibit broadband emission (420-710 nm) that covers most of the visible spectrum while not suffering from self absorption. Study of the fundamental properties of this material could lead to the development of more economical and environmentally friendly materials with similar preperties, eventually leading to the next generation of solid state lighting technologies. [189]. However, nanostructure characterization is prerequisite knowledge to help in understanding the material’s properties. Using the developed methodology we learned that, the core structure of the ultra—small CdSe 125 nanoparticles can not be just described in terms of locally disordered wurtzite atomic structure with estimated stacking fault density of ~ 50% . The PDF data reveal structure modification happens in the smallest particle, an additional structural peak appears gradually around 1' = 3.5 A in the nanoparticle PDFs data as the nanoparticle size decreases. 6.1.1 Combining complementary probes Many probes exist for characterizing nanostructured materials. Some are inherently local, such as transmission electron microscopy (TEM) and scanning probe micro— scopies. Others are bulk average probes that are sensitive to local structure such as the atomic pair distribution function (PDF) method or extended x-ray absorption fine structure analysis (EXAFS). Investigation of the structure of nanoscale materials at. different length scales, using the PDF technique, is the the central theme of this thesis work, covering both technical developments and scientific applications. This thesis work is illustrated as part of the solution to the (nanostructure problem) [7]. The atomic arrangement inside the new materials is prerequisite knowledge to help in understanding the material’s properties. The nanostructure problem can be simply stated as “the need to determine atomic arrangements inside nanostructured materials, quantitatively and with high precision” [7] The strength of the PDF technique lies in the local and intermediate range struc- ture information. Thus the conventional crystallographic methods of probing the long range ordered structure can be combined naturally with the PDF analysis. This has been routinely exercised, providing complementary and unique insights into the real structure of nanomaterials. Combining imaging techniques such as transmission electron microscopy (TEM) and local structural PDF methods developed here, can help move forward towards a robust method to determine atomic arrangements inside 126 nanostructured materials, quantitatively and with high precision. Ch.3 is a textbook example of how these complimentary methods can be combined to great effect. The combination of TEM and PDF methods added more understand- ing to a complicated nanoscale structure of a novel compound, such as Na3BiO4. 6.1.2 Applying the PDF-based-Methodology in new nano- systems The purpose of the PDF technique is to answer significant scientific and / or techno- logical questions suitable to this method. One very effective way is to bring the PDF technique to a wider user community. Widespread use of the PDF-based-methodology could bring in a large number of new users and have significant impact in various sci— entific disciplines. On the other hand, the capability and application scope of the PDF technique needs also to be expanded. For that I have written (~ 50 pages) technical manual about the RAPDF technique (rapid acquisition pair distribution function), as “11011Specific user friendly technique” to study local distortion in nano scale materials at different length scales. On the other hand a user friendly software for PDF analysis (PDFgui) have been just released for public [124]. Technical manual that describes the RAPDF technique cab be found in the Appendixes (A, B and C ). As the high energy sources, such as APS, becoming more accessible and the data collection time get more faster (with the usage of RAPDF technique) and user friendly software [124] joined with technique manual (presented in the Appendix), this will make the PDF as “nonspecific user friendly” technique and useful tool for nanostruc- tural analysis in the next decades. The user friendly software [124] with technique manual are available for public. 127 6.2 Future work Future work would be applying the developed PDF—based—methodology on nanostruc- tured materials under extreme conditions, such as low temperature or high pressure. 6.3 Acknowledging funding agencies This work presented here was supported in part by National Science Foundation (NSF) grants DMR-0304391 and DMR—0703940. Data were collected at the 61DD beamline of the MUCAT sector at the Advanced Photon Source (APS). Use of the APS is supported by the US. DOE, Office of Science, Office of Basic Energy Sciences, under Contract No. W ~31-109-Eng-38. The MUCAT sector at the APS is supported by the US. DOE, Office of Science, Office of Basic Energy Sciences, through the Ames Laboratory under Contract No. VV—7405-Eng—82. 128 Appendix A RAPDF Data Collection A. 1 Sample preparation The easiest sample to handle in RAPDF technique is a fine grinded powder packed into a flat plate disk that is 1 mm thick and 10 mm in diameter sealed between Kapton tape, the optimal thickness is usually 0.9 times the attenuation length of the sample, thus varies from sample to sample, Fig A.1. If sample volume is an issue, sample disks with a 5 mm diameter can be used. A sample disk with less than 5 mm diameter hole may cause problems with aligning the sample with the incident beam, due to the RAPDF technique experiment setup. If sample volume is extremely small (i.e. no large samples available) an issue, the RAPDF setting can handle a capillary sample holder but extra attention will need to be paid in order to align it with the incident beam. Flat plate geometry is therefore a preferred choice. RAPDF sample geometry vs data quality have been tested using both, flat plate and capillary geometries using the same sample (Ni-300 mesh). Data quality was identical and gave the same results, which indicates that RAPDF technique can handle both sample geometries, see Figs A2 and A.3. RAPDF data quality is sensitive to the particle size of the measured powder sample. Since the beam size used in RAPDF setting is rather small 0.5x0.5 mm. the 129 A . od-~r— .. (b‘. 1 A. Q as) Figure A.1: Sample holders used in rapid acquisition PDF (RA-PDF) experiment. (a) Flat plat sealed between Kapton tap and (b) 1.0 mm diameter Kapton capillary. sample should be a. very fine average powder. The preferred particle size is 300 mesh, otherwise the presence of large particles may cause spotty raw data, with saturated detector pixels occurring sometimes (Section. 3.2.2). Another possible problem could happen due to poor powder averaging is preferred orientation, which can affect the data quality as well. A.2 Requirements of the experiment To collect powder diffraction data suitable for PDF analysis, a few requirements need to be met: A.2.1 Extended Q range The finite Q space measurement range directly determines the experimental real space resolution of the PDFs [90]. Since we can only measure the scattering up to a max- imum value in Q, which is called Qmar, the resulting PDF will show termination 130 F (11“) 21 28 14 F01") 2+; 4 5 6 7 0 l 27 _>. . >.- Db > > P 5 3‘ —3 -2 —1 20 21 22 23 24 25 26 27 28 29 o (r‘) (b) Figure A2: (a) The experimental reduced total scattering structure function F (Q) = Q(S (Q) — 1) of Ni powder for two different sample geometries: capillary (solid blue line) vs flat plate (red symbols). (1)) The high Q region on expanded scale. 131 16 S!” l 1 I i I i 1 I ‘9' I i r I , A , l , I ; A I ‘?+~ 5 1.: ;.:__i.§ 5 : gig; 5:} ::§ l (9° t 1 : ¥ ‘9 ”1‘ u 1». 111111111141 +_ 1 . I O ‘ 3 “'3. N WV‘MWJ‘WJWMWWWVMW T- I I I I I I I I I O 2 4 6 8 10 12 14 16 18 20 '01) (a) w- I I l I ' I I l I N_ . 1 ' l I I . co- 1 : l ; A I . I l T v- ' l I L 5 i l . °< : u - i :’ 1 : V . :1 ‘ ,: l: i l o l 3 $1113: cl 1! . 1 °_1"1-1 my -4 2.: v.« ”j.“ 0 |. . N WWW/“WWW T” 1 . . . . .4._L . . ' 0 2 4 6 8 10 12 14 16 18 20 r (A) (b) Figure A. 3. The experimental reduced pair distribution funttion C(r ) of Ni powdel with Qmm- of 28 7 A 1 (solid blue symbols) and the calculated PDF from refined strurtural model (solid red line). The difference curve is Sll()\\11 offset below (solid green line). The data were collected at room temperature. (a) Flat plate geometry and (b) capillary geometry. 132 ripples due to the cutoff in Q space. In practice, the required Qmam is set by par- ticular experimental needs. However, to reduce the effect of the termination ripples caused by the cutoff in Q space, QmaxZ 18.0 A"1 is generally necessary, with much higher Qmam, (e.g. 2 30.0 A’l) required in case of highly crystalline materials. For the RAPDF technique as the physical size of the image plate (IP) is fixed, the avail- able Q range is defined by the relationship between X-ray energy and the sample to detector distance (will be discussed in more details in Section. 