1. CYL729: Materials Characterization
Diffraction
Microscopy
Thermal Analysis
A. Ramanan
Department of Chemistry

2. Reference Books
George M. Crankovic (Editor)

3. Electro-magnetic Spectrum

4. History of X-rays
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

5. History of X-rays - cont’d
1912 von Laue, Friedrich and Knipping observe X-ray diffraction (Nobel Prize to von Laue in 1914)
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

6. 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

7. 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).

8. 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)

9. 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)

10. X-ray Crystallography
DIFFRACTION

11. 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).

12. 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.

13. 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.

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

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

16. 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)

17. 7 Crystal Systems
Lattice parameters:
a, b, c; a, b, g

18. 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

19. 14 Bravais lattices

20. 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 !!

21. 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...

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

23. 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….

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

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

26. 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

27. Orthorhombic: P, I, F, C C F

28. 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

29. Trigonal: P : 3-fold rotation

30. Hexagonal
Monoclinic
Triclinic

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

32. 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 [=8 x 1/8]
I [=(8 x 1/8) + (1 x 1)]
F [=(8 x 1/8) + (6 x 1/2)]
C [=(8 x 1/8) + (2 x 1/2)]

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

34. Fractional Coordinates
(0,0,0)
(0, ½, ½)
(½, ½, 0)
(½, 0, ½)

35. Cs (0,0,0)
Cl (½, ½, ½)

36. 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

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

39. 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

40. 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)

41. Miller Indices

42. Miller Indices

43. 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}

44. (100)
(111)
(200)
(110)

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

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

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

49. (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

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

51. The Generating of X-rays
Bohr`s model

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

53. 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

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

55. 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

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

57. 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).

58. 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.

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

60. X-Ray Diffraction
35KeV ~ A
Cu K A
Mo:

61. 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.

62. Diffraction
Interference fringes
Light
Constructive
Destructive

63. Diffraction Conditions

64. 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, (d sin q ) = nl.

66. Lattice spacing
For cubic system

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

69. Powder diffraction
X-Ray
(211)
(200)

70. Powder X-ray Diffraction
Film
Tube
Powder

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

72. 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

73. 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.

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

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

76. Powder Diffraction Pattern

77. 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

78. 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

79. 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.

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

81. 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

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

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

84. 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

85. 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
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