Class 17 – X-ray Diffraction
Reading
• Chapter 12 in DeGraef and McHenry
• Chapter
Ch t 3 in
i Engler
E l and
d Randle
R dl
• Chapter 6 in DeGraef and McHenry (I WILL ASSUME THAT YOU
KNOW ABOUT THIS CONCEPT ALREADY! IF NOT, READ IT!)
Prof. M.L. Weaver
Review: X-Ray Diffraction
•
XRD is a powerful experimental technique used to determine the
– crystal structure and its lattice parameters (a
(a, b,
b c,
c )
  ) and
– spacing between lattice planes (hkl Miller indices) this interplanar spacing (d) is the distance
between parallel planes of atoms or ions.
•
Diffraction is result of radiation’s being scattered by a regular array of scattering centers
whose
hose spacing is abo
aboutt same as the  of the radiation
radiation.
•
Diffraction gratings must have spacings comparable to the wavelength of diffracted
radiation.
We know that atoms and ions are on the order of 0.1
0 1 nm in size,
size so we think of
crystal structures as being diffraction gratings on a sub-nanometer scale.
•
•
For X-rays, atoms are scattering centers (photon interaction with an orbital electron in the
atom). Spacing (dhkl) is the distance between parallel planes of atoms……
2
XRD to Determine Crystal Structure
& Interplanar Spacing
Recall incoming X-rays diffract from crystal planes:
reflections must
be in phase for
a detectable
d
bl signal
i l
is scattering
(Bragg) angle
extra
distance
traveled
by wave “2”



d
Adapted from Fig. 3.37,
Callister & Rethwisch 3e.
i.e., for diffraction to occur, x-rays
scattered off adjacent crystal planes
must be in phase:
spacing
between
planes
Measurement of critical angle,
c, allows computation of
interplanar spacing (d)
X-ray
intensity
(f
(from
detector)
d

2 sin c
B
Bragg’s
’ L
Law(1) d hkl 
cubic
c
a
h2  k 2  l 2
(2)
 Combine (1)+(2):
(Bragg’s Law is
not satisfied)
3
Recall the interplanar spacings (dhkl)
for the 7 crystal
y
systems
y
•As crystal symmetry
decreases, the number
of peaks observed
increases:
•Cubic crystals, highest
symmetry, fewest peaks.
•Triclinic crystals, lowest
symmetry, large number of
peaks.
4
Geometry of XRD (F.Y.I)
•A single wavelength of x-ray radiation is
often used to keep the number of diffraction
peaks to a small workable number, since
samples often consists of many small crystal
grains orientated randomly.
•The diffracted beam intensity is monitored electronically
by a mechanically driven scanning radiation detector.
Usually graphite to
focus beam like a
parabolic mirror
Collimate
beam



High Z to max.
absorption. Removing
unwanted radiation
X-ray
Source
(fi ed)
(fixed)

Bragg angle = 
Diffraction angle, what’s measured experimentally = 2
•Counter/detector is rotated about O-axis; 2 is its angular position.
•Counter and specimen are mechanically coupled such that a rotation of the specimen through  is
accompanied by a 2 rotation of the counter.
•This assures that the incident and reflection angles are maintained equal to one another.
More on Bragg’s Law
•Bragg’s Law is a necessary but insufficient condition for diffraction.
•It only defines the diffraction condition for primitive unit cells, e.g. P cubic, P tetragonal, etc., where
atoms are onlyy at unit cell corners.
•Crystal structures with non-primitive unit cells have atoms at additional lattice (basis) sites.
•These extra scattering centers can cause out-of-phase scattering to occur at certain Bragg angles.
•The net result is that some of the diffraction predicted by Bragg’s Law (eq. 1) does not occur, i.e.
certain sets of p
planes do not exist ((forbidden reflections).
)
•Selection (or Reflection) rules:
Bravais Lattice
Example Compounds
Allowed Reflections
Forbidden Reflections
Primitive Cubic
Simple Cubic (-Po)
Any h,k,l
None
Body-Centered Cubic
Body-Centered Cubic metal
h+k+l even
h+k+l odd
Face-Centered Cubic
Face-Centered Cubic metal
h,k,l all odd or all even
h,k,l mixed odd or even
Face-Centered Cubic
NaCl-rocksalt, ZnS-zincblende
h,k,l all odd or all even
h,k,l mixed odd or even
F
Face-Centered
C t dC
Cubic
bi
Si Ge
Si,
G - Diamond
Di
d cubic
bi
As FCC,
A
FCC but
b t if allll even
and h+k+l≠4n, then
absent (n is integer)
h,k,l
h
k l mixed
i d odd
dd or even
and if all even and
h+k+l≠4n
Primitive Hexagonal
Hexagonal closed packed metal
All other cases
h+2k=3n, l odd
These rules are calculated based on atomic scattering factors and structure factors,
which we will discuss next class and next week (you will also see on final exam).
6
Selection Rules for Cubic Crystals
d hkl 
a
h k l
2
2
2