2.3). A.2.2 Achieving good counting statistics To obtain quantitatively reliable structural information, high counting statistics is necessary. As the scattering is a random event, the measured counts follow the Poisson (,‘listribution. When the number of measured scattered photons is large, its uncertainty is simply the square root of the measured counts. An empirical rule of thumb for sufficient statistics for PDF study is to collect around one million counts per inverse Angstrom. In RAPDF technique during the data collection, for each sample, usually the image plate exposes for S seconds, and repeated N times for a total data collection time S x N seconds. This approach avoids detector saturation whilst allowing sufficient statistics to be obtained. Since the multiple data sets are measured under identical conditions, we directly sum them together to get the total counts. To avoid too large intensity values, we divide the total counts by the number of exposures. A.2.3 Achieving good signal to noise ratio This requirement is of special importance, and hard to prepare for it in advance as well, mostly because more than one scattering process contributes to the measured total counts and only elastic coherent scattering is used in the PDF analysis [48]. To achieve a good signal to noise ratio, it is necessary to have a good estimate of 133 what fraction of the total counts is the elastic signal sought after. For example, at Q just as high as 25.0 A_1 for Si atoms, the Compton scattering intensity is 4 times the elastic intensity. If 1% is the desired relative uncertainty. you will need as many as 160,000 counts. Still chances are you are looking for even smaller changes in the elastic scattering intensities, which calls for more counts. In principle, it is difficult to know in advance the expected signal. The best approach is to analyze the data during the experiment, then decide whether more collection time is necessary. A.3 Setup of the RAPDF experiment For practical reasons, the description of the procedure to set. up the experiment will follow the flight path of the X-rays, from the source towards the detector (upstream to dmvnstream). The first. task is to choose. the optimal beam-line with proper X-ray characteristics for your specific experiment. Experimental setup in the early stages is similar to the conventional PDF X-ray experiments. The schematic diagram of the RA-PDF experimental setup is shown in Fig. A.4. 1. Setting the energy : The X—ray energy usually set. based on particular experimental needs. Few ways to calibrate the energy are briefly explained here. In principle, we can compute the X-ray energy given the monochroma- 1 tor tilting angle and the reflection plane used, starting from Bragg’s equation 2d,,“ 2 However, those calculated values are usually only good ap- n/\ 81“ 9hk1' prozr'imations due to the various uncertainties in the tilting angle, etc. One commonly used approach is to scan across the absorption edge of a standard material, such as Pb, or Au K edges, and use their well known absorption edge energy values as references. The other way is to use multiple channel analyzer (MCA) to locate the incident beam energy. One complication of this method lThe Bragg equation relates to the spacing between the crystal planes, (1).“, to the particular Bragg angle, 0111-1. at which reflections from these planes are observed. 1311 Area detector Sample Monochromator > 60.0 keV a [I Vlicon (111) 1 SlitS Storage ring Figure A.4: Schematic diagram of the rapid acquisition PDF (RAPDF) experiment layout. [0 is that the MCA needs to be calibrated first, usually with a set of standard ra— dioactive materials which have characteristic emission lines. Timing the energy to desired value usually done by the beam scientist. This step does not need to be the first step, it is possible to change the X—ray energy sometimes during the experiment. . Setting the beam size: Set the upstream slit openings to give you the appro- priate beam size. For example. a beam size of 0.5x0.5 rmn at 61D-D at APS was used in this work. The primary considerations for beam size are usually the Q space resolution and incident X-ray flux. With some approximations, starting from Q = 1&1 sin 9, the reciprocal space resolution fimction can be expressed as, AQ~AA cos2(0)cos(2e) Ad Ar Q A 1+sin2(26) X d r) (A'l) where 0 = % tan'1(5) is the angle between incident and the scattered X-rays. d is the sample to detector distance, 1' is the distance of the pixel from the center 135 of the image plate. In typical RAPDF experiments, the maximum 26 angle is about 30.0 degrees. we can show that the first term 9%— is the smallest, usually of the order of 10—4 determined by the characteristics of the monochromator(s). The prefactor of the Q dependent second term varies from 1.0 to 0.65 as 26 goes from 0.0 to 30.0 degrees. The typical value of 93‘! is 2715(90 = 0.005 with 1 mm thick sample and 200 mm sample to detector distance. Ar can be estimated as the sum of beam dimension and image plate pixel size (0.1x0.1 mm), while r value ranges from 0.0 to 172.5 mm. If we simply take 100.0 mm as the mean value of 'r, the 9,1 becomes comparable to Add with beam dimension of 0.5x0.5 mm. A much larger beam size than this would make it the limiting factor to the instrument resolution. There are also some other considerations as well, such as the intrinsic beam size, the incident. beam intensity, the scattering power of the sample and so on. 0.5x0.5 nnn beam size was used in this work. Eq. Al is explaining the limiting factors to the instrument resolution for RAPDF AA Ad Ar experiment setup. Factors such 7’ T’ —,.—, sample thickness and beam width need to be optimized in order to achieve a better instrument resolution (Fig. A.5 (b))- . Align the center of the diffractometer with the beam center: locate the center of rotation of the diffractometer, usually with help from a telescope: A sharp metal pin is mounted on the goniometer (at sample position), and the goal is to get the tip of the pin in the same (fixed) position, when any one circle of the diffractometer is rotated. When the tip of the pin does not move irrespective of the goniometer circle positions the alignment of the goniometer center is achieved. Once you have done this, use the motor to adjust position of the diffractomcter to set the tip of the pin on the beam center (allowing the goniometer center to be aligned with the beam center). You can use a beam monitor or a piece of a burn paper or put. a fluorescent screen to do it. The 136 easiest way is to place a silicon diode detector right after the pin and perform :r-y scanning of the diffractometer. Once this is done, mark the current position of the pin tip with the center cross of the telescope, so it easy to remember. Make SURE NOT to move the telescope at any time during the experiment. Now the position of the center cross of the telescope is representing the beam center which is aligned with the center of the diffractometer. Installing the image plate (IP): Before you install the IF, you need to have an estimated value for the sample to detector distance, d(mm). For a given energy value E (keV), the value of (I(mm) can be calculated based on the value of Qmm; you want to achieve. In most cases, you set first the energy you will be working at and the wanted Qmar value, then the corresponding (I(Hrm.) can be calculated as following: (a) Find the. corresponding A in A using the relation MA) = fig}??? (b) Find the corresponding 29 value of the preferable Qma 1: value using the A value obtained in (a). (c) As the dimension of the image plate (e.g. 345 mm in diameter for Mar345 detector) is fixed, the sample to detector distance can be found using (1(mm) = 172.5/ tan(26). For example, if you want to achieve Qmax of 35.0 A—1 at 100 keV with the Mar345 detector, what is the needed sample to detector distance in mm? First, calculate the corresponding wavelength in A for the given energy. Second, to achieve Qmam of 35.0 A‘1 the sample to detector distance needs to be 202.5 mm. Put the image plate detector in its approximate position. It will require adjusting later. 137 5. Installing the beam stop and aligning itzThe beam stop is usually located a few mm in front of the center of the IP detector. The point from installing the beam step is to protect the IP detector from the incident direct beam which goes through the sample. Hence miss aligning the beam stop could damage the IP detector permanently. To protect the IP detector from damage can happen during the beam stop alignment, before you open the beam shutter, make sure to have Pb shielding right in the front of the image plate detector. A direct beam WILL damage the IP permanently. A good way is to put a silicon diode monitor aligned with the incident beam, right after the beam stop which is installed on a translation stage. Through scanning the beam stop positions (x,y) you can quickly position the beam stop to fully intercept. the incident beam. After you are done with beam stop alignment measure the air scattering (dont load any sample) for a. long time (like 1000 seconds) to make sure the air scattering background is symmetric all over the 2D image, give more attention to region around the 2D image center. If you found the 2D image is asyrrnnetric then you need to re—align the beam stop more accurately. The beam stop position somehow forming the air scattering background, so that if the beam stop position has been changed for some reason, then a NEW background run need to performed. 6. Align the image plate detector: One goal is to set the beam center, which already aligned with the beam stop, close to the detector center. The other is to achieve precise orthogonality of the IP plane to the beam direction. The beam center and the orthogonality of IP plane, they dont need to be very precise, the resulted calibration parameters will take care of that. Program F IT2D [196] provides an easy-to—use interface to do calibration. This requires the use of a calibrant, for example, fine standard powder like Si. Calibration method is 138 explained in more details in (Section 2.3). . Reducing the background: The RAPDF experiment is subject to rather high background levels. Most. background intensities come from air scattering (Fig A.5 (a)). Thus the first step here is to shield out the air scattering before the sample by placing a thick Pb plate with a hole at the center for the incident beam. If there is a considerable gap between the Pb shield and the sample, a collimator right in front of the sample is preferred. During experiments, usually more than 50% of the incident beam is allowed to pass through the sample. The rather long sample to beamstop distance (2 150 mm) creates a significant amount of air scattering. The collimation after the sample is in principle possible, but. in practice hard to implement. One way is to shorten the sample to beam stOp distance. However, this would increase the blocked scattering angle by the beam stop. A very effective way would be to use an evacuated secondary flight path. This however is more difficult to achieve in practice. A test has been performed by inserting a balloon filled with He gas in the secondary flight—path (Fig A.5 (a)). The test shows that the air scattering provides a greater background at low angles than at high angle. In practice, the background scattering only becomes an issue for weakly scattering samples where the collection time increases. . Install additional equipment required for your experiment: For exam- ple, the displex with cryostat needs to be set up for low temperature measure— ments. In the case of high temperature measurements, the furnace should be installed. This topic will be addressed in details, somewhere in this manual, under chapter describing parametric studies using the RAPDF technique. 