CYL729: Materials Characterization Diffraction Microscopy Thermal Analysis A. Ramanan Department of Chemistry aramanan@chemistry.iitd.ac.in

Reference Books George M. Crankovic (Editor)

Electro-magnetic Spectrum

History of X-rays 1885-1895 Wm. Crookes sought unsuccessfully the cause of repeated fogging of photographic plates stored near his cathode ray tubes. X-rays discovered in 1895 by Roentgen, using ~40 keV electrons (1st Nobel Prize in Physics 1901) 1909 Barkla and Sadler discovered characteristic X-rays, in studying fluorescence spectra (though Barkla incorrectly understood origin) (Barkla got 1917 Nobel Prize) 1909 Kaye excited pure element spectra by electron bombardment

History of X-rays - cont’d 1912 von Laue, Friedrich and Knipping observe X-ray diffraction (Nobel Prize to von Laue in 1914) 1912-13 Beatty demonstrated that electrons directly produced two radiations: (a) independent radiation, Bremsstrahlung, and (b) characteristic radiation only when the electrons had high enough energy to ionize inner electron shells. 1913 WH + WL Bragg build X-ray spectrometer, using NaCl to resolve Pt X-rays. Braggs’ Law. (Nobel Prize 1915) n l = 2d sin q

History of X-rays - cont’d 1913 Moseley constructed an x-ray spectrometer covering Zn to Ca (later to Al), using an x-ray tube with changeable targets, a potassium ferrocyanide crystal, slits and photographic plates 1914, figure at right is the first electron probe analysis of a man-made alloy T. Mulvey Fig 1.5 (in Scott & Love, 1983). Note impurity lines in Co and Ni spectra

History of X-rays - cont’d Moseley found that wavelength of characteristic X-rays varied systematically (inversely) with atomic number Z l Using wavelengths, Moseley developed the concept of atomic number and how elements were arranged in the periodic table. The next year, he was killed in Turkey in WWI. “In view of what he might still have accomplished (he was only 27 when he died), his death might well have been the most costly single death of the war to mankind generally,” says Isaac Asimov (Biographical Encyclopedia of Science &Technology).

Historical Summary of X-rays 1859 Kirchhoff and Bunsen showed patterns of lines given off by incandescent solid or liquid are characteristic of that substance 1904 Barkla showed each element could emit ≥1 characteristic groups (K,L,M) of X-rays when a specimen was bombarded with beam of x-rays 1909 Kaye showed same happened with bombardment of cathode rays (electrons) 1913 Moseley found systematic variation of wavelength of characteristic X-rays of different elements 1922 Mineral analysis using X-ray spectra (Hadding) 1923 Hf discovered by von Hevesy (gap in Moseley plot at Z=72). Proposed XRF (secondary X-ray fluorescence) 1923 Manne Siegbahn published The Spectroscopy of X-rays in which he shows that the Bragg equation must be revised to take refraction into account, and he lays out the “Siegbahn notation” for X-rays 1931 Johann developed bent crystal spectrometer (higher efficiency)

Summary of X-ray Properties X-rays are considered both particles and waves, i.e., consisting of small packets of electromagnetic waves, or photons. X-rays produced by accelerating HV electrons in a vacuum and colliding them with a target. The resulting spectrum contains (1) continuous background (Bremsstrahlung;“white X-rays”), (2) occurrence of sharp lines (characteristic X-rays), and (3) a cutoff of continuum at a short wavelength. X-rays have no mass, no charge (vs. electrons)

X-ray Crystallography DIFFRACTION

What is a Unit Cell? © 1993 American Chemical Society A unit cell is a building block of a crystalline solid. Shifting the unit cell along any of its edges by the length of the edge will generate identical cells to build up the entire crystal. If not chosen properly, shifts of the cell will not give an identical cell (upper right unit cell). Unit cells do not need to be squares or rectangles (lower right unit cell). Unit cells facilitate the counting of atoms comprising the solid (empirical formula).

Unit cell can be chosen in different ways! © 1993 American Chemical Society Use to teach stoichiometry. Two possible unit cells for the structure are shown with dashed lines. The dots could represent the Na+ ions and the open circles Cl– ions. Smaller unit cell has one each of the two types of atoms. The larger unit cell has four each of the two types of atoms. Both give a stoichiometric ratio of 1:1, the ratio of atoms in the empirical formula.

