Diffraction Techniques and Microscopy PDF

Slide Set 2 Diffraction Techniques and Microscopy X-ray diffraction (XRD) Neutron diffraction (ND) check Electron diffraction (ED) Scanning electron microsc...

Slide Set 2 Diffraction Techniques and Microscopy X-ray diffraction (XRD) Neutron diffraction (ND) check Electron diffraction (ED) Scanning electron microscopy (SEM) Transmission electron microscopy (TEM) The Figures with figure numbers are from respective chapters in Materials Characterization: Introduction to Microscopic and Spectroscopic Methods, Yang Leng, Second Edition (Wiley-VCH, 2013) 1 Energy of Different Electromagnetic Radiations E = h = hc/ = hc h: Planck’s constant : frequency in s-1 or Hz c: velocity of light (m s-1) : wave length (m) : wave number in (m-1) 2  X-ray Diffraction (XRD) Most common and effective method to determine crystal structures X-rays are short-wavelength, high-energy beams of electromagnetic radiation X-rays are produced by high-speed electrons accelerated by a high-voltage field colliding with a metal target 3 Generation of X-rays 4  Generation of X-rays Quantum Electron X-Ray Degeneracy Numbers Subgroup Notation n l j K 1 0 1/2 1s 2 LI 2 0 1/2 2s 2 LII 2 1 1/2 2p 2 LIII 2 1 3/2 2p 4 MI 3 0 1/2 3s 2 MII 3 1 1/2 3p 2 MIII 3 1 3/2 3p 4 MIV 3 2 3/2 3d 4 MV 3 2 5/2 3d 6 NI 4 0 1/2 4s 2 NII 4 1 1/2 4p 2 NIII 4 1 3/2 4p 4 NIV 4 2 3/2 4d 4 NV 4 2 5/2 4d 6 NVI 4 3 5/2 4f 6 NVII 4 3 7/2 4f 8 OI 5 0 1/2 5s 2 Selection rule OII 5 1 1/2 5p 2 OIII 5 1 3/2 5p 4 n = ±1, 2, 3, … l = ±1 j = 0, ±1, but not 0 0 OIV 5 2 3/2 5d 4 ml = 0, ±1 OV 5 2 5/2 5d 6 5  Most Common X-ray Radiations Critical Anode Atomic Kα Excitation Optimum Material Number (nm) Potential Voltage (kV) (keV) Cr 24 0.22291 5.99 40 Fe 26 0.1937 7.11 40 Cu 29 0.1542 8.98 45 Mo 42 0.0710 20.00 80 X-rays generated by copper are approximately the following: λKα1 = 0.15406 nm λKα2 = 0.15444 nm λKβ = 0.13922 nm We often simply call Kα1 and Kα2 radiations the Kα doublet. The Kα doublet is the most widely used monochromatic X-Ray source for diffraction work. The above table shows the most common Kα doublets, of which the wavelengths are an average of the Kα1 and Kα2. 6  X-ray Diffraction: Bragg’s Law Sin  = SQ / dhkl = QT / dhkl n = Path difference = SQ + QT = dhkl sin dhkl sin  n = 2dhkl sin (n = 1, 2, 3, …..) This is Bragg’s law. Callister and Rethwisch, Materials Science and Engineering, 8th Edition 7  X-ray Diffraction Pattern 1 ℎ 𝑘 𝑙 = + + 𝑑 𝑎 𝑏 𝑐 Structure Factor, 𝐹 = 𝑓 𝑒 = 𝜉 𝑓 cos 2𝜋 ℎ𝑥 + 𝑘𝑦 + 𝑙𝑧 + 𝜉 𝑖𝑓 sin 2𝜋 ℎ𝑥 + 𝑘𝑦 + 𝑙𝑧 𝑓 is atomic scattering factor; 𝑥 , 𝑦 , 𝑧 are atom positions 𝐼 = 𝐹 8 Interplanar Spacings and Unit Cell Volumes 9  Crystal Systems Callister and Rethwisch, Materials Science and Engineering, 8th Edition 10  Structure Factor Calculation First calculate 𝐹. From 𝐹 ,𝐼 can be calculated by multiplying 𝐹 by its complex conjugate 𝐼 = 𝜉 𝑓 cos 𝛿 + 𝜉 𝑖𝑓 sin 𝛿 𝜉 𝑓 cos 𝛿 − 𝜉 𝑖𝑓 sin 𝛿 𝐼 = 𝜉 𝑓 cos 𝛿 + 𝜉 𝑓 sin 𝛿 𝛿 = 2𝜋 ℎ𝑥 + 𝑘𝑦 + 𝑙𝑧 Since i2 = -1 These refer to after summing up for all atoms in the structure. Do not square for individual atoms, i.e., Do not directly calculate Ihkl. First calculate Fhlk by summing up for all atoms, and then take the complex conjugate to get Ihkl. 11  Structure Factor Calculation for FCC (100) FCC: 0,0,0 , , ,0 , ,0 , , 0, , 𝑺𝒕𝒓𝒖𝒄𝒕𝒖𝒓𝒆 𝒇𝒂𝒄𝒕𝒐𝒓, 𝑭𝟏𝟎𝟎 1 1 = 𝑓 cos 2π 0 + 0 + 0 + 𝑖sin 2π 0 + 0 + 0 + cos 2π +0+0 + 𝑖sin 2π +0+0 2 2 1 1 + cos 2π +0+0 + 𝑖sin 2π +0+0 + cos 2π 0 + 0 + 0 + 𝑖sin 2π 0 + 0 + 0 2 2 𝐹 = 𝑓 cos 0 + 𝑖sin0 + cos π + 𝑖sin π + cos π + 𝑖sin π + cos 0 + 𝑖sin 0 𝐹 = 𝑓 1 − 1 − 1 + 1 + 𝑖𝑓 [0 + 0 + 0 + 0] 𝐹 = 𝑓 