139 I I I l I v I Empty Kapton 200 sec (Bkg) .— Air seat. in balloon 200 sec - — —~ Air scat. 200 sec ------ Figure A.5: (a) Air scattering intensity coming from the secondary flight path region with balloon (dashed red line) and without balloon (doted green line), vs the typical background originating from two windows of empty kapton, 0.1 mm thick, separated by 10 mm of air in between (blue solid line). (b) The experimental reduced pair distribution function (7(7) of CdSe powder with Qmax of 28.7 A_ . The data were collected at room temperature with the same geometry. (solid blue dot ) Experiment setup factors were optimized for achieving a better instrument resolution , (solid red line) experiment setup factors were not optimized. 140 A.4 Diffraction intensity Diffracted intensity in a powder diffraction experiment is observed on a set of Debye- Scherrer rings. Detection of the diffracted radiation can geometrically be seen as the intersection of the set of Debye-Scherrer cones with a surface that represents all possible locations of the detector. The intersection of the Debye—Scherrer cones with a plane, which can be regarded as the detection plane of an imaging plate detector that is aligned perpendicular to the axes of the Debye-Scherrer cones, results in a set of concentric rings (denoted Del.>ye-Scherrer Rings or powder rings) on the detection plane, (Fig. A6). The radii of the rings contain information about the diffraction angles and in an ideal case the intensity on a ring is homogeneous and does not dependent on the azimuth angle (p' . Real powder rings however, are more or less spotty or discontinuous or overexposed. A real powder rings are shown in Figs. A.7 (a) where you can see dark spots all over the rings, this is due to the presence of large particles. Implementing a sample rocking during the data collection can help reducing the spottiness of recorded data Figs. A.7 (b). Having spotty data can be a source of peak broadening or the reason of different relative peak intensity (Fig. A.8). Overexposed can be a source of peak broadening or the reason of irregular peak shapes (Section 3c.2.2). A.5 Collecting the RAPDF data Here we briefly describe some common actions during the RAPDF measurements. The RAPDF data collection mostly involves changing samples and controlling the data acquisition system. Operation of data acquisition system varies form one beam- line to another. However, sample-changing is essentially the same. 141 800 l 000 2600 2400 2200 1 800 Rows l 600 l200 800 1000 1200 l 400 1600 [800 2000 2200 2400 2600 2600 2400 2200 2000 l 800 l 600 1400 l 200 1200 1400 1600 I800 2000 2200 2400 2600 Columns Hi r f :a::rfi"'"‘"r ej [000 3000 10000 30000 Intensity Figure A.6: Two dimensional contour plot from the Mar345 Image Plate Detector. The XRD data are from nickel powder measured at ambient conditions. The wave— length of the radiation was 0.1270 AThe concentric circles represent intersections of different colors with the area detector (Debye-Scherrer rings). The sample was corr— tained in a flat plate, 1.0 mm thickness, irradiated volume 0.25 1111113, beam size 0.5x 0.5 mm2. The small dark area. in the center of the image is a shadow cast by the beam stop assembly. (b) Figure A.7: X-ray powder diffraction data of fci-HoMan at ambient conditions, as it was recorded on a MAR3450 imaging plate detector for 150 seconds. The sample was contained in a flat plate of 1.0 mm thickness (irradiated volume 0.25 mm3, beam size 0.5x0.5 rnrn2).(a) before sample rocking, (b) after sample rocking. The wavelength of the radiation was 0.0955 A. The small white area in the center of the image is a shadow cast by the beam stop assembly. 143 15 "—1 I N- .. ,_ m. - A‘D T 5n LI. n n ' 5 ’ l co. ril‘h... . ..., malt- | ..., , 1r 1,, il‘fi'f}?\i".).","félr:‘g‘fv‘vf"1"~\»:‘*~-’~“"“~‘a"‘"‘~‘“‘#hfl‘"\\‘"f“‘hlj|l 0,3303%?” on _ ‘ . I I I I I I O 4 8 12 16 20 24 28 32 Q (11") (a) so . . ‘- 4 6 (r2) ) 5;»; a» . El, 5:» ? 3 3 :3:— 0 2 4 5 8 1O 12 14 16 18 20 r (A) (b) Figure A.8: The corresponding total scattering function F (Q) (a), and the corre- sponding atomic PDF, C(r) (b) for data shown in Fig. A.7, (dot symbol) before sample rocking (solid line) after sample rocking. The difference curve is shown offset below. 144 A.5.1 Sample position As the sample to detector distance is obtained from the calibration runs (with Si or LaBG), calibration method will be discussed in (Section 3.3), then the sample has to be placed at the same position as the calibrant, in order to ensure sample alignment. This is usually done with the help of the telescope indicating both the center of the beam and center of the diffractomcter. However, the RAPDF setup requires a very close sample-to—detector distance, e.g. 200 mm. It might happen that the telescope view of the sample is blocked by the image plate detector. If this is the case, alternative ways of ensuring the same sample positions should be implemented, such as a. monitor screen, identical sample holders which go in an aligned sample pocket etc.. If possible, a small telescope can be mounted on the diffractometer (the Chi circle) for sample alignment. When changing the sample, be sure not to bump 2 the beam stop. A significant shift of the beam stop requires realignment of the beam stop, and new set of measurements of the background scattering. After changing the sample make sure also that the sample is aligned with the incident beam. To check this you can use a laser beam which can be mounted on the beam path and trajects the laser beam through the upstream slits tracing the X-ray beam path. If you are using a small sample diameter (3 5.0 mm) then extra attention will need to be paid in order not to hit the sample holder. A.5.2 Varying the setup During the data collection, if the X-ray energy needs to be changed with a new sample to detector distance,then extra attention needs to be paid in order to benefit from the calibration of the current setup. This is because the sample to detector distance and the X-ray energy are strongly correlated in the calibration. ONLY one of them can be refine during the calibration method to get sensible results for the refined parameters. I) - *bump means here accidentally move stuff To vary the setup, you can either change the X-ray energy or the sample to detector distance first, then measure a. standard fine powder to do calibration with fixing the one which was NOT changed (energy OR distance), write down the calibration results. Now you can change the other one which was fixed (energy OR distance) and measure a standard fine powder to do calibration one more time with fixing the one which was NOT changed. If only room temperature measurements are performed, moving the sample (if possible) is much simpler than moving the detector because the later always requires a new alignment for the detector. A.5.3 Exposure time Beam exposure time depends on the scattering power of the sample and the incident beam intensity. The rule of thumb is to keep all the counts in the linear response regime of the image plate. In our case, the maximum count of in each pixel is 64,000, set by the electronic readout device. A reasonable maximum intensity for our data is advised not to exceed 60,000. Grainy samples may cause sparse pixel saturations even with very short exposure times because of crystals with considerable sizes. Certainly the best solution is to have fine powdered samples. As grinding samples may be not possible during the experiments, we may try to implement sample rocking, or try another section of the sample (Figs A.7). However, we should not worry about a small number of saturated pixels in our data. These can be removed during the data processing (Chapter 2). Also, the relative error would be rather small compared to the total number of pixels, e.g. ~ 10,000,000 pixels in the Mar345 detector. A.5.4 Measuring the background To properly analyze the RAPDF data, we also need to measure the background and other calibration data if necessary. Here the background refers to the sum of the 146 container scattering and the empty instrument scattering (mostly air scattering). The background intensities will be subtracted from the sample total intensities during the data processing. When the background level is rather high, the absorption correction due to the existence of the sample in the beam (when collecting sample data) becomes necessary. This correction is usually rather difficult since it depends on the details of the experimental setup. Strictly speaking, different absorption corrections also become necessary when the container itself scatters (absorbs) considerably. For the most commonly used kapton tape, the scattering from the container itself is small compared to the air scattering, (Fig A.5 (a)) and there is no need to distinguish between the container and empty instrument scattering. If a glass capillary is used, the scattering from it may be very strong. we may need to measure the empty instrument separately. At the end, you should have. the data for the sample. the data for the background, and the data for the calibration. 