Electronic Materials
EECE 7296 – FALL 2023
V.G. Harris
Introduction to X-ray Diffraction
• Generation of x-rays
• EM spectrum
• Bragg’s law
• Equipment
• Consider-single crystal
• Consider-polycrystal
• Let’s look at bcc Fe – breakout problem
1
Hexaferrite (magnetoceramics) for RF nonlinear devices (unit cells)
X-type (Ba2Me2Fe28O46)
X-type
b)
U-type (Ba4Me2Fe36O60)
a)
U-type
Iron (Fe)
The Utility of X-ray Diffraction
• Identifies reflected atomic planes
• Allows for calculation of atomic dimensions of the sample’s phase
• Provides identification of the sample’s crystal phase
• Identifies crystal symmetry
• Allows evaluation of defects and strain
• Allows evaluation of crystal orientation
• 40 Nobel Prizes awarded for X-ray research
Electron collisions and transitions
https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book
%3A_University_Physics_III__Optics_and_Modern_Physics_(OpenStax)/08%3A_Atomic_Structure/8.06%3A_Atomic_Spectra_and_Xrays
X-ray Tube (provides an affordable source of x-rays)
Electromagnetic spectrum
7
Detector
X-rays
1
1
2
2

d
d
Constructive interference
A

dsin
dsin
B
Atomic planes
Crystal
(c)
x + y = λ = nλ
dsin = x
dsin = y
2dsin = nλ
n is the “order” of the wave
n=1 is the 1st harmonic
Monochromators are used to limit the X-ray beam
to be monochromatic in most cases
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
http://www.upscale.utoronto.ca/GeneralInterest/Harrison/DoubleSlit/DoubleSlit.html
8
Detector
X-rays
1
1
2
2

d
A

dsin
dsin
B
d
1)
(c)
Atomic planes
Crystal
Bragg’s Law
2dsin = nλ
hc
λ= E
E = energy of X-ray, KeV
λ = wavelength of X-ray, nm
hc = 1.240 KeV*nm
Relate “d” to specific crystal structure (cubic systems):
a
d=
a = lattice constant
2
2
2
½
(h + k + l )
(hkl) = miller indices
1) From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
9
Bragg condition for a cubic crystals
From
2dsinθ = nλ
d=
a
(h2 + k2 + l2)½
We obtain
sin  
2
n 2
2
2

h k l
2
4a
2 2


10
X-ray Diffraction Equipment
12
X-ray diffraction from a single crystal
X-ray
source
Monochromatic
X-rays
Single
crystal
sample
Goniometer Stage
Hybrid Pixel
Array Detector
(HPAD)
Output
13
X-ray powder diffraction
Consider diffraction from a collection of misoriented
crystals (polycrystalline)
Intensity (Counts)
Traditional -2 powder diffraction
2 (degrees)
(for illustration- not a one to one comparison)
15
Protein X-ray Crystallography (game changer for rapid
drug design)
Theory
X-rays
Crystal
Phase
Diffraction
Pattern
Fitting
Electrondensity map
Atomic model
"for their outstanding achievements in the development of direct
methods for the determination of crystal structures"
Herbert A. Hauptman
Jerome Karle
Break out problem
• At what angles will these planes appear as peaks in the x-ray
diffractogram if Cu ka radiation is used (=0.154 nm) and the
lattice parameter of the sample (body centered cubic iron) is
0.2866 nm?
18
Solution for the first one is (110)
a
0.2866nm
d

2
2
2
2
h k l
n
0.154nm
sin  

2d 2 * 0.2026nm
  arcsin(0.3799)  22.30
2  44.60
sin  2
n 2 2 2
2
2

h

k

l
4a 2


This expression of course provides the same answer
19
Consider bcc-Fe
44.67
65.02 82.33
20
Intensity (Counts)
Consider:
2 (degrees)
a
What is scattering the x-rays?
Explain why (110) intensity is greater than (211)
which is greater than (200) – draw and calculate
areal densities of planes
XRD of bcc iron
Triclinic
XRD of SrTiO3
K2Ce(PO4)2
Triclinic structure
XRD Study of BaM/MgO/AlN/SiC samples
(008)
Figure. Room temperature Cuka radiation q-2q diffraction pattern acquired from BaM film PLD grown on MgO/AlN/SiC
heterostructure. The result demonstrates strong crystallographic texture with (00𝑙) diffracting planes aligned orthogonal to the
growth axis. Inset pole figures for (008) planes confirm heteroepitaxy while similar data collected from the (107) peak confirm
six-fold in-plane symmetry. “X” denotes unidentified low-angle diffraction peaks that have less than 1% relative intensity.