Aula XRD V02

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Introduction to 
X-Ray Diffraction 
With Applications to 
Carbons and other Materials 
Summary 
• Crystal structures 
• X-rays 
• Instrumentation 
• Diffraction and Bragg’s Law 
• The diffractogram 
• Applications 
 
3 
• CRYSTAL 
– A solid composed by atoms, ions, or molecules, arranged in 
a three-dimensional periodic array 
 Sólido composto de átomos, íons ou moléculas, arranjados 
em um padrão periódico em três dimensões. 
 (Cullity & Stock, Elements of X-Ray Diffraction) 
 
• The unit cell 
 Smallest repeating unit possessing the same symmetry of the 
crystal 
 Menor unidade de repetição que possui a mesma simetria da 
estrutura cristalina. (West, BSSC) 
– 1D case 
– 2D case 
4 
• The unit cell 
Example: sodium chloride (a.k.a. rock salt or halite) 
5 
• The unit cell 
6 
• The unit cell 
7 
• The unit cell 
8 
• SYMMETRY 
– Symmetry operations and elements (West, BSSC) 
 
Link for introductory concepts (English): 
http://www.materials.ac.uk/elearning/matter/crystallography/3dcrystallography/index.html 
9 
• Seven crystal systems 
10 
• 14 Bravais 
 lattices 
11 
• Space symmetry elements 
 
– Screw axis 
 (Eixo helicoidal ) 
 
 
– Glide plane 
 (Plano de reflexão-translação ou plano deslizante) 
12 
• 230 Space 
groups 
13 
• Directions 
[ u v w ] 
 
• Planes (families of planes) 
 ( h k l ) 
14 
• Other examples 
15 
• Exercício 
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– Obs.: The symbol { } is used to indicate sets of 
equivalent crystalographic planes. 
 
– For example, the set of planes (100), (010) and 
(001), in a cubic system, are collectively 
represented as { 100 }. 
 
– The equivalent directions can also be represented 
collectively; in the cubic system, for example, the 
directions [100], [010], and so on, are represented 
as < 100 >. 
17 
• Miller indices 
 
 
• Distance between planes 
– Perpendicular distance between any two consecutive 
planes 
– Importance → Bragg’s Law (diffraction) 
– Examples 
• u.c. for NaCl: a = 5.625 Å; 
• Calculate d for the planes (200), (111) and (123) 
– For other crystals, particularly monoclinic and 
triclinic, the formulas are more complicated... 
18 
• Formulas for the 
interplane distances 
(d-spacing) and u.c. 
volumes. 
The nature of X-rays 
The Electromagnetic Spectrum 
• Dual behavior 
 (quantum mechanics) 
• Absorption by matter 
• Difraction and interference 
• Wavelength: 
– Visible ~ 4000-7000 Å 
– X-rays ~ 1 Å 
 
X-rays – emission 
Continuous and Characteristic Spectrum 
• Beam of electrons hitting a 
 metallic target: deceleration of 
 the charges generates X-rays 
• Electron energy: 
EK = eV = ½(mv
2) 
– For V = 30 kV 
 → v ~ ⅓ c 
– < 1% of EK converted to X-rays 
• Continuous spectrum (white 
 radiation or Bremsstrahlung) 
• Short wavelength limit 
X-rays – emission 
Continuous and Characteristic Spectrum 
• Beam of electrons hitting a 
 metallic target: deceleration of 
 the charges generates X-rays 
• Above a certain value of V 
 (characteristic of the metal) 
 → Appearance of well- 
 defined maxima 
• K, L, M, etc sets 
 - Electronic transitions 
X-rays – emission 
Continuous and Characteristic Spectrum 
• Beam of electrons hitting a 
 metallic target: deceleration of 
 the charges generates X-rays 
• Above a certain value of V 
 (characteristic of the metal) 
 → Appearance of well- 
 defined maxima 
• K, L, M, etc sets 
 - Electronic transitions 
Wavelength dependence 
→ Dependence with atomic number ( Moseley: Z↑ => λ↓ ) 
Element λ k(α1), Å 
 Fe 1.936 
 Ni 1.658 
 Cu 1.5406 
 Mo 0.7093 
 Ag 0.559 
 W 0.209 
 Au 0.180 
Generation and Instrumentation 
(less than 1% of electron kinetic energy is converted to X-rays) 
Schematics of an X-ray tube 
Generation and Instrumentation 
- Detection: 
 Electronic detectors 
 Photographic films 
 Fluorescent screens 
• Difference in optical path 
– Phase difference 
– Change in emerging amplitude 
• When the path difference equals 
na integer numbher of 
wavelengths constructive 
interference takes place (the 
amplitudes are added) 
• When the path difference equals 
half wavelength, there is 
destructive iterference 
• Path differences happen naturally 
when X-rays interact with crystals 
(diffraction) 
Diffraction 
• A crystal can be viewed as a 3-D diffraction grating 
• For constructive interference: 
2d´senθ = nλ Bragg’s Law 
• Scattering can take place in 
several directions 
• Certain directions may lead 
to phase relations 
λ = 2d senθ 
More convenient: 
Reflections of any 
order are considered 
as 1st order for planes 
with d = 1/n 
Diffraction 
Diffraction 
Diffraction 
• Exercício: Determinar, em uma rede 2D quadrada, a expressão que 
relaciona a separação interplanar com os índices de Miller. 
 
