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X-Ray DiffractionBackground and Fundamentals
Prof. Thomas Key
School of Materials Engineering
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Crystalline materials are characterized by the orderly periodic arrangements of atoms.
• 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.
The (200) planes of atoms in NaCl
The (220) planes of atoms in NaCl
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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 is fixed.– Consequently, a family of planes produces a diffraction peak only at a specific angle .– Additionally, the plane normal must be parallel to the diffraction vector
• 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.
sin2 hkld
dh
kld
hkl
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Our powder diffractometers typically use the Bragg-Brentano geometry.
X-ray tube
Detector
Φ
• Angles– The incident angle (ω) is between the X-ray source and the sample.– The diffracted angle (2) is between the incident beam and the detector. – In plane rotation angle (Φ)
• In “Coupled 2θ” Measurements:– The incident angle is always ½ of the detector angle 2 . – The x-ray source is fixed, the sample rotates at °/min and the detector
rotates at 2 °/min.
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Coupled 2θ Measurements
X-ray tube
Detector
Φ
• In “Coupled 2θ” Measurements:– The incident angle is always ½ of the detector angle 2 . – The x-ray source is fixed, the sample rotates at °/min and the detector
rotates at 2 °/min.• Angles
– The incident angle (ω) is between the X-ray source and the sample.– The diffracted angle (2) is between the incident beam and the detector. – In plane rotation angle (Φ)
Motorized Source Slits
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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.
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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.
KL
M
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Why does this sample second set of peaks at higher 2θ values?
• Hints:– It’s Alumina– Cu source
– Detector has a single
channel analyzer
006
113
Kα1
Kα2
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Diffraction Pattern Collected Where A Ni Filter Is Used
To Remove KβK1
K2
What could this be?
K
W L1
Due to tungsten contamination
02.6hchkeVE
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Wavelengths for X-Radiation are Sometimes Updated
Copper
Anodes
Bearden
(1967)
Holzer et al.
(1997)
Cobalt
Anodes
Bearden
(1967)
Holzer et al.
(1997)
Cu K1 1.54056Å 1.540598 Å Co K1 1.788965Å 1.789010 Å
Cu K2 1.54439Å 1.544426 Å Co K2 1.792850Å 1.792900 Å
Cu K 1.39220Å 1.392250 Å Co K 1.62079Å 1.620830 Å
Molybdenum
Anodes
Chromium
Anodes
Mo K1 0.709300Å 0.709319 Å Cr K1 2.28970Å 2.289760 Å
Mo K2 0.713590Å 0.713609 Å Cr K2 2.293606Å 2.293663 Å
Mo K 0.632288Å 0.632305 Å Cr K 2.08487Å 2.084920 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)
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Calculating Peak Positions
sin2 hkld
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Lattice Parameters & Atomic Radii
3
4Ra
Common BCC Metals– Chromium
– Iron (α)
– Molybdenum
– Tantalum
– Tungsten
a
Body Centered Cubic (BCC)
a
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Lattice Parameters & Atomic Radii
3
4Ra
Common BCC Metals– Chromium
– Iron (α)
– Molybdenum
– Tantalum
– Tungsten
(110)
a
Body Centered Cubic (BCC)
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Lattice Parameters & Atomic Radii
22Ra Common FCC Metals
– Aluminum– Copper– Gold– Lead– Nickel– Platinum– Silver
Face Centered Cubic (FCC)
a
a
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Lattice Parameters & Atomic Radii
22Ra Common FCC Metals
– Aluminum– Copper– Gold– Lead– Nickel– Platinum– Silver
(100) Face Centered Cubic (FCC)
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Planes and Family of Planes
(011) (110) (101)
(111) (111) (111)
(001) (010) (100)
a
b
c<001>
(planes) <directions>
Cubic
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Other Families of Planes
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Higher Order Planes (Half Planes)
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Not all Planes Produce Peaks
Structure Factors Only if [Fhkl]2 ≠0 does a peak
appear
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.
These calculations are easily doable for simple structures
– 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.
2 hklP0hkl FpCLII
Peak Intensity
N
1nnnnnhkl lwkvhu i 2πexpfF
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Structure Factors: Useful Knowledge
• Atomic scattering factors vary as a function of atomic number (Z) and diffraction angle (θ)
• Values can be looked up in tables– 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:
xix
nin
nin
iii
iii
n
cos2expixexp
integeran is where,expinexp
integeran is where,1)exp(
16exp4exp2exp
15exp3expexp
(1)
(2)
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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
lkhiflkh
ififFhkl
exp1
2222exp02exp 21 . . . (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 22 4 fFhkl
(ii) If the sum (h + k + l) = odd in Equation (5), Fhkl = 0 and 02 hklF
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.
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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
222exp
222exp
222exp02exp
lkif
lhif
khififFhkl
lkilhikhifFhkl expexpexp1 . . . . . (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.
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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:
FCCBAhkl FlkhffF
2exp
khilhikhilkhffF BAhkl
expexpexp12
exp
The structure factors for this structure are:
F = 0 if h, k, l mixed (just like FCC) F = 4(fA ± ifB) if h, k, l all odd F = 4(fA - fB) if h, k, l all even and h+ k+ l = 2n where n=odd (e.g. 200) F = 4(fA + fB) if h, k, l all even and h+ k+ l = 2n where n=even (e.g. 400)
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X-Ray Diffraction Patterns
• BCC or FCC?• Relative intensities determined by:
111
200
220 311222
2 hklP0hkl FpCLII
sin2 hkld
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A random polycrystalline sample that contains thousands of crystallites should exhibit all possible diffraction peaks
2 2 2
• 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 crystallites, not just one or two.
111
200
220
311
222
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Why are peaks missing?
111
200
220
311
222
•The sample is a cut piece of Morton’s Salt
•JCPDF# 01-0994 is supposed to fit it (Sodium Chloride Halite)
JCPDF# 01-0994
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It’s a single crystal(a big piece of rock salt)
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At 27.42 °2, Bragg’s law fulfilled for the (111) planes, producing a diffraction peak.
The (200) planes would diffract at 31.82 °2; however, they are not properly aligned to produce a diffraction peak
The (222) planes are parallel to the (111) planes.
111
200
220
311
222
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Questions
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Multiplicity (p) Matters
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Example 5
Radiation from a copper source -
Is that enough information?
“Professor my peaks split!”
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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.
Cu
H2O In H2O Out
e-
Be
XRAYS
windowBe
XRAYS
FILAMENT
ANODE
(cathode)
AC CURRENT
window
metal
glass
(vacuum) (vacuum)
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Spectral Contamination in Diffraction Patterns
K1
K2
KW L1
K1
K2 K1
K2
• The K1 & K2 doublet will almost always be present– Very expensive optics can remove the K2 line– K1 & K2 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 X-ray source
• W and K lines can be removed with optics
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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.
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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”
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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 aspects of XRPD– Matching intensities to those in the PDF reference database– Crystal structure refinements– Quantitative phase analysis
• This assumption is not necessarily valid for thin films or small quantities of sample on a ZBH
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One by-product of the beam divergence is that the length of the
beam illuminating the sample becomes smaller as the incident
angle becomes larger. • The length of the incident beam is determined by the divergence slit, goniometer radius, and incident angle.
• This should be considered when choosing a divergence slits size:– 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 sample at higher angles
– Appendix A in the SOP contains a guide to help you choose