0 + 𝑖𝑓 𝐹 =0 𝐼 = 𝐹 𝐼 = 𝜉 𝑓 cos 𝛿 + 𝜉 𝑖𝑓 sin 𝛿 𝜉 𝑓 cos 𝛿 − 𝜉 𝑖𝑓 sin 𝛿 𝐼 = 𝜉 𝑓 cos 𝛿 −𝑖 𝜉 𝑓 sin 𝛿 𝐼 = 𝜉 𝑓 cos 𝛿 + 𝜉 𝑓 sin 𝛿 =0 +0 =0 12  Structure Factor Calculation for FCC (110) FCC: 0,0,0 , ,0 , 0, 0, , 𝐒𝐭𝐫𝐮𝐜𝐭𝐮𝐫𝐞 𝐟𝐚𝐜𝐭𝐨𝐫, 𝑭𝟏𝟏𝟎 1 1 1 1 = 𝑓 cos 2π 0 + 0 + 0 + 𝑖sin 2π 0 + 0 + 0 + cos 2π + +0 + 𝑖sin 2π + +0 2 2 2 2 1 1 1 1 + cos 2π +0+0 + 𝑖sin 2π +0+0 + cos 2π 0 + +0 + 𝑖sin 2π 0 + +0 2 2 2 2 𝐹 = 𝑓 [cos 0 + 𝑖𝑠𝑖𝑛 0 + cos 2𝜋 + 𝑖𝑠𝑖𝑛 2π + cos 𝜋 + 𝑖𝑠𝑖𝑛 π + cos 𝜋 + 𝑖𝑠𝑖𝑛(π) 𝐹 = 𝑓 cos 0 + cos 2𝜋 + cos π + cos π + 𝑖𝑓 [sin 0 + sin 2π + sin π + sin π ] 𝐹 = 𝑓 1 + 1 − 1 − 1 + 𝑖𝑓 [0 + 0 + 0 + 0] 𝐹 = 𝑓 0 + 𝑖𝑓 0 = 0 Or I110=|F110|2 = 𝑓 0 + [𝑖𝑓 0 ] = 0 13  Structure Factor Calculation for FCC (111) FCC: 0,0,0 , ,0 , 0, 0, , 𝐒𝐭𝐫𝐮𝐜𝐭𝐮𝐫𝐞 𝐟𝐚𝐜𝐭𝐨𝐫, 𝑭𝟏𝟏𝟏 1 1 1 1 = 𝑓 cos 2π 0 + 0 + 0 + 𝑖sin 2π 0 + 0 + 0 + cos 2π + +0 + 𝑖sin 2π + +0 2 2 2 2 1 1 1 1 1 1 1 1 + cos 2π +0+ + 𝑖sin 2π +0+ + cos 2π 0 + + + 𝑖sin 2π 0 + + 2 2 2 2 2 2 2 2 𝐹 = 𝑓 [cos 0 + 𝑖𝑠𝑖𝑛 0 + cos 2𝜋 + 𝑖𝑠𝑖𝑛 2π + cos 2𝜋 + 𝑖𝑠𝑖𝑛 2π + cos 2𝜋 + 𝑖𝑠𝑖𝑛(2π) 𝐹 = 𝑓 cos 0 + cos 2 𝜋 + cos 2π + cos 2π + 𝑖𝑓 [sin 0 + sin 2π + sin 2π + sin 2π ] 𝐹 = 𝑓 1 + 1 + 1 + 1 + 𝑖𝑓 [0 + 0 + 0 + 0] 𝐹 = 𝑓 4 + 𝑖𝑓 0 ≠ 0 Or I111 =|F111|2 = 𝑓 4 + [𝑖𝑓 0 ] ≠ 0 14  Diffraction Peaks for NaCl and KCl 300 (hkl) 100 110 111 200 210 211 --- 220 310 221 h2 + k2 + l2 1 2 3 4 5 6 8 9 10 Simple cubic Yes Yes Yes Yes Yes Yes --- Yes Yes Yes Body-Centered Cubic …. Yes …. Yes …. Yes --- Yes …. Yes Face-Centered Cubic …. …. Yes Yes …. …. --- Yes …. …. Sodium Chloride …. …. Yes Yes …. …. --- Yes …. …. 600 Potassium Chloride 200 220 222 400 420 422 --- 440 620 442 SC: all with h2 + k2 + l2 = an integer; BCC: h + k + l = even; FCC: hkl all odd or all even 15  Applications of XRD Phase identification - Comparison of the pattern with JCPDS database can help identify the phase(s), presence of impurity phases, semiquantitative analysis of different phases ….. Unit cell parameter determination -Obtained by least square fitting of the peak positions in the XRD pattern - Based on unit cell parameters, theoretical density can be obtained -Comparison of theoretical and observed density can provide info on porosity -Composition can be determined based on solid solution lattice parameter Crystal structure determination - Determined by a least square fitting of the intensities of the peaks - Normal spinel vs. inverse spinel vs. mixed configurations can be determined - Na2MoO4 (normal): (Mo)tet[Na2]octO4 Na+: 0.102 nm, Mo6+: 0.0.041 nm - FeGa2S4 (inverse): (Ga)tet[GaFe]octS4 Fe2+: 0.078 nm, Ga3+: 0.0.047 nm - MgCr2S4 (normal): (Mg)tet[Cr2]octS4 - Fe3O4 (inverse): (Fe3+)tet[Fe2+Fe3+]octO4 Co3O4 (normal): (Co2+)tet[Co3+]octO4 16 Applications of XRD 17 ATOMIC ORBITALS 18  CRYSTAL FIELD SPLITTING ex > c ex < c Tetragonal (Jahn-Teller) Distortion 19  CATION DISTRIBUTION IN SPINEL Fe3O4 MgAl2O4 is a mineral and is called spinel. Any other material having the structure similar to that of MgAl2O4 is called spinel material. Eight MgAl2O4 formulas per unit cell Spinel AB2O4: oxygen forms fcc and half of the octahedral sites (alternate) are occupied by B ions and one-eighth of the tetrahedral sites are occupied by A ions: (A)tet[B2]octO4, i.e., (A)8a[B2]16dO4 (normal spinel); (B)8a[AB]16dO4 (inverse) Fe3O4 spinel Fe2+ (high spin): 3d6, t2g4eg2 (octahedral), e3t23 (tetrahedral) Crystal field stabilization energy, CFSEoct = (- 4 Dqoct x 4) + (6 Dqoct x 2) = - 4 Dqoct CFSEtet = (- 6 Dqtet x 3) + (4 Dqtet x 3) = - 6Dqtet = (- 6 Dqtet) (4 Dqoct / 9 Dqtet) = - 2.66 Dqoct Octahedral site stabilization energy, OSSE = (- 4 Dqoct) – (2.66 Dqoct) = - 1.33 Dqoct So Fe2+ will be more stable in