147 Appendix B RAPDF Raw Data Processing The goal of this Chapter is to demonstrate the methodology of obtaining a two column data set equivalent of the conventional angle dispersive scan from two dimensional RAPDF image plate raw data, where every data point represents a sum of scattering intensities over the same solid angle. Thus, the raw data processing here refers to the procedure of obtaining intensity versus diffraction angle, 20, from the two dimensional RAPDF raw data shown in Fig. A.6. As described in Section 1.4, diffracted intensity in a powder diffra(,:tion experiment is observed on a set of Debye—Scherrer cones. Detection of the diffracted radiation can geometrktally be seen as the intersection of the set of Debye-Scherrer cones with a surface that represents all possible locations of the detector. The intersection of the Debye-Scherrer cones with a plane results in a set of concentric rings on the detection plane. These cones can be regarded as the detection plane of an imaging plate detector that is aligned perpendicular to the axes of the Debye-Scherrer cones. The radii of the rings contain information about the diffraction angles. In the ideal case, the intensity on a ring is homogeneous and is independent of the azimuth angle ((45) as shown in Fig. A.6. Real powder rings however, are more or less spotty or disc-(mtinuous as shown in Fig. A.7. 1'18 B.1 Startup with FIT2D The program F IT2D [115] is the choice of software used for performing image plate data reduction described in this manual. After starting the program, the user has to specify the amount of memory which will be used by FIT2D to store the image plate recording. The x-dimension of the memory array also defines the maximum number of points of the resulting one—dimensional data of intensity versus diffraction angle. The allocated memory should correspond to the size of the image in pixels (Table. 8.1). A larger amount of memory can be allocated, but in FITZD’S standard graphical representation, the excess memory appears as a band of zero intensity around the images. This leads to erroneous intensity values, because the integration is extended into this zero—intensity area. Reproducible results can only be obtained when the dimensions of the allocated memory exactly match the size of the image and the full image is displayed (FULL UN-ZOOM). Image plate recordings in several different MAR3450 MAR2300 automatically recognized yes yes image size — x [pixels] 3450 2300 image size - y [pixels] 3450 2300 pixel size -x [,u] 100 150 pixel size -x [it] 100 150 Table 8.1: Comparison of the image plate formats Mar3450 and Mar2300: both formates are generated by the Marresearch Mar3450 detector. Mar3450 format was used in this work. Marresearch formats can be read by FIT2D program (Table. 8.1). If the image is not displayed correctly, the reason is most probably a wrong value for one of these parameters. At the data-input stage, one can also select whether a flat field correction and a spatial distortion correction will be applied to the image. The flat field correction needs to be applied if different areas of the image plate detector have different sen- sitivity. The correction is calculated using an image of the detection area uniformly 14 9 exposed with radiation of the same wavelength, which has been used in the acqui- sition of the image for that the correction is intended. For more information about these corrections see Ref. [196]. The image plate recordings used in this work were obtained using MAR3450 format. B.2 RAPDF data reduction B.2. 1 Data homogeneity For homogeneous Debye-Scherrer rings the intensity as a function of the ring radius can be measured by scanning a narrow stripe of the image along radial direction (from the center outward). The resulting table of intensity versus ring radius (r) is equivalent to the data collected in a conventional powder diffraction experiment with a point detector mounted on the ‘26-arm of a goniometer. If the same procedure is applied to the 2D data consisting of inhomogeneous or discontinuous Debye-Scherrer rings,then the obtained intensities would dependent on the direction of the radial scan. To get reliable values of the intensities, averaging over several radial scans taken along different azimuthal directions, or over larger continuous segments of the Debye- Scherrer rings is required. The most accurate intensities are obtained when complete or nearly complete rings are used. In general, the intensity of a ring at certain radial distance (7) should not depend on the azimuthal coordinate ((1‘)), in the ideal case of homogeneous Debye-Scherrer rings. Fig. A.6 shows an ideal case of homogeneous Debye-Scherrer ringsl. 1At high angles, polarization effects may be seen which give a stronger scattering intensity per- pendicular to the X-ray polarization plane. B.2.2 Overexposed spots The components of an image plate detector system, especially the photomultiplier, impose a limitation to the dynamic range of the detector. The detection process in an image plate detector works as follows: X—rays interact with Eu2+ ions, which are present in a layer of (usually) BaFBrzEu within the image plate. The 153-112+ are ionized to Eu3+ and fast electrons are generated. These electrons become trapped close to their point of origin at sites with anion vacancies. The amount of trapped electrons is proportional to the intensity of the X—rays which caused the ionization. During the read—out process, the image plate is scanned pixel by pixel with red light from a He—Ne laser. The red light causes the electrons to leave the traps and to recombine with the Eu3+; thereby emitting blue luminescence radiation. The intensity of the luminescence radiation is proportional to the amount of previously trapped electrons, i.e. it is proportional to the intensity of the X—rays. For each pixel the intensity of the luminescence light is measured with a photomultiplier; the output signal of the photomultiplier is digitized and stored. The MAR3450 detector uses data records with a length of 17 bits (0 to 131071) to store the intensity information of one pixel. The level of intrinsic noise of an image plate detector system is very low and weak signals (single photons) can be detected. The detector shows a linear response to the detected signal over its complete dynamic range, which encompasses five or six orders of magnitude (for more details see appendix B). The non—linearities of the photomultiplier are compensated during digital data processing. However, above a. specific intensity level, the response of the photomultiplier be- comes independent of the initial intensity and a constant output is obtained. In this case photomultiplier is in saturated regime. The same intensity is found for all pixels where a saturation of the photomultiplier has occurred. This saturated intensity is corresponding to the maximum value expressible in a data record. Affected pixels are denoted as overexposed pixels; an area consisting of overexposed pixels is denoted as 151 overexposed spot or overexposed area. It is observed that in overexposed areas blooming can occur2. Therefore, pixels adjacent. to overexposed areas also appear more intense than they would be based on the detected amount of radiation. Overexposed spots can be much more intense than average spots on the powder rings. It is possible that the intensity at the non-overexposed flanks of overexposed spots is higher than at the center of average spots. Therefore, even if there are. only few overexposed spots on each powder ring, they can have significant impact. on the peak intensities of the integrated data of intensity versus diffraction angle and they are a source of systematic errors. Overexposed spots are also a source of diffraction peak broadening, and the reason for irregular peak shapes. It is thus necessary to exclude not only overexposed pixels, but also the regions round the overexposed pixels; i.e., in the case of a well defined diffraction spot with an overexposed center, the complete spot must be excluded. Significant differences of the peak intensities are observed prior to and after the exclusion of overexposed spots. The effect of the overexposed spots on the integrated intensity, for long and short radial scans can be seen in Figs 8.1 and B.2 respectively. Finding overexposed areas with FIT2D is straightforward if a threshold mask is used. The value of the threshold depends on dynamic range of the detector. For the MAR3450 detector a value of 100000 counts has been used3. After the overexposed areas have been found, the environment of the overexposed spots can be masked using a threshold mask, a peak mask4 or a polygon mask, for more information about mask tools, see Ref. [196]. gBlooming denotes the migration of the electrons to adjacent pixels, where they cause secondary ionization of Eu. In an imaging plate this may occur when a large amount of electrons are generated simultaneously, e.g. when the signal is strong enough to cause saturation of the detector. 3T his value was smaller than maximum possible value to account for noise and the fading of the image. 4The user is prompted to select graphically the center of each peak to be masked out with a circle of diameter of 5, 9, 15 or '27 pixels (a) Mask Peaks(5) Used . O t 3 1 Threshold Mask Used - [ No Mask Used -- 9 ' . N i ’58 a? . 8 '- .i - o i ..i N 5 - 3 it _ t 3 9 g.’ i n; _ i ' $.55: ~ 3 ' ' ‘ ‘ . fl 0 4 8 12 16 20 24 28 tth (deg) (b) Figure B.1: (a) XRD data of Ti1,5Zr0,5Sb at ambient conditions, recorded by MAR3450 image plate detector using 60 seconds exposure. Solid red line extending from the center outward represents scanned radial direction which is equivalent to the {1? axis in right panel. (b) The integrated one dimensional XRD data of the image shown in the left panel. Data. integrated for different mask tools: mask peaks(5)3 (blue crosses), treshold mask (red empty squares), and no n‘rask used (green solid line). 