1. X-Ray Diffraction
Background and Fundamentals
Prof. Thomas Key
School of Materials Engineering
1

2. Crystalline materials are characterized by
the orderly periodic arrangements of atoms.
The (200) planes
of atoms in NaCl
The (220) planes
of atoms in NaCl
• The unit cell is the basic repeating unit that defines a
crystal.
• Parallel planes of atoms intersecting the unit cell are
used to define directions and distances in the crystal.
– These crystallographic planes are identified by Miller indices.
2

3. Bragg’s law is a simplistic model to understand
what conditions are required for diffraction.
•
For parallel planes of atoms, with a space dhkl between the planes, constructive
interference only occurs when Bragg’s law is satisfied.
–
–
–
In our diffractometers, the X-ray wavelength λ is fixed.
Consequently, a family of planes produces a diffraction peak only at a specific angle θ.
Additionally, the plane normal must be parallel to the diffraction vector
•
•
•
•
θ
d hkl d hkl
λ = 2d hkl sin θ
θ
Plane normal: the direction perpendicular to a plane of atoms
Diffraction vector: the vector that bisects the angle between the incident and diffracted beam
The space between diffracting planes of atoms determines peak positions.
The peak intensity is determined by what atoms are in the diffracting plane.
3

4. Our powder diffractometers typically
use the Bragg-Brentano geometry.
Detector
X-ray
tube
Φ
ω
•
θ
2θ
Angles
– The incident angle (ω) is between the X-ray source and the sample.
– The diffracted angle (2θ) is between the incident beam and the detector.
– In plane rotation angle (Φ)
•
In “Coupled 2θ” Measurements:
– The incident angle ω is always ½ of the detector angle 2θ .
– The x-ray source is fixed, the sample rotates at θ °/min and the detector
rotates at 2θ °/min.
4

5. Coupled 2θ Measurements
X-ray
tube
Detector
Motorized
Source Slits
Φ
ω
•
θ
2θ
In “Coupled 2θ” Measurements:
– The incident angle ω is always ½ of the detector angle 2θ .
– The x-ray source is fixed, the sample rotates at θ °/min and the detector
rotates at 2θ °/min.
•
Angles
– The incident angle (ω) is between the X-ray source and the sample.
– The diffracted angle (2θ) is between the incident beam and the detector.
– In plane rotation angle (Φ)
5

6. The X-ray Shutter is the most important
safety device on a diffractometer
• X-rays exit the tube through X-ray
transparent Be windows.
• X-Ray safety shutters contain the
beam so that you may work in the
diffractometer without being
exposed to the X-rays.
• Being aware of the status of the
shutters is the most important
factor in working safely with X rays.
6

7. •
•
•
The wavelength of X rays is determined
by the anode of the X-ray source.
Electrons from the filament strike the target anode, producing
characteristic radiation via the photoelectric effect.
The anode material determines the wavelengths of characteristic radiation.
While we would prefer a monochromatic source, the X-ray beam actually
consists of several characteristic wavelengths of X rays.
K
L
M
7

8. Why does this sample second set
of peaks at higher 2θ values?
• Hints:
113
– It’s Alumina
– Cu source
– Detector has a single
channel analyzer
Kα1
Kα2
006
8

9. Diffraction Pattern Collected
Where A Ni Filter Is Used
To Remove Kβ
Kα1
Kα2
What could
W Lα1
this be?
Due to tungsten
contamination
Kβ
E ( keV ) = hυ =
hc 6.02
=
λ λ ( Α)
9

10. Wavelengths for X-Radiation are
Sometimes Updated
Copper
Anodes
Bearden
(1967)
Holzer et al.
(1997)
Cobalt
Anodes
Bearden
(1967)
Holzer et al.
(1997)
Cu Kα1
1.54056Å
1.540598 Å
Co Kα1
1.788965Å
1.789010 Å
Cu Kα2
1.54439Å
1.544426 Å
Co Kα2
1.792850Å
1.792900 Å
Cu Kβ
1.39220Å
1.392250 Å
Co Kβ
1.62079Å
1.620830 Å
Cr Kα1
2.28970Å
2.289760 Å
Molybdenum
Anodes
Chromium
Anodes
Mo Kα1
0.709300Å
Mo Kα2
0.713590Å
0.713609 Å
Cr Kα2
2.293606Å
2.293663 Å
Mo Kβ
0.632288Å
0.632305 Å
Cr Kβ
2.08487Å
2.084920 Å
•
0.709319 Å
Often quoted values from Cullity (1956) and Bearden, Rev. Mod. Phys. 39
(1967) are incorrect.
– Values from Bearden (1967) are reprinted in international Tables for X-Ray
Crystallography and most XRD textbooks.
•
Most recent values are from Hölzer et al. Phys. Rev. A 56 (1997)
10