a
S
where “S” is reflection or
line #,
e.g. (100) is 1st order
reflection and
(200) is 4th order
reflection
reflection.
As we will determine
later when we calculate
the structure factors, these
selection rules also hold
for other Bravais lattices,
e.g.
I-tetragonal,
F-orthorhombic, etc.
P
I
F
S
((hkl))
S
((hkl))
S
((hkl))
1
100
-
-
-
-
Odd+even
2
110
2
110
-
-
Odd+even
3
111
-
3
111
All odd
4
200
4
200
4
200
All even
5
210
-
-
-
-
Odd+even
6
211
6
211
-
-
Odd+even
7
-
-
-
-
-
8
220
8
220
8
220
9
221,300 -
All even
-
Odd+even
Odd+even
Odd
eve
10 310
10
310
11
-
-
11
311
All odd
12
222
12
222
All even
311
12 222
7
Example of XRD Pattern
•The unit cell size and geometry may be resolved from the angular positions of the diffraction peaks.
•When the distance between the peak spacings are all pretty much the same (2) it is likely cubic.
•The arrangement of atoms within the unit cell is associated with the relative intensities of these peaks
z
Inntensity (relative)
c
a
z
(110)
c
y (110)
a
b
x
z
(200)
b
c
y
x
Diffraction
pattern for
ppolycrystalline
y y
-iron (BCC)
a
x
d-spacing decreases
according to Bragg’s Law
(211)
b
y
(211)
(200)
Diffraction angle 2
h+k+l even are only allowed according to previously shown selection rules for BCC.
8
Review of Systematic Absences in the
Diffraction Patterns of 4 Cubic Structures
=S=
 When indexing XRD data for your material always try cubic first (least amount of
diffracting planes since most symmetric lattice parameters).
9
X-Ray Diffraction for Tetragonal,
e go andd Orthorhombic
O o o b c Crystals
C ys s
Hexagonal
Bragg’s Law (1):

d
(1)
2 sin c
a
(2) Plane spacing
d hkl 
h 2  k 2  l 2 for cubic crystals
Plane spacings for:
(4)
(5)
(6)
Combined (1) and (2):
2
sin 2 
2 2 2 2 If crystal is tetragonal with a=a≠c then (1) and (4) become:
 n 
2
or sin   2 (h  k  l )
2
2
   2
2
2


2
2
2
4
a
2
a
h

k

l
 
(7)
(3)
sin   2 h  k   2 l 2 
4a
4c
For a particular incident x-ray wavelength and cubic
with a=a≠c then (1) and (5) become:
crystal of unit cell size a,
a this equation predicts all possible If crystal is hexagonal
2

2 2
Bragg angles at which diffraction can occur from planes
2
2
2
sin   2 h  k  hk  2 l
(8)
(hkl).
3a
4c
If crystal is orthorhombic with a≠b≠c then (1) and (6):
Diffraction planes are determined solely by the shape
2 2
2 2
2 2
2
sin   2 (h )  2 (k )  2 (l ) (9)
and size (lattice parameters) of the unit cell.


 
4a
4b
4c
• Intensities of diffracted beams are determined by positions of the atoms within the unit cell. Thus, we must
measure the intensities if we are able to obtain any information at all about atom positions. (Intensities (I)
); i.e.,, I is p
proportional
p
to F).
)
are related to Structure Factor ((F);
• We will determine that for many crystals there are particular atomic arrangements which reduce the
intensities of some diffracted beams to zero, i.e. no diffracted beam at the angle () predicted by Equations
(3), (7), (8), (9), etc., which means F=0.
10
The Effect of Cell Distortion (Symmetry)
on XRD Patterns
• As the symmetry of crystal decreases more peaks are seen, e.g.
cubic  triclinic.
• Similar with cell distortion. If distort cubic I along [001] by 4%
so c is
i now, 4.16Å
4 16Å gett I tetragonal.
l
• This decreases symmetry more, and shifted diffraction lines are
formed (shown in the middle for tetragonal).
• If now stretch 8% along [010] axis with a=4.00Å, b= 4.32Å, and
c= 4.16Å
4 16Å to get orthorhombic I which lowers symmetry even
more thus adding more diffraction lines (shown right).
• The increase in number of lines is due to the introduction of new
hkl plane d-spacings, caused by non-uniform distortion.
• Thus in cubic cell,
cell the (200)
(200), (020) and (002) planes all have the
same spacing and only one line is formed, called the (200) line.
However, this line splits into two when the cell becomes tetragonal
since now the (002) plane spacing differs from the other two.
• When the cell becomes orthorhombic, all three spacings are
different and three lines are formed.
• Common in phase transformations, e.g. carbon steel slowly
cooled get Ferrite (BCC) Cementite (orthor.), and quench from
austentite ((FCC)) martensite ((I-tetragonal).
g
)
• In disorder-order reactions (CuAu), cubic in disordered state but
becomes either tetragonal or orthorhombic (depending on
temperature) when ordered.
11
11
Example: Phase Transformations in ZrO2
Low-pressure forms of
ZrO2.
Red=O Blue=Zr
Property
P
Lattice parameters (Å)
a
b
c
Temperature (K)
Coordination
Volume (Å3)
Density (g/cc)
Space Group
C bi ZrO
Cubic
Z O2
T
Tetragonal
l ZrO
Z O2
M
Monoclinic
li i ZrO
Z O2
5.034
5.034
5.034
>2570
Zr=8;
O1=4; O2=4
127.57
6.17
Fm-3m
5.037
5.037
5.113
1400 to 2570
Zr=8;
O1=4; O2=4
129.73
6.172
P42/nmc
5.098
5.171
5.264
<1400
Zr=7;
O1=3; O2=4
136.77
5.814
P21/c
XRD scan of a tetragonal ZrO2
thin film taken from UNT’s
Rigaku Ultima III high-resolution
XRD
12
Monoclinic
Cubic
13