 
 
 
 
• Expressão geral para sistemas com eixos ortogonais: 
 
 
 
• Exercício: Sabendo que em um cristal cúbico a reflexão dos planos (1 1 1) 
(Cu Kα = 1.54 Å) ocorre a 11.2°, determinar a. 
 
 a = 6.87 Å 
Positions and Intensities 
• Given the diffractogram below: 
What information can be extracted? 
Positions and Intensities 
• Simple patterns usually indicate high symmetry 
• Use Bragg’s Law to determine the values of d 
• Look for systematic absences 
 → Possible to determine Bravais Lattice 
• Calculate values for sin2θ and try to index the reflections 
 (e.g. For a cubic system, sin2θ = (λ2 /4a2)(h2+k2+l2) 
• Be careful with similar scattering factors 
 Isoelectronic systems 
 Example: KCl (K, Z = 19; Cl, Z = 17) 
Positions and Intensities 
Systematic Absences 
- For centered cells, classes of 
reflections will be absent due to 
scattering taking place exactly out 
phase between certain atomic 
planes. 
- For primitive cubic cells, all integer 
values of h, k, l are possible. 
Systematic Absences 
- For centered cells, classes of 
reflections will be absent due to 
scattering taking place exactly out 
phase between certain atomic 
planes. 
 
The specific values of 2θ and 
d will generally be different 
according to the crystal. 
Systematic Absences 
- For centered cells, classes of 
reflections will be absent due to 
scattering taking place exactly out 
phase between certain atomic 
planes. 
- Specific symmetry elements may 
cause other, additional absences 
(translational symmetry). 
Systematic Absences 
Systematic Absences 
- Specific symmetry elements may 
cause other, additional absences 
(translational symmetry). 
• Back to the experimental diffractogram, and using these 
concepts: 
Positions and Intensities 
Positions and Intensities 
• Back to the experimental diffractogram, and using these 
concepts: 
XRD for rock salt, a = 5.64 Å 
Structure: fcc 
Exercise: Utilize sin2θ and assemble the table. 
• Exercícios: 
Seja uma cela cúbica de corpo centrado com a = 5 Å. Calcule o difratograma. 
 
 
 
Suponha que o composto acima sofreu um alongamento de 5% ao longo de c. 
Como ficaria o novo difratograma? 
Positions and Intensities 
• Why do the diffractograms change so drastically when 
the crystal symmetry changes? 
BaTiO3 - Cubic 
BaTiO3 - monoclinic 
Positionsand Intensities 
• Multiplicity 
– For cubic systems: (120), (210), (012), (201), etc... 
 ⇒ same d 
– Peaks are undistinguishable 
a
lkh
d
2
222
2
1 

For example.: 
Cubic symmetry ⇒ maximum multiplicity (48) 
Orthorhombic symmetry ⇒ multipl = 8 ( a≠b≠c ) 
 