153 B.2.3 Objects in the beam path When scattered or diffracted radiation hits a solid object (non—transparent to X— rays), the radiation is blocked and a shadow of the object is cast on the detection area. One of these shadow sources is the beam stop assembly, which appears as small white area in the center of the image in Fig. B.2 (a). Since these objects (such as Beryllium window displex or diamond anvil cell) are usually made of non—transparent to X-rays metals, all radiation -including sections of the Debye-Scherrer coues- that interacts with the objects is absorbed, but depending on the distance of the object. from the detector, some amount of scattered radiation reaches the detector in the shaded areas. The intensities in the shaded areas are completely unrelated to the intensities in the non—shaded areas of the image. In the experiment, where the image shown in Fig. B.2 (a) was recorded, the beam stop was several mm away from the detector and intensities of 4530 counts are measured in the shaded area versus 1100 counts in the background outside the shadow. Circular shaped shadows centered at the beam center causes problems, if Debye— Scherrer rings exist under the shaded area. If not treated, the corresponding Bragg reflections would have an intensity of zero (missing Bragg peaks). Therefore, careful calculations of the distance between the beam stop and the detector have to be performed in advance to estimate the diameter of the shadow cast by the beam stop assembly in the center of the image which is equivalent to the minimal accessible 29min- B.2.4 Usage of masks As already indicated in the previous two sections, it is sometimes necessary to ex- clude certain areas of the image plate recording from the subsequent steps of the data reduction. These areas can cause problems mostly due to the phenomena like overexposed spots and presence of objects in the beam path. Several tools, denoted as masks, exist in FIT2D, which are used to define an overlay for an image. Pixels 154 located below the overlay are subsequently excluded from any data processing pro- cedures, but the image itself is not changed. There are circular masks (denoted as “peak masks”5) in several different sizes. These masks are most convenient to ex— clude small, isolated areas of the image. Larger, continuous areas can be excluded by drawing a free-form polygonal region using the polygon mask tool. These two types of masks have to be applied manually, but there is also an automatic mask, the so—called treshold mask, which was used in this work. When this mask is used, the image is searched for pixels, whose intensities are above or below (can be specified) a given treshold value and all pixels that fulfill the criterion are masked. Two tools (unmask peak and unmask polygon) are available to delete or modify already defined masks. For more information about mask tools, see Ref. [196]. B.3 Calibration The image plate data set is essentially a two dimensional matrix of the intensities. Thus we first need to know the 20 or Q value of each matrix element, which requires the following parameters: the pixel size, the beam center position (170, yo) which is also denoted as the center of the Debye-Scherrer rings on the image, the sample—to— detector distance, the X—ray energy, and the tilt of the image plate relative to the incident beam. In practice, several parameters are known (in advance) before hand: the pixel size that depends on the laser scanning mode during read-out, and either the X-ray energy or the sample to detector distance. The other parameters are obtained through the calibration process. Four parameters describing the geometry of the Debye-Scherrer rings on the detec- tion plane: two coordinates of the beam center, tilt angle and tilt plane rotation angle. Two more parameters describing the instrumental setup: wavelength of the radiation 5The user is prompted to select graphically the center of each peak to be masked out with a circle of diameter 5, 9, 15 or 27 pixels and sample-to—detector distance. All of these six parameters need to be determined prior the integration of the 2D image plate —— recording of powder diffraction data -— can be performed, which gives 1D data of intensity versus diffraction angle 26’ or d-spacing can be obtained. Each time the sample-to-detector distance or the wavelength of the radiation is changed, a new set of these six parameters is required. For a series of measurements, which were all made using the same setup, it is sufficient to do the calibration only once and then use the values obtained in the calibration for all other measurements. This applies to all measurements where the position of the sample, the wavelength and the alignment of the detector were not changed. There are two different ways to determine these parameters in FIT2D, as we father explain. B.3.1 Calibration using an external standard All six parameters can be (.letermined in single step using an external standard, i.e. using a material whose lattice parameters and symmetry are known, to be able to use the procedure described here, an additional powder diffraction experiment has to be performed (calibration measurement) in which a sample is the standard material. The instrumental conditions must be the same for the calibration measurement and the actual experiment. Therefore, each time the sample-to—detector distance or the wavelength of the radiation is changed, a new calibration measurement is required. limitation to calibration standards exist. F IT 2D only allows standard materials, for which information about the indexing was coded into the program. The list currently includes A1203, LabG, NaCl (beware of moisture), C802, silicon and paraffin wax. For best results a very high quality powder sample should be used, and well exposed image should be obtained. However dont expect high accuracy using paraffin wax. For more information, see Section 11.5 Ref. [196]. After the image plate recording is read into the program and the corrections for parasitic diffraction phenomena (Section 3.2) have been performed, the ”calibrant” subroutine is invoked and the standard is specified. The user has to provide starting values for the wavelength of the radiation and the sample to detector distance and one can select the parameters to be determined in the calibration procedure. However, the wavelength and the sample-to—detector distance are highly correlated and they cannot be determined together if a cubic standard material, e.g. silicon or LaB6, is used. Therefore one of them, usually the wavelength, is measured by different methods and it kept fixed. At the MUCAT beamline at the APS, the wavelength calibration mea- surement is usually performed by the beamline scientist. This measurement involves a time-consuming procedure and usually a single wavelength is used for all experi- ments. The sainple—to—detector distance can be obtained by measuring it manually using a ruler, the value obtained in this way is sufficiently accurate to be used as a starting value for calibration routine. During the calibration procedure, the position of the innermost (well formed) powder ring of the standard needs to be marked on the diffraction image by selecting several points of the ring in the graphical representation of the image on the com- puter monitor. FIT2D only needs a few points, but to increase the accuracy, greater than the requested number of points should be given. About ten points appears to be sufficient. The points should be distributed evenly over the powder ring. The procedure continues automatically and the program calculates and suggests two sets of parameters, which are written to the terminal (in the UNIX version). The user is required to select one, usually the one which has smaller standard uncertainties. One must be aware that the calibration is strictly correct only for the standard measurement. When the sample is replaced, the sample-to—detector distance can change by an unspecified, although very small amount comparing with the value obtained during the calibration, small systematic shifts of the diffraction angles and the lattice parameters can be introduced. This does not apply to the relative shifts of lattice parameters, which were obtained from a series of measurements taken at different temperatures or pressures, as long as the sample has not been moved. For RAPDF measurements of a sample series using the same experiment setup, it. is recommended to perform calil.)ration measurement every period of time, say every 12 hours, to ensure the same parameters are constantly obtained. B.3.2 Standard-free calibration If a calibration measurement does not exist, the four parameters describing the geom- etry of the Debye—Scherrer rings can be determined in FIT2D for any diffraction image. from the image itself. However, the wavelength of the radiation and the sample-to- (letector distance need to be known beforehand (in advance). The alternative calibra- tion procedure is invoked by the ”tilt” command and the information required on the first two screens of these routine has to be provided by the user. Several methods to find the beam center are available here, with the ”circle coordinates” or the ”ellipse coordinates” procedures being most appropriate. Both routines work in the same way. As a first step, one has to identify a powder ring in the image. The procedure is similar to specifying the innermost powder ring in the calibration using a standard. Here, however, it is possible to select any powder ring, so a non-overlapping strong ring should be selected, which is completely on the image. After an appropriate num- ber of points on the ring and a cutoff range, which specifies two limit radii between those the ring is found, additional powder rings can be indicated to be included in the calculation to obtain higher accuracy. The procedure concludes automatically and presents two solutions. From these solutions the better one, having smaller errors, should be chosen. B.4 RAPDF data integration B.4.1 Debye-Scherrer rings geometry After completion of the data reduction (section 3.2) and obtaining corresponding calibration parameters (section 3.3), the next step is to perform data integration. The data integration is a process of obtaining intensity versus diffraction angle 20 from a two dimensional RAPDF raw data as shown in Fig. ??. The integration of two dimensional image plate data is made along circles with radius 7‘ around the center of the Debye-Scherrer rings for (r) values ranging from zero up to the border of the detector. The zero here is the position of the center of the Debye-Scl'ierrer rings on the image (denoted as beam center) and is represented by two parameters (.170 and yo coordinates). The ideal case (integration along circles) only applies if the detector is perfectly aligned perpendicular to the incident beam. Normally this is not the case and the Debye-Scherrer rings on the image are slightly distorted and thus have an elliptical shape. The integration has to be made along curves that represent the shapes of the Debye-Scherrer rings. The sizes of the ellipses are given by one independent variable: an effective radius (7) that replaces the radius of the circles. Two parameters are required to describe the shape and the orientation of the integration ellipses. One of them is the angle between the principal axes of the ellipses and the coordinate axes of the image in the detector plane (tilt plane rotation angle). The other parameter is the angle between the normal of the detection plane and the direction of the incident beam (tilt angle). If wrong values for these two angles or the beam center coordinates are used for the integration, the Bragg peaks in the integrated powder pattern would suffer from peak broadening, irregular peak shapes, and incorrect intensities. 