Unit Cells? White and black birds by the artist, M. C. Escher. © 1993 American Chemical Society White and black birds by the artist, M. C. Escher.

A unit cell chosen such that it contains minimum volume but exhibit maximum symmetry

Translational vector {R = n1 a1 + n2 a2 + n3 a3}

Crystal Structure Ideal Crystal: Contain periodical array of atoms/ions Represented by a simple lattice of points A group of atoms attached to each lattice points Basis LATTICE = An infinite array of points in space, in which each point has identical surroundings to all others. CRYSTAL STRUCTURE = The periodic arrangement of atoms in the crystal. It can be described by associating with each lattice point a group of atoms called the MOTIF (BASIS)

7 Crystal Systems Lattice parameters: a, b, c; a, b, g

Bravais Lattice: an infinite array of discrete points with an arrangement and orientation that appears exactly the same from whichever of the points the array is viewed. Crystal System Bravais Lattice Essential Symmetry Conditions Cubic P, F, I 4 C3 a=b=c ==900  Tetragonal P, I 1 C4 along [c-axis] ==900 Hexagonal P 1 C6 along [c-axis] Rhombohedral R 1 C3 along body diagonal  = =   900 Orthorhombic P, F, I, C 3 C2 mutually perpedicular along the three axes a  b  c Monoclinic P, C 1 C2 along [b-axis] a b  c ==900 &   900 Triclinic C2 or inversion centre       900

14 Bravais lattices

Unit cell symmetries - cubic 3 C4 - passes through pairs of opposite face centers, parallel to cell axes 4 C3 - passes through cubic diagonals A cube need not have C4 !!

Copper metal is face-centered cubic Identical atoms at corners and at face centers Lattice type F also Ag, Au, Al, Ni... -Iron is body-centered cubic Identical atoms at corners and body center (nothing at face centers) Lattice type I Also Nb, Ta, Ba, Mo...

periodic table Hexagonal closed body-centered cubic (bcc) packed (hcp) face-centered cubic (fcc)

Sodium Chloride (NaCl) - Na is much smaller than Cs Caesium Chloride (CsCl) is primitive cubic Different atoms at corners and body center. NOT body centered, therefore. Lattice type P Also CuZn, CsBr, LiAg Sodium Chloride (NaCl) - Na is much smaller than Cs Face Centered Cubic Rocksalt structure Lattice type F Also NaF, KBr, MgO….

Diamond Structure: two sets of FCC Lattices Z = 8 C atoms per unit cell

Tetragonal: P, I one C4 Yellow and green colors represents same atoms but different depths. Why not F tetragonal?

Example 2- CaC2 - has a rocksalt-like structure but with non-spherical carbides C C Carbide ions are aligned parallel to c  c > a,b  tetragonal symmetry

Orthorhombic: P, I, F, C C F

Side centering Side centered unit cell Notation: A-centered if atom in bc plane B-centered if atom in ac plane C-centered if atom in ab plane

Trigonal: P : 3-fold rotation

Hexagonal Monoclinic Triclinic

Unit cell contents Counting the number of atoms within the unit cell Many atoms are shared between unit cells

lattice type cell contents P 1 [=8 x 1/8] I 2 [=(8 x 1/8) + (1 x 1)] Atoms Shared Between: Each atom counts: corner 8 cells 1/8 face center 2 cells 1/2 body center 1 cell 1 edge center 4 cells 1/4 lattice type cell contents P 1 [=8 x 1/8] I 2 [=(8 x 1/8) + (1 x 1)] F 4 [=(8 x 1/8) + (6 x 1/2)] C 2 [=(8 x 1/8) + (2 x 1/2)]

e.g. NaCl Na at corners: (8  1/8) = 1 Na at face centres (6  1/2) = 3 Cl at edge centres (12  1/4) = 3 Cl at body centre = 1 Unit cell contents are 4(Na+Cl-)

Fractional Coordinates (0,0,0) (0, ½, ½) (½, ½, 0) (½, 0, ½)

Cs (0,0,0) Cl (½, ½, ½)