octahedral site than in tetrahedral site Fe3+ (high spin): 3d5, t2g3eg2 (octahedral), e2t23 (tetrahedral) CFSEoct = 0, CFSEtet = 0, OSSE = 0 So Fe3+ will not have any preference for octahedral or tetrahedral site Cation distribution: (Fe3+)tet[Fe2+Fe3+]octO4, i.e., (Fe3+)8a[Fe2+Fe3+]16dO4 (inverse) Magnetic measurements confirm the above cation distribution 20 PERIODIC TABLE 21  Applications of XRD Particle size analysis - XRD can be used to determine particle size if the size is < 200 nm - Line broadening increases with decreasing particle size - Below 2 – 10 nm size, the lines are so broad that they effectively disappear - Small angle X-ray scattering (SAXS) can be used to determine inhomogeneities of the order of 1 – 100 nm; X-ray scattering at low angles reflects electron density fluctuations and contrast that might exist over the range of 1 – 100 nm. - Factors influencing line broadening: (i) Instrumental factor: related to resolution of incident X-ray wavelength; broadening (linewidth) will vary smoothly with 2 or d spacing (ii) Sample factor: crystallite size and nonuniform microstrain; crystallite size dependence is given by Scherrer equation:  =  /  cos   is full width at half maximum is wavelength of X-ray is crystallite size is half the diffraction angle 22  Applications of XRD Microstrain determination  =  /  cos rearrangement gives  cos =  /  (1) n = 2d sin  (2) Differentiating equation (2) with respect to d and gives 0 = 2d cos   2 sin  d (3) d / d = - (cos / sin )  (4) Taking d / d as microstrain and 2  = , equation (4) gives = - ( / 2) (cos / sin ) = - 2 (sin / cos ) = k (sin / cos ) k is a constant = - 2)  cos  = k sin   The combined effect of crystallite size  and microstrain  on  can be obtained by combining equations (1) and (5):  cos  = k sin   The crystallite size and strain effects on line broadening can be separated by plotting  cos  vs. sin , in which the slope k will be related to microstrain and the intercept ( / ) will be related to the crystallite size. 23  Applications of XRD Determination of thermal expansion coefficient - Change in lattice parameter with temperature can be used to determine Assessment of phase transitions with temperature Crystal structure determination with single crystals - Unknown structures, atom positions, space groups, bond lengths, coordination numbers, electron distribution, bonding (ionic vs. covalent) can be determined from systematic absences and peak intensities Application of X-ray to polymers - Crystalline polymers would give diffraction peaks while completely amorphous polymers would give a broad halo that would represent the average separation between polymer chains - Semi-crystalline polymers will have a superposition of crystalline reflections on amorphous halo; crystalline peaks can give info on unit cell parameters, lattice strain, various phases present etc. 24  Electron Diffraction (ED) Advantages XRD needs crystals of at least 0.05 mm, which may not be available, as intensities of the diffracted beams may be too weak with small crystals Scattering efficiency of electrons is high, so small samples can be used for ED ED will give series of spots, from which structure can be determined Unit cells parameters and space group info can be obtained with < 0.01 mm size Inhomogeneities with small amounts of impurity phases can be analyzed Disadvantages High scattering efficiency leads to secondary diffraction peaks (extra peaks), so care should be taken to interpret data Intensities of peaks are unreliable, so cannot be used for quantitative analysis More expensive (TEM