159 B.4.2 Integration parameters All four geometric 1.)arametcrs have to be known prior to the integration. They can be determined using the shapes of the Debye-Scherrer rings, i.e. information present on the diffraction images. Therefore the integration of the diffraction images is pos— sible without any additional inforn'iation, but then it would provide data of intensity versus effective radius (r). This data format needs to be converted to intensity versus diffraction angle 26 or d-spacinG, in order to be used in PDFgetX2 [90] or Rietveld refinement crystallogratmic programs. To achieve the conversion, two additional parameters are required: the wavelength of the radiation and the distance between the sample and the detector. To determine these two parameters, additional information, not present in the image, is required - the lattice parameters of the measured standard compound. Errors at the stage of conversion introduce a systematic shift to the positions of the Bragg peaks in the integrated pattern and this systematic shift also affects the lattice parameters. The intensities of the Bragg peaks are unaffected. In the next section it will be described, how all six geometric parameters can be obtained with FIT2D. B.4.3 Integration Process The goal of this part of the data processing is to obtain a two column data set equiv- alent to the conventional angle dispersive scan, where every data point represents the sum of scattering intensities over the same solid angle. In the image plate data different Q values have different number of corresponding pixels. This necessitates an additional geometric correction to properly normalize the data. The X-ray polariza- tion correction should also be done at this step, because it depends not only on the distance of the pixel from the beam center, but also on the azimuthal angle. Data integration can be straightforwardly done using FIT2D with the appropriate choice of above corrections. The integrated data should be saved in “C HI” file format. It is 160 also recommended to save data in the GSAS [122] format needed for later Rietveld analysis. For each sample, typically multiple exposures are taken to improve counting statistics. Since multiple data sets are collected under identical conditions, they are summed directly together to get the total counts. To avoid excessively large values, we divide the total counts by the number of exposures . Automatic integration of multiple data sets is possible with the F IT‘ZD GUI macro functionality. This becomes necessary when there is a large number of data files to integrate, as is typically the case in the RA-PDF experiments. A more efficient way is to use the keyboard mode in which program FIT2D can be driven by command line inputs. In this case, one can put all the commands in a macro file, and give the macro file name to FIT2D to achieve the same result. Certainly the macro file should contain information such as the required processing parameters and the related image plate data file names. Many samples are usually measured with the same experimental setup, thus only the data file names need to be changed in the macro file. In consideration of this, the macro file is designed to take parameter values from variables, and the variable values can be passed when calling FIT2D (for more details see appendix A) . B.4.4 Normalization by the monitor counts The last step of the data preprocessing before using PDFgetX2 is to normalize the data by the total monitor counts. This is very important because sample and back- ground are usually measured with different exposure time. Even with the same ex- posure time, the incident beam intensities are most likely still differth due to the self-decay and re-filling processes of the synchrotron storage ring. The normalization constant should be the number of counts at the incident beam monitor that is closest to the sample. There should be no beam size defining slits/collimation’s after this 161 monitor, otherwise its reading does not reflect the real incident beam intensity on the sample. In the case of multiple image plate data being averaged, the corresponding total monitor counts should also be averaged,i.e. divided by the number of data files (exposures). Once normalization is done, we are ready to use PDFgetX2 in order to obtain the PDFs (see the next chapter ). ° Mask Peoks(5) Usgd . 8 Threshold Mask Used in ‘ No Mask Used bWs. 9 1 _ N I o i ll 3 ll - ’3 fl 3; 33 S - _ i‘fi El 0 'a m . s S . S .. 3' [flay '0 1 4 5 6 7 a 9 1o tth (deg) (b) Figure B.2: Data from Fig. B.1 shown on an expanded scale. (a) Overexposed pixels are appearing on the inner rings (small green dots). The small white area in the center of the image is a shadow cast by the beam stop assembly. The red cross denotes the center of image plate. The red line from the center outward is representing the scanned radial direction which is equivalent to the .L‘ axis in right. panel. (b) The integrated one dimensional data of the image shown in the left panel. Data integrated for different mask tools: mask peaks(5)3 (blue crosses), treshold mask (red empty squares), and no mask used (green solid line). 163 Appendix C RAPDF Data Analysis RAPDF data analysis here refers to the procedure of obtaining the total scattering function, S(Q), and the atomic pair distribution function (PDF), 00'), from the normalized integrated RAPDF data (Section 3.4). For this the data needs to be corrected using the standard methods [$18. 6]. Fig C.1 shows steps in the analysis of Ni RAPDF data to obtain the I (Q) ._ F (Q) and C(r) after applying the necessary corrections to the integrated raw data shown in Fig. A.6. Data corrections are done using PDFgetX2 software package [90]. C. 1 Data Analysis The conventional way to obtain PDFS typically starts from the SPEC file format. data collected with a solid state detector (SSD), and is described in detail in the PDFgetX2 program tutorial [90, 197]. A simplified description is provided here. The raw data. from the SSD are in SPEC data. file format. The program PDFgetX2 [90] is used to preprocess the raw data to obtain the intensity versus 29 or Q (two column data). In the preprocessing, the measured scattering intensities were first corrected for detector dead time (e.g., 0.35 as), then normalized by the corresponding incident beam intensities (monitor counts collected by one ion chamber right before the sam- 1611 $ I I I . I I I o a .— c, H A _ l0 :3 H _ 30 _ H H _. Lo _. O l l I L l I l I I I I I I I o_ b _ AN <) I _ _ ss- — o I I . I I I . I . I . I . I 4 8 12 16 20 24 28 —1 Q (13- ) w I I I I I I I I I I I lf‘li .. 1 m (\I’P—Ij (c) _ on: co— - v — —I <0; I ' . I I I I I I I m_l . I . I — 4 8 12 16 20 24 r (A) N cn Figure 01: (a) The corrected experimental intensity of the Ni raw data shown in Fig. A.6, (b) the experimental reduced structure function F (Q) = Q(S(Q) — 1) of data in (a), (c) the experimental 0(7') obtained by Fourier transforming the data in (b) with Qmuxcut of 30.0 A‘ . 165 plc). Further, all the scans (with different. ranges) were merged together to yield the whole range. The same procedure was repeated on the background scattering data. After the preprocessing, the background data are subtracted from the sample data to obtain the sample only scattering intensity versus 29. Further, corrections are applied for the sample absorption, polarization, and Compton scattering to obtain the elastic scattering only (for more details about these corrections see Chapter 5 of Ref. [18]) . The last step is to normal- ize the elastic scattering by the Q dependent average scattering power of the material to get the total scattering structure function S (Q), which is then transformed to ob- tain the experimental C(r) using the formula: C ('1') = % 100C Q[S (Q) — 1] sin Qr dQ. The data coming from the image plate are two dimensional images, and thus need to be integrated first to give the multiple column ASCII data (Section 3.4). One column should be the intensity, while the other should be either 26 or Q. The integrated data can then be fed to program PDFgetX2 to obtain the PDF as described al_)ove. C.2 Additional corrections to the RAPDF data The RAPDF data collection is much quicker than the conventional one dimensional data collection (three to four orders of magnitude faster) and requires a simpler experimental setup. However, the data analysis becomes more difficult, in addition to the extra steps required to integrate the image plate data. The characteristics of the image plate as well as the high background levels require some additional corrections. We will focus on two aspects, the lack of energy resolution and the energy dependent response. 