Density Calculation Calculate the density of copper. n = 4 atoms/cell, n: number of atoms/unit cell A: atomic mass VC: volume of the unit cell NA: Avogadro’s number (6.023x1023 atoms/mole) Calculate the density of copper. RCu =0.128nm, Crystal structure: FCC, ACu= 63.5 g/mole n = 4 atoms/cell, 8.94 g/cm3 in the literature

Miller Indices describe which plane of atom is interacting with the x-rays

How to Identify Miller indices (hkl)? [001] direction: [hkl] family of directions: <hkl> planes: (hkl) family of planes: {hkl} c b a [010] [001] to identify planes: Step 1 : Identify the intercepts on the x- , y- and z- axes. Step 2 : Specify the intercepts in fractional coordinates Step 3 : Take the reciprocals of the fractional intercepts

Miller indices (hkl) to identify planes: Step 1 : Identify the intercepts on the x- , y- and z- axes (a/2, ∞, ∞) Step 2 : Specify the intercepts in fractional co-ordinates (a/2a, ∞, ∞) = (1/2,0,0) Step 3 : Take the reciprocals of the fractional intercepts (2, 0, 0) e.g.: cubic system: (210) (110) (100) (111)

Miller Indices

Miller Indices

Crystallographic Directions And Planes Lattice Directions Individual directions: [uvw] Symmetry-related directions: <uvw> Miller Indices: 1. Find the intercepts on the axes in terms of the lattice constant a, b, c 2. Take the reciprocals of these numbers, reduce to the three integers having the same ratio (hkl) Set of symmetry-related planes: {hkl}

(100) (111) (200) (110)

In cubic system, [hkl] direction perpendicular to (hkl) plane

Wilhelm Conrad Röntgen Wilhelm Conrad Röntgen discovered 1895 the X-rays. 1901 he was honoured by the Noble prize for physics. In 1995 the German Post edited a stamp, dedicated to W.C. Röntgen.

The Principles of an X-ray Tube Cathode Fast electrons Anode focus

(slowed down and changed direction) The Principle of Generation of X-ray Ejected electron (slowed down and changed direction) nucleus Fast incident electron electrons Atom of the anodematerial X-ray

The Principle of Generation the Characteristic Radiation Emission Photoelectron M K L K Electron L K

The Generating of X-rays Bohr`s model

The Generating of X-rays energy levels (schematic) of the electrons M Intensity ratios KKK L K K K K K

The Generating of X-rays Anode Mo Cu Co Fe (kV) 20.0 9.0 7.7 7.1 Filter Wavelength (Angström) K1 : 0,70926 K2 : 0,71354 K1 : 0,63225 Zr 0,08mm K1 : 1,5405 K2 : 1,54434 K1 : 1,39217 Ni 0,015mm K1 : 1,78890 K2 : 1,79279 K1 : 1,62073 Fe 0,012mm K1 : 1,93597 K2 : 1,93991 K1 : 1,75654 Mn 0,011mm

The Generating of X-rays Emission Spectrum of a Molybdenum X-Ray Tube Bremsstrahlung = continuous spectra characteristic radiation = line spectra

Interaction between X-ray and Matter incoherent scattering Co (Compton-Scattering) coherent scattering Pr(Bragg´s-scattering) wavelength Pr absorbtion Beer´s law I = I0*e-µd intensity Io fluorescense > Pr photoelectrons

C. Gordon Darwin C. Gordon Darwin, grandson of C. Robert Darwin developed 1912 dynamic theory of scattering of X-rays at crystal lattice

P. P. Ewald P. P. Ewald 1916 published a simple and more elegant theory of X-ray diffraction by introducing the reciprocal lattice concept. Compare Bragg’s law (left), modified Bragg’s law (middle) and Ewald’s law (right).

Bragg’s Description The incident beam will be scattered at all scattering centres, which lay on lattice planes. The beam scattered at different lattice planes must be scattered coherent, to give an maximum in intensity. The angle between incident beam and the lattice planes is called . The angle between incident and scattered beam is 2 . The angle 2 of maximum intensity is called the Bragg angle. W.H. Bragg (father) and William Lawrence.Bragg (son) developed a simple relation for scattering angles, now call Bragg’s law.