11. Calculating Peak Positions
λ = 2d hkl sin θ
11

12. Lattice Parameters & Atomic Radii
Body Centered Cubic
(BCC)
4R
a=
3
a
Common BCC Metals
a
–
–
–
–
–
Chromium
Iron (α)
Molybdenum
Tantalum
Tungsten
12

13. Lattice Parameters & Atomic Radii
(110)
Body Centered Cubic
(BCC)
4R
a=
3
a
Common BCC Metals
–
–
–
–
–
Chromium
Iron (α)
Molybdenum
Tantalum
Tungsten
13

14. Lattice Parameters & Atomic Radii
Face Centered Cubic
(FCC)
a
a = 2R 2
Common FCC Metals
a
–
–
–
–
–
–
–
Aluminum
Copper
Gold
Lead
Nickel
Platinum
Silver 14

15. Lattice Parameters & Atomic Radii
(100)
Face Centered Cubic
(FCC)
a = 2R 2
Common FCC Metals
–
–
–
–
–
–
–
Aluminum
Copper
Gold
Lead
Nickel
Platinum
Silver 15

16. Planes and Family of Planes
<001>
(001)
Cubic
c
(010)
(100)
b
(011)
(110)
(101)
a
(111)
(111)
(111)
(planes)
<directions>
16

17. Other Families of Planes
17

18. Higher Order Planes
(Half Planes)
18

19. Not all Planes Produce Peaks
Peak Intensity
I hkl = I 0pCL P [ Fhkl ]
Structure Factors
2
 Only if [Fhkl]2 ≠0 does a peak appear
N
where
– Io = Intensity of the incident X-ray beam
– p = Multiplicity factor (a function of the
crystallography of the material)
– C = Experimental constant (related to temperature,
absorption, fluorescence, and crystal imperfection).
Temperature factor=e-2M; Absorption factor = A(θ).
– LP = Lorentz-Polarization factor.
Fhkl = ∑ fn exp[ 2π i ( hu n + kv n + lw n ) ]
n =1
– ƒn = atomic scattering factor of atom ‘n’ is a
measure of the scattering efficiency
– u,v,w are the atomic positions in the unit cell
– h,k,l are the Miller indices of the reflection.
– N is number of atoms in the unit cell
 The summation is performed over all atoms in
the unit cell.
These calculations are easily doable for
simple structures
19

20. Structure Factors: Useful Knowledge
(1)
• Atomic scattering factors vary as a
function of atomic number (Z) and
diffraction angle (θ)
• Values can be looked up in tables
–
(2)
Linear extrapolations are used for
calculating the values between those
listed.
Calculating structure factors involves complex exponential functions. Use the following
relationships to determine the values of the exponential:
20

21. 2. BCC Structure
Consider the BCC lattice with single atoms at each lattice point; its unit cell can be reduced to
two identical atoms. Atom #1 is at 0,0,0 and atom #2 is at ½, ½, ½.
For this case we have
.
.
.
(5)
Note: For atoms of the same type, f1 = f2 = f.
Observations:
(i)
If the sum (h + k + l) = even in Equation (5), Fhkl = 2f and
(ii)
If the sum (h + k + l) = odd in Equation (5), Fhkl = 0 and
Thus, diffractions from BCC planes where h + k + l is odd are of zero intensity. They are
forbidden reflections. These reflections are usually omitted from the reciprocal lattice.
21

22. 3. FCC Structure
The FCC unit cell has four atoms located at (0,0,0), (½,½,0), (½,0,½), and (0, ½,½).
It follows that, for the same kind of atoms, the structure factor the FCC structure is given by the
expression
.
.
.
.
.
(6)
If h, k, and l are all even or all odd (i.e. unmixed) then the sums h + k, h + l, and k + l are all
even integers, and each term in Equation (6) equals 1. Therefore, Fhkl = 4f. However, if h, k, and l
are mixed integers, then Fhkl = f(1+1-2) = 0.
22