166 C.2.1 Oblique incident angle This correction accounts for the angular dependence of the scattered photon effective path length in the IP phosphor layer [121]. as a direct result of its incomplete ab- sorption of the scattered photons. This correction becomes very significant at high X-ray energies and large incident angles (both present in the RAPDF experiments). Two parameters are used here. The scattered photon absorption coefficient of the IP phosphor layer and the incident angle (equal to '20). The X-ray energy used in our experin’ients is highly penetrating with the absorption coefficient less than 1%. 0.2.2 Compton intensities at high Q and fluorescence The image plate counts only give the detected X-ray intensities, with no knowledge of the energy of the X-ray. The lack of energy resolution makes it impossible to distinguish between elastic, Compton, and fluorescent photons. This causes rather serious problems for PDF analysis. In the low Q region, elastic scattering is usually the strongest signal, thus removal of the parasitic Compton and fluorescence intensities does not introduce significant error into the analysis. However, elastic scattering decays exponentially with Q; Compton intensity increases with Q; and fluorescent intensity is constant. As a consequence, in the high Q region (QmaxZ 30.0 A‘l), the situation is quite different. For example, even in the case of Ni (2:28) atoms, the Compton scattering is about 4 times as large as the elastic scattering at Q = 30 A"1. This ratio increases with decreasing Z values. The fluorescent intensities can be efl’ectively suppressed by going to higher incident. X—ray energies for low Z elements or just below the edge for high Z elements. However, for the medium to high Z elements, the current routinely achievable highest energy X-rays still give rather significant. fluorescent intensities. Therefore, the extraction of elastic signal in the high Q region is not a trivial task. Small errors on the estimation of Compton or fluorescent intensities would result in significant deviations to the elastic signal. 167 Another complication is the statistical uncertainty of the extracted elastic intensities, as the uncertainty of both Compton and fluorescent intensities will be added to it. To achieve the same counting statistics as when measuring elastic scattering separately, 2 the total counts from the image plate would have to be ~ n. times as many (if elastic scattering accounts for 1/n of the total intensities). C.2.3 Energy dependence of the IP response Due to the detection mechanism of the image plate, X-rays with the same mnnbcr of photons but (:lifferent energies will result in different counts. Though elastic scattering has the same energy at all Q values, the energy of Compton scattering decreases with increasing Q and the fluorescence intensities usually comprise of several characteristic energy lines. The high Q region data again are sensitive to this effect due to the dominating Compton and / or fluorescent intensities. This energy dependent detection efficiency is not. a simple linear function, and in fact is difficult to measure accurately. In program PDFgetX2, either a linear or quadratic empirical formula is used as the energy dependence of the image plate detection efficiency while ensuring the detection efficiency decreases with irnrreasing X—ray energy. C.3 Estimate of standard deviations Proper estimate of the standard deviations (BSD) and their propagation along the data corrections are rather important, as the experimental PDFs are commonly used in least square regression programs for structure refinement [158, 157]. For X-ray photon counting detectors, the BSD of the observed count (after dead time correction) is simply its square root. when the count is large (more than a few hundreds). However, the IP detector is not a photon counter. Quite a few experimental factors contribute to the relative uncertainties of IP response, such as the read-out system, and the non— 168 " 2 5 ‘1 l' l 1,0 "19 .5. ~,. E3 ‘3\\\\\ g 2 b X \\\ 3 \g\ E \‘N\ E l ' \\\\\ \U§g\ (I) \\ g 0.5 . \\\\ \35x :22 3 4 \:fl__ _1 l 10 105 107 Photon Number Absorbed in NxN pixels Area Figure C2: Estimated relative standard deviations of the IP response among areas of N x N pixels as a function of photon number absorbed in the area. The scale of N is shown in the upper part. Open circles are for the drum scanner and crosses for BASQOOO. Dashed lines represent the relative standard deviation of the photon number absorbed in the area. Taken from [120]. uniformity of the IP, etc. [198, 199]. To complicate the situation, in the published literature, the ESD is calculated with respect to the noise level of the IP system under study [200, 1‘20]. Fig. C? shows the ESD of the IP response as a function of exposure level. It was found that. the relative ESD of the IP response is rather close to an ideal photon counter at low exposure levels, while it saturates to a little less than 1% at. high exposure levels. The IP response relative to the noise is found to be close to 1-2 per incident photon at the high energies of our interest. Therefore, based on the specifications of the IP system in our experiment (Mar345, sensitivity of 1 X-ray photon per ADC-unit at 8 keV, intrinsic noise 1-2 photon equivalents), we approximate the ESD of each pixel count to be the larger value between its square root and its one hundredth. Once the ESD of each pixel is determined, we can propagate the ESD through the 169 integration from the two dimensional raw image and geometric corrections. However, program FIT2D does not generate/ propagate ESDs, and only gives the two column data: intensity versus the scattering angle ‘26. Thus we need to correctly retrace the procedure done in F IT2D to obtain proper ESDS. The steps are the following. The integrated data from FIT2D are the averaged intensities among the pixels within each bin grid corrected by the geometric factor. As each pixel has the same area, the geometric factor is simply cos3(26). So the first step is to multiply the intensities by cos3(20) to get the averaged total counts. Then, we need to compute the number of pixels in each bin grid. For the scattering angle bin width of (529, the area for this bin on the IP is approximately 27rd2 sin(20)/ cos3(26) - 620, where d. is the sample to detector distance. Dividing the area by each pixel size gives the number of pixels. Given the integrated count c and pixel area A we obtain pixel, (/c . cos3(29) EASD = (C‘l) \/27rd2 sin(26)/ cos3(20) - 629/AI,.,jICl/ cos3('26)l noting that ‘/c - cos3(26) should be replaced by 0.01 'c-cos3(20) if the latter is larger due to the 1% saturation of relative uncertainties of the IP. The resulted PDF, C(r), is in a compatible file format for the PDF refinement program PDFFIT [94]. Please refer to the user’s guide of the program PDFgetX2 [197] for details. 170 Appendix D Image plate characteristics as an X-ray detector D.1 Use of the image plate as an X-ray detector The detection media of an image plate (IP) is one layer of very small crystalline grains of photo-stimulable phosphor mixed with organic binders. The grain size is about 5 run; the layer thickness is usually around 150 am. There is usually a protective layer (about 10 pm) on the top and one support layer on the bottom (about 250 pm). A polyester backing plate is often used to make it reusable and easy to handle as a flexible plastic plate. The IP’s sensitive surface can be easily identified by its all-blue- white or all-white appearance, while the backingr side is usually black or gray. D.2 Principle of X—ray image plate detection The photo—stimulable phosphor used nowadays is BaF(Br,I):Eu2+ (previously it was BaFBrzE112+) [201]. This material is capable of storing a fraction of the absorbed X-ray energy, and emitting photo—stimulated luminescence (PSL) later when stimu- lated by visible laser light. The mechanism of the PSL is illustrated by the energy 171 level scheme of BaFBrrEu2+ shown in Fig. D.1. The energy level of interest is the Eu2+ ground state, 6.5 eV below the conduction band, and the vacant F+(Br+, 1+) centers 2.0 eV below the conduction band. The ground state has E112+ and vacant F+ centers. The incident X-ray will pump electrons from the valence band to the conduction band. Despite being a complicated process, the X-ray irradiation results in a certain number of Eu3+ and F pairs proportional to the absorbed X—ray energy. The F centers are caused by the absence of halogen anions from their designated positions in the lattice. The energy trapping state of F centers is meta-stable with a long lifetime. However, the trapped electrons can be easily excited by visible light (2 2 eV) to return to the conduction band. One of the occurring processes is the re- combination of Eu3+ with one election to E112+, along with the emission of a photon of 3.2 eV (blue light). A red He-Ne laser is usually used for read-out. Its wavelength (632.8 nm) is considerably separated from the PSL wavelength (390 nm). A con- ventional high-quanturn-efficiency photo-multiplier tube (PMT) is used to collect the photostimulated photons. The signal is then amplified and digitalized to be processed by computers. The remaining F centers in the phosphor after read-out can be further erased by exposing to visible light. In practice, it is more efficient to bleach it with a powerful halogen lamp (500W). D.3 Characteristics of image plate detectors The performance of the image plate as an X-ray area detector has been reviewed by Amemiya and others [203, 201, 202, 198, 204, 205, 199, 206] in great detail. Here we briefly summarize those characteristics for completeness. Detective quantum efficiency The definition of detective quantum efficiency (DQE) is DQE = (.S‘O/i’\f0)2/(Sz-/N,-)2 where S is signal and N is noise (the standard deviation of the signal), and subscripts o and 2' refer to the output and input, 172 conduction band Pfi 6.5 eV 4.6 eV 33 eV 3.2 6V V t l Eu2+ “A Eu 3+ ‘-_- valence band Excitation Figure D.1: Energy level scheme of image plate phosphor material BaFBr:Eu2+ pro- posed in Ref. [202]. Arrows indicate the excitation and PSL process as suggested by Takshashi et al. [203]. 