Bragg’s Law A powder sample results in cones with high intensity of scattered beam. Above conditions result in the Bragg equation or

X-Ray Diffraction 35KeV ~ 0.1-1.4A Cu K 1.54 A Mo:

Structure Determination © 1993 American Chemical Society The structure of a crystalline solid can be determined using X-ray diffraction. X-rays from an X-ray tube are directed through a collimator at a crystal. The X-rays interact with the electron clouds of the atoms and diffract because the wavelength of the X-rays is about the same magnitude as the spacing between atoms. Photographic film can be used to detect the diffracted X-rays. The experiment can be scaled up by factors of thousands: diffraction can be illustrated using visible laser light, an array of laser-written, photographically-reduced dots on a 35-mm slide, and a projection screen.

Diffraction Interference fringes Light Constructive Destructive

Diffraction Conditions

Diffraction Conditions © 1993 American Chemical Society In Fraunhofer diffraction the diffraction equation is d sin f = nl. In Bragg diffraction, a similar diffraction equation results, 2 (d sin q ) = nl.

Lattice spacing For cubic system

Bragg’s Law For cubic system: But not all planes have the diffraction !!!

Powder diffraction X-Ray (211) (200)

Powder X-ray Diffraction Film Tube Powder

The Elementary Cell a = b = c o = = = 90 c a b

Relationship between d-value and the Lattice Constants Bragg´s law The wavelength is known Theta is the half value of the peak position d will be calculated Equation for the determination of the d-value of a tetragonal elementary cell h,k and l are the Miller indices of the peaks a and c are lattice parameter of the elementary cell if a and c are known it is possible to calculate the peak position if the peak position is known it is possible to calculate the lattice parameter

D8 ADVANCE Bragg-Brentano Diffractometer A scintillation counter may be used as detector instead of film to yield exact intensity data. Using automated goniometers step by step scattered intensity may be measured and stored digitally. The digitised intensity may be very detailed discussed by programs. More powerful methods may be used to determine lots of information about the specimen.

The Bragg-Brentano Geometry Tube Detector   2 focusing-circle Sample measurement circle

The Bragg-Brentano Geometry Mono- chromator Antiscatter- slit Divergence slit Detector- slit Tube Sample

Powder Diffraction Pattern

What is a Powder Diffraction Pattern? A powder diffractogram is the result of a convolution of a) the diffraction capability of the sample (Fhkl) and b) a complex system function. The observed intensity yoi at the data point i is the result of yoi =  of intensity of "neighbouring" Bragg peaks + background The calculated intensity yci at the data point i is the result of yci = structure model + sample model + diffractometer model + background model 5

Which Information does a Powder Pattern offer? peak position - dimension of the elementary cell peak intensity - content of the elementary cell peak broadening - strain/crystallite size/nanostr. 6

Powder Pattern and Structure The d-spacings of lattice planes depend on the size of the elementary cell and determine the position of the peaks. The intensity of each peak is caused by the crystallographic structure, the position of the atoms within the elementary cell and their thermal vibration. The line width and shape of the peaks may be derived from conditions of measuring and properties - like particle size - of the sample material.

Powder diffraction X-Ray (110) (211) (200)

Example: layered silicates mica growth oriented along c-axis 2*theta d 7.2 12.1 14.4 6.1 22 4.0 (hkl) (001) (002) (003) (00l) C~12.2 A

What we will see in XRD of simple cubic, BCC, FCC?

What we will see in XRD of simple cubic, BCC, FCC?

Observable diffraction peaks Ratio SC: 1,2,3,4,5,6,8,9,10,11,12.. BCC: 2,4,6,8,10, 12…. FCC: 3,4,8,11,12,16,24…. Simple cubic

Determine:(a) Crystal structure?(b) Lattice constant? Ex: An element, BCC or FCC, shows diffraction peaks at 2q: 40, 58, 73, 86.8,100.4 and 114.7. Determine:(a) Crystal structure?(b) Lattice constant? (c) What is the element? 2theta theta (hkl) 40.0 20 0.117 1 (110) 58.0 29 0.235 2 (200) 73.0 36.5 0.3538 3 (211) 86.8 43.4 0.4721 4 (220) 100.4 50.2 0.5903 5 (310) 114.7 57.35 0.7090 6 (222) a =3.18Å, BCC,  W