Difractogram with few peaks Difractogram with many peaks 
Positions and Intensities 
• Multiplicity - Exercise 
2-THETA h k l 
27.44 1 1 0 
36.08 0 1 1 
39.19 0 2 0 
41.24 1 1 1 
44.04 1 2 0 
54.32 1 2 1 
56.63 2 2 0 
62.76 0 0 2 
64.05 1 3 0 
65.51 2 2 1 
A rutila (TiO2) cristaliza no sistema P 42/m n m (136, tetragonal), com a = 4.59 
Å, c = 2.96 Å, e Z = 2. 
Dê as multiplicidades das dez primeiras reflexões da rutila, listadas na tabela 
abaixo. 
Positions and Intensities 
• Intensities: the structure factor Fhkl 
– Expresses the amplitude and phase of a reflection hkl. 
– It is the result of the waves scattered by all atoms in the 
unit cell, in the direction hkl. 
– Three factors 
• Direction 
• Amplitude 
• Phase 
– The interaction of the X-rays with the crystal happen 
through the electrons; the more electrons an atom has, 
the more strongly it will scatter X-rays. 
 → The scattering factor f0 
 → Depends on the atomic number, the Bragg angle, and the 
wavelength of the X-ray 
Positions and Intensities 
• Intensities: the structure factor Fhkl 
Positions and Intensities 
• Intensities: the structure factor Fhkl 
– Expresses the amplitude and phase of a reflection hkl. 
– It is the result of the waves scattered by all atoms in the 
unit cell, in the direction hkl. 
– Three factors 
• Direction 
• Amplitude 
• Phase 
– The interaction of the X-rays with the crystal happen 
through the electrons; the more electrons an atom has, 
the more strongly it will scatter X-rays. 
 → The scattering factor f0 
 → Depends on the atomic number, the Bragg angle, and the 
wavelength of the X-ray 
Positions and Intensities 
• Mechanisms of Interaction 
• Neutrons interact with the atomic nuclei via very short range 
forces (fm) 
• Neutrons also interact with unpaired eletrons via magnetic 
dipole interaction 
• Phase identification 
– Databases (ex. ICDD) 
– Search & match 
 
Applications 
• Crystallite size 
 
• The larger the “scattering region” (coherente), the better-
defined the diffraction peaks 
 
 
 
 
 
 
• The Scherrer equation 
 
 
Applications 
• Preferential orientations 
– Ex: MoO3 
 → cristal;izes as platelets 
 → tend to align with the substrate surface 
– Preparation of the sample and acquisition method 
Applications 
epswww.unm.edu/xrd/xrdbasics.pdf 
Applications 
• Residual Stress (tension) • Cristallinity in polymers 
Application - GIC 
Journal of Nanomaterials, vol. 2014 (2014) 
T Wada, T Yasutake, A Nakasuga, T Kinumoto, T Tsumura, M Toyoda 
Evaluation of Layered Graphene Prepared via Hydroxylation of Potassium-Graphite Intercalation 
Compounds (Open Access) 
Also: AFM, Raman 
Application - GIC 
Phys. Rev. B 24, 3505 (1981) 
S. Y. Leung, M. S. Dresselhaus, C. Underhill, T. Krapchev, G. 
Dresselhaus, B. J. Wuensch 
Structural studies of graphite intercalation compounds using 
(00l) x-ray diffraction 
Application - Graphene 
Chemical Communications 2010 
Zhigang Xiong, Li Li Zhang, Jizhen Ma, X. S. Zhao 
Photocatalytic degradation of dyes over graphene-gold nanocomposites under visible light irradiation 
Application - Graphene 
Journal of Electron Spectroscopy and Related Phenomena 195 (2014) 145–154 
L. Stobinski, B. Lesiak, A. Malolepszy, M. Mazurkiewicz, B. Mierzwa, J. Zemek, P. Jiricek, I. Bieloshapka 
Graphene oxide and reduced graphene oxide studied by the XRD, TEM and electron spectroscopy methods 
Also: TEM, XPS, REELS 
Application - Graphene 
International Centre for Diffraction Data 2012 ISSN 1097-0002 
Thomas N. Blanton and Debasis Majumdar 
X-RAY DIFFRACTION CHARACTERIZATION OF POLYMER INTERCALATED GRAPHITE OXIDE 
Application - Graphene 
International Centre for Diffraction Data 2012 ISSN 1097-0002 
Thomas N. Blanton and Debasis Majumdar 
X-RAY DIFFRACTION CHARACTERIZATION OF POLYMER INTERCALATED GRAPHITE OXIDE 
Application - Graphene 
International Centre for Diffraction Data 2012 ISSN 1097-0002 
Thomas N. Blanton and Debasis Majumdar 
X-RAY DIFFRACTION CHARACTERIZATION OF POLYMER INTERCALATED GRAPHITE OXIDE 
Application - Carbon Dots 
Sci. Technol. Adv. Mater. 13 (2012) 045008 
N Puvvada, BNP Kumar, S Konar, H Kalita, M Mandal, A Pathak 
Synthesis of biocompatible multicolor luminescent carbon dots for bioimaging applications