173 respectively. As scattering events are random following the Poisson distribu- tion, the (Si/N21)2 term is just the Si- The background noise level of the IP is usually less than 3 X—ray photons/ (100 ,um)2, which in practice largely de- pends on the noise level of the IP readout system. The relative uncertainty of the IP deviates from an ideal detector at high exposure levels (2 100 X- ray photons/ (100 ,um)2). The “system fluctuation noise” saturates the relative uncertainty of the IP results around the 1% level. The origin of the system fluctuation noise includes non-uniformity of absorption, non—uniformity of the color-center density, fluctuation of the laser intensity, non-uniformity of PSL collection, and fluctuation of the high-voltage supply to the PMT. DynaJnic range and linearity of the IP response The dynamic range refers to the detectable weakest and strongest signal with acceptable distortion. One dimensionless definition is taken as the ratio between the maximum counts in the linear regime and the lowest detectable signal (determined by the intrinsic noise of the detector system). The IP has rather good dynamic range of 105:1, with linear response range from 8x 101 to 4x 104 photons/ (100 ,um)2. The error rate is less than 5%. Sometimes, two sets of PMTs are necessary to cover the entire dynamic range of the IP. Spatial resolution and active area size The determining factor of the spatial res- olution of the IP is the laser-light scattering in the phosphor during the readout. The laser-light scattering originates from a mismatching of the refractive indices at the boundaries of phosphor crystal grains. When the IP is read with a 100 pm square laser size, the spatial resolution is limited by the linear spread function with 170 pm full width at half maximum (FWHM). The point spread function of IP systems has been studied by Bourgeois et al. [200] in detail. The active area size of the image plate is rather important for PDF studies which need large IPs. Commercially available IPs have various standard sizes ranging from 1 74 127x127 mm2, 201x252 mm2, to 201x400 1111112. The largest automated IP system currently contains the 345 mm diameter IP disk (MAR345). Energy dependence The measured intensity on the IP is proportional to the de— posited energy in the phosphor layer. This depends on the absorption efficiency per incident photon and the deposited energy of one absorbed photon. Unfor— tunately, the IP response is energy dependent, and its energy dependence is shown in Fig. D.2 from 4.0 to 60.0 keV. We are only concerned with the IP response per incident photon (lower curve). Except for the discontinuity at the Ba, K edge, the detection efficiency decreases with photon energy in the high energy regime of our interest. This necessitates an energy dependent detection efficiency correction discussed later in this chapter. Fading and other factors The IP stores the absorbed energy in the meta-stable F centers and thus will fade with time after exposure. Exposing the IP to visible light or high temperature will expedite the fading. At room temperature, the IP is usually stable within one hour. The fading lifetime does not depend on the X-ray energy during exposure, as it is the F centers that store the energy. The surface roughness and the non-uniformity of the IP is about 1-2%. A calibration image is usually obtained by exposing the IP to a flood field from a radioactive source, and then used to correct the recored images. In principle, the IP, being an integrating-type counter, is not counting rate limited. However, extremely intense X-rays seem to cause permanent damage to the phosphor layer. With the exercise of common precautions, the IP has proved to be a very reproducible and reusable X-ray area detector. To summarize, the IP detector system meets various requirements of X—ray area detectors. They are, high detective quantum efficiency, a wide dynamic range, a linearity of response, a high spatial resolution, a large active area size, and a high 175 IP response (D.S.U) 2 ‘Q . O o A l . L A g n l L l A I n I O 8 1 6 24 32 4O 48 56 Energy (keV) Figure D2: The IP response as a function the the energy of an X-ray photon. (i) is the IP response per incident X-ray photon. (a) is the IP response per absorbed X-ray photon. The unit of the ordinate corresponds toe the background noise level of the IP scanner (the data were digitized from the the corresponding figure in reference [120]). counting rate capability. D.4 The IP and PDF method Recent developments have shown the utility of 2D detector technology in scattering studies of liquids. A recent report by Crichton et al. [116], has made use of integrated two-dimensional IP data for in-situ studies of scattering from liquid 08882. This study, and others more recently, demonstrate the feasibility of using IPs for diffuse scattering measurements though the measurements have been limited in real-space resolution. with Qmax S 13.0 A71 [117, 118], making thorn less suitable for the study .1 76 of crystalline and nanocrystalline materials. Image plates have also been successfully used to study diffuse scattering from single crystals [119]. A Debye-Scherrer camera utilizing IPs has also been tested in our group and shows promise for lower energy X—ray sources such as laboratory and second generation synchrotron sources [1111]. Successful application of IP technology to the measurement of quantitatively re- liable high real-space resolution PDFs requires that a number of issues be resolved. For example, it is necessary to correct for contamination of the signal from Compton and fluorescence intensities and for angle and energy dependencies of the IP detec- tion efficiency [120, 121]. Here we show that high quality medium-high real-space resolution PDFs can be obtained by applying relatively straightforward corrections. As expected, the quality of the PDFs is lower in samples comprising predominantly low atomic-number elements. Nonetheless, even the PDFs of these samples prove admuate. Further quality studies were also carried out to investigate the RA-PDF capabilities. We also compared the IP data with the conventional solid state detector data to verify that quantitatively reliable structure information can be obtained from the RA-PDF method. 177 Appendix E The PDF of a single spherical particle Finite size effects of nanoparticles on the atomic pair distribution function have been coverd in detail by Kodama et al. [127]. The materails here have been adapted from [83]. The microscopic pair density gives a distribution of atomic pair distances r in a sample, weighted by the pair’s scattering lengths pm = 4,32,, ,2 like — m). (El) #2 The following method of constructing p(r) will be useful in determining its form with respect to spherical particles. For each 7", define a spherical shell with this radius. Let the center of this sphere coincide with the position of an atom 2', and record as weighted 6 functions every atom j that intersects the spherical shell. Finally, divide the result by its surface area and the total number of atoms N. The calculation of [)(r) for a spherical particle limits the position of the center of the sphere to the atoms within the particle itself, whereas the sphere’s surface can extend beyond. Note that p(r) for a single particle may only be a part of the total microscopic pair density of 178 a solid or solution. The contribution to p(r) of all atomic pairs {1 j } within a spherical particle is limited to the range 0 < 71,-]- < 2R, where R is its radius. If the particle is in solution, or we consider an ensemble of identical particles with random orientations embedded within a host lattice, then p(’r) 2 p0 for r > 2R, where p0 is the constant atomic number density outside the particle. To simplify the following, let p0 be equal to the number density of the particle itself, or take p0 = O for empty space. The essence of the problem addressed hereafter is to quantify the relative population of atomic pair distances between any two atoms within the particle and pair distances where one atom resides outside of the particle. That is, p(r) will have r-dependent contributions from both the microscopic pair density pc(r) of an infinite crystal and the uncorrelated outside structure p0, Pf”) = fel'ra Rl/)c("') + l1 ‘ fe("‘~. Rll P0: (B2) where 0 S fe(r, R) S 1. When R —+ 00, fe(r, R) = 1 for all 'r and p(r) is that of an infinite crystal. W'hen R —> 0, fe(1', R) = 0 for all r and p(r) = p0. fe('r, R) is the envelope function that we now derive when R is between these limits. Consider a point within the particle whose position is given by the vector r', with the center of the particle defining the origin. Orient the point and the particle so that. r’ aligns with the z axis, as shown in Fig. E]. A spherical shell of radius r around this point will either be enclosed by the particle if r < R — 7", intersect the surface of the particle if R — r’ S r S R + 'r’, or enclose the particle if r > R + 7". When R — r’ S r g R + 7", a line from any point on the circle of intersection and the position 1" will meet the z axis at an angle a. The fraction of the surface of radius 'r around this point that is enclosed within the particle is I 1 27f 7r f('r ,7‘,R) = i/ rdo/ rsinOdO / 7rr 0 an 179 Figure E. 1: A spherical particle with radius R. An atom a distance r' from the center of the particle can have a shell of radius 7" that is only partially embedded within the particle. = $(1+ cos a). (B.3) Using the two right triangles from Fig. E1, the angle a can be expressed as R2 _ 'I‘l2 __ 7.2 21")" cos a = (E4) The contribution of all such spheres enclosed within the particle is obtained by in- tegrating f (r' ,r, R) over the remaining positions 7" in the region occupied by the particle, taking care to consider when the shell of radius r extends outside the region. Using Eq. BA, for 7' g 2R, R 47r f(r, R) = 3(R—r)3+41r/ f(r,,'r,R)r"2 (11" R—r 47r 3 3r 1 'r 3 = _ _ __ __ _ _ E. 3R]1 4R+16(R)] (5) 180 Finally, dividing by the total particle volume gives the envelope function ‘ r r 3 fe(7',(1) = [1— 3:2 + $- (71) ] 9((1—1‘), (E6) where d 2 2R is the particle diameter and 9(3) = 0(1) for negative (positive) .r is the Heaviside step function. f¢(r, (1) and its derivative are continuous for all positive r. 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