vs. XRD) 25  Neutron Diffraction (ND) Neutron scattering power consists of nuclear scattering and magnetic scattering: - Scattering by the nuclei - depends on nuclear structure - Scattering by magnetic moments of atoms – depends on magnetic structure X-ray scattering is a simple function of atomic number, so light atoms cannot be located; with neutrons, atomic nuclei are responsive, so light atoms can be located. Light atoms like H, Li, and O can be detected in presence of heavy atoms Neutrons can be used to distinguish between atoms with close atomic numbers - MnFe2O4: (Mn0.8Fe0.2)tet[Mn0.2Fe1.8]octO4; FeCo2O4: (Co2+)tet[Fe3+Co3+]octO4 Neutrons can be used to determine oxygen contents: YB2Cu3O7- (0.06 ≤  ≤ 1; but will be prone to larger error! The magnetic dipole of neutron can interact with unpaired electrons and give additional scattering effect, so magnetic structures, magnetic ordering, ordering temperatures, cation distribution in spinels can be determined Ferromagnetic, antiferromagnetic, ferrimagnetic etc. can be determined 26 Neutron Diffraction of MnO Neutron has a magnetic dipole, which leads to magnetic scattering in addition to scattering by electrons MnO has sodium chloride structure with a Neel temperature of 120 K At 80 K, Mn2+ ions undergo antiferromagnetic ordering with the Mn2+ spins alternating up and down along the three axes, resulting in a doubling of the unit cell for neutron scattering, but not for X-ray scattering At 293 K, the magnetic ordering is broken down due to thermal energy 27  Scanning Electron Microscopy (SEM) Scanning electron microscopy (SEM) examines microscopic structure by scanning the surface of materials, with much higher resolution and much greater depth of field SEM provides a three-dimensional appearance of its images because of its large depth of field. The depth of field can reach the order of tens of micrometers at 103 × magnification and the order of micrometer at 104 × magnification The magnification of an SEM is determined by the ratio of the linear size of the display screen to the linear size of the specimen area being scanned, not as in a TEM where magnification is determined by the power of the objective lens. SEM can provide image magnification from about 20× to > 100,000×. For a probe size of 10 nm, effective magnification is 20,000× SEM can enable to obtain chemical information with an X-ray energy-dispersive spectrometer (EDS) 28  Optical Arrangement in SEM Field emission gun, 1-40 kV high beam brightness Electron probe formation 29  Signal Detection in SEM Elastic scattering produces backscattered electrons (BSEs), which are incident electrons scattered by atoms in the specimen. BSEs are typically deflected from the specimen at large angles and with little energy loss; they typically retain 60- 80% of the energy of incident electrons. Inelastic scattering produces secondary electrons (SEs), which are electrons ejected from atoms in the specimen. SEs are typically deflected at small angles and show much lower energy compared with incident electrons. SEs are the primary signals for achieving surface topographic contrast/images, while BSEs are useful for formation of elemental composition contrast. Topographic contrast in an SEM refers to variation in signal levels that corresponds to variations in geometric features on the specimen surface. Energy in SEM can be varied from 5 to 20 keV; low keV can give surface features and high kV can give bulk info as well. 30  Topographic Contrast in SEM More electrons are emitted by an edge than a flat surface 5 µm 31 Particle Shape and Elemental Distribution from SEM 5 µm 32 Compositional Contrast in SEM Compositional contrast arises because of the capability of BSEs to escape from the specimen depends on the atomic numbers of the specimen atoms. 