23. Compound Structure Factors
Consider the compound ZnS (sphalerite). Sulphur
atoms occupy FCC sites with zinc atoms
displaced by ¼ ¼ ¼ from these sites. The unit cell
can be reduced to four atoms of sulphur and 4
atoms of zinc.
Consider a general unit cell for this type of
structure. Many important compounds adopt this
structure; examples include ZnS, GaAs, InSb, InP
and (AlGa)As. It can be reduced to 4 atoms of
type A at 000, 0 ½ ½, ½ 0 ½, ½ ½ 0 i.e. in the
FCC position and 4 atoms of type B at the sites ¼
¼ ¼ from the A sites. This can be expressed as:
The structure factors for this structure are:
F=0
F = 4(fA ± ifB)
F = 4(fA - fB)
F = 4(fA + fB)
if h, k, l mixed (just like FCC)
if h, k, l all odd
if h, k, l all even and h+ k+ l = 2n where n=odd (e.g. 200)
if h, k, l all even and h+ k+ l = 2n where n=even (e.g. 400)
23

24. X-Ray Diffraction Patterns
111
λ = 2d hkl sin θ
200
220
•
•
311
222
BCC or FCC?
Relative intensities determined by:
I hkl = I 0pCL P [ Fhkl ]
2
24

25. A random polycrystalline sample that contains thousands of
crystallites should exhibit all possible diffraction peaks
200
220
111
222
311
2θ
2θ
2θ
• For every set of planes, there will be a small percentage of crystallites that are properly
oriented to diffract (the plane perpendicular bisects the incident and diffracted beams).
• Basic assumptions of powder diffraction are that for every set of planes there is an equal
number of crystallites that will diffract and that there is a statistically relevant number of
25
crystallites, not just one or two.

26. Why are peaks missing?
JCPDF# 01-0994
200
220
111
222
311
•The sample is a cut piece of Morton’s Salt
•JCPDF# 01-0994 is supposed to fit it (Sodium Chloride Halite)
26

27. It’s a single crystal
(a big piece of rock salt)
200
220
111
222
311
2θ
At 27.42 °2θ, Bragg’s law
fulfilled for the (111) planes,
producing a diffraction peak.
The (200) planes would diffract at 31.82
°2θ; however, they are not properly
aligned to produce a diffraction peak
The (222) planes are parallel to the (111)
planes.
27

28. Questions
28

29. 29

30. Multiplicity (p) Matters
30

31. 31

32. 32

33. 33

34. 34

35. 35

36. Example 5
Radiation from a copper source -
Is that enough information?
“Professor my peaks split!”
36

37. X-radiation for diffraction measurements is
produced by a sealed tube or rotating anode.
• Sealed X-ray tubes tend to
operate at 1.8 to 3 kW.
• Rotating anode X-ray tubes
produce much more flux because
they operate at 9 to 18 kW.
– A rotating anode spins the anode
at 6000 rpm, helping to distribute
heat over a larger area and
therefore allowing the tube to be
run at higher power without
melting the target.
• Both sources generate X rays by
striking the anode target wth an
electron beam from a tungsten
filament.
– The target must be water cooled.
– The target and filament must be
contained in a vacuum.
H2O In
H2O Out
Cu
Be
window
ANODE
Be
window
eXRAYS
XRAYS
FILAMENT
(cathode)
metal
glass
(vacuum)
(vacuum)
AC CURRENT
37

38. Spectral Contamination in
Diffraction Patterns
Kα1
Kα1
Kα2
Kα2
Kα1
Kα2
• The Kα1 & Kα2 doublet will almost always be present
Kβ
W Lα1
– Very expensive optics can remove the Kα2 line
– Kα1 & Kα2 overlap heavily at low angles and are more
separated at high angles
• W lines form as the tube ages: the W filament
contaminates the target anode and becomes a new Xray source
• W and Kβ lines can be removed38 optics
with

39. Divergence slits are used to limit the
divergence of the incident X-ray beam.
•
The slits block X-rays that have too
great a divergence.
• The size of the divergence slit
influences peak intensity and peak
shapes.
• Narrow divergence slits:
– reduce the intensity of the X-ray beam
– reduce the length of the X-ray beam
hitting the sample
– produce sharper peaks
• the instrumental resolution is improved so
that closely spaced peaks can be
resolved.
39