33  Charging effect in SEM and its Prevention Surface charging occurs when examining electrically nonconductive surfaces. Excessive electrons accumulated on the specimen surface where it is impinged by the electron beam. Accumulation of electrons on the surface builds up charged regions. Electric charges generate distortion and artifacts in SEM images because charged regions deflect the incident electron probe in an irregular manner during scanning. Surface charging is a concern for nonconductive specimens, such as ceramics, polymers, and specimens containing biological substances. Can be prevented by coating a conductive film onto specimen surface. 34  Electron Backscatter Diffraction in SEM Electron backscatter diffraction (EBSD) can determine crystalline materials properties in SEM 35  Transmission Electron Microscopy (TEM) Electron microscopes generate images of material microstructures with much higher magnification and resolution than light microscopes The wavelength of electrons in electron microscopes is about 10,000 times shorter than that of visible light The resolution of electron microscopes reaches the order of 0.1 nm if lens aberrations can be minimized Transmission electron microscope (TEM) is similar to the conventional transmission light microscope. Scanning electron microscope (SEM) is more like a scanning confocal laser microscope. 36  Transmission Electron Microscope TEM has the following: light source, condenser lens, specimen stage, objective lens, and projector lens http://www.phy.cuhk.edu.hk/centrallaboratory/TecnaiF20/TecnaiF20.html 37  Image Modes in TEM TEM contrast relies on deflection of electrons from their primary transmission direction when they pass through the specimen The contrast is generated when there is a difference in the number of electrons being scattered away from the transmitted beam There are two mechanisms by which electron scattering creates images: mass-density contrast and diffraction contrast 38 Mass-density Contrast in TEM The deflection of electrons can result from interaction between electrons and an atomic nucleus 39  Diffraction Contrast in TEM Diffraction contrast is the primary mechanism of TEM image formation in crystalline specimens Electrons can be scattered collaboratively by parallel crystal planes similar to X-rays. Bragg’s law applies to electron diffraction. The diffraction angle (2θ) in a TEM is very small (≤1°) and the diffracted beam from a crystallographic plane (hkl) can be focused as a single spot on the back- focal plane of the objective lens. When the transmitted beam is parallel to a crystallographic axis, all the diffraction points from the same crystal zone will form a diffraction pattern (a reciprocal lattice) on the back-focal plane 40 Diffraction Contrast in TEM 41  Phase Contrast in TEM TEM phase contrast produces the highest resolution of lattice and structure images for crystalline materials. Thus, phase contrast is often referred as to high-resolution transmission electron microscopy (HRTEM). Phase contrast must involve at least two electron waves that are different in wave phase. The transmitted beam and diffraction beam. Recombination of transmitted and diffracted beams will generate an interference pattern with periodic dark-bright changes on the image plane because of beam interferences. 42 Phase Contrast in TEM 43 Phase Analysis by TEM 44