40. Varying Irradiated area of the
sample
• the area of your sample that is illuminated by the
X-ray beam varies as a function of:
– incident angle of X rays
– divergence angle of the X rays
• at low angles, the beam might be wider than
your sample
– “beam spill-off”
40

41. The constant volume
assumption
• In a polycrystalline sample of ‘infinite’
thickness, the change in the irradiated
area as the incident angle varies is
compensated for by the change in the
penetration depth
• These two factors result in a constant
irradiated volume
– (as area decreases, depth increase; and vice
versa)
• This assumption is important for many
41
aspects of XRPD

42. divergence is that the length of the
beam illuminating the sample
becomes smaller as the incident
• The length of the
incident angle becomes larger.
beam is
determined by the
divergence slit,
goniometer radius, and
incident angle.
• This should be
considered when
choosing a divergence
slits size:
40.00
35.00
L
I
e 30.00
r
n
r
g 25.00
a
t
d
h 20.00
i
a
15.00
t
m
e
m 10.00
d
2°DS
1°DS
(
)
– if the divergence slit is
too large, the beam
may be significantly
longer than your sample
at low angles
– if the slit is too small,
you may not get enough
intensity from your
185mm Radius Goniometer, XRPD
0.5°DS
5.00
0.15°DS
0.00
0
20
40
60
2Theta (deg)
42
80
100

43. 43

44. Detectors
• point detectors
– observe one point of space at a time
• slow, but compatible with most/all optics
– scintillation and gas proportional detectors count all
photons, within an energy window, that hit them
– Si(Li) detectors can electronically analyze or filter
wavelengths
• position sensitive detectors
– linear PSDs observe all photons scattered along a
line from 2 to 10° long
– 2D area detectors observe all photons scattered
along a conic section
– gas proportional (gas on wire; microgap anodes)
• limited resolution, issues with deadtime and saturation
– CCD
44

45. Sources of Error in XRD Data
• Sample Displacement
– occurs when the sample is not on the focusing
circle (or in the center of the goniometer circle)
– The greatest source of errorcosmost data
2s in θ
∆ 2θ = −
(in radians )
– A systematic error:
R
• S is the amount of displacement, R is the goniometer
radius.
• at 28.4° 2theta, s=0.006” will result in a peak shift of
0.08°
– Can be minimized by using a zero background
sample holder
45

46. Other sources of error
• Axial divergence
– Due to divergence of the X-ray beam in plane with the sample
– creates asymmetric broadening of the peak toward low 2theta
angles
– Creates peak shift: negative below 90° 2theta and positive
above 90°
– Reduced by Soller slits and/or capillary lenses
• Flat specimen error
– The entire surface of a flat specimen cannot lie on the focusing
circle
– Creates asymmetric broadening toward low 2theta angles
– Reduced by small divergence slits; eliminated by parallel-beam
optics
• Poor counting statistics
– The sample is not made up of thousands of randomly oriented
crystallites, as assumed by most analysis techniques
46
– The sample might be textured or have preferred orientation

47. sample transparency error
• X Rays penetrate into your sample
– the depth of penetration depends on:
• the mass absorption coefficient of your sample
• the incident angle of the X-ray beam
• This produces errors because not all X rays
are diffracting from the same location
– Angular errors and peak asymmetry
– Greatest for organic and low absorbing (low atomic
number) samples
•
sin 2θ
∆ 2θ =
Can be eliminated by using parallel-beam
2 µR
optics or reduced by using a thin sample
µ is the linear mass absorption coefficient for a specific sample
47

48. Techniques in the XRD SEF
• X-ray Powder Diffraction (XRPD)
• Single Crystal Diffraction (SCD)
• Back-reflection Laue Diffraction (no
acronym)
• Grazing Incidence Angle Diffraction
(GIXD)
• X-ray Reflectivity (XRR)
• Small Angle X-ray Scattering (SAXS)
48

49. Available Free Software
• GSAS- Rietveld refinement of crystal structures
• FullProf- Rietveld refinement of crystal structures
• Rietan- Rietveld refinement of crystal structures
• PowderCell- crystal visualization and simulated
diffraction patterns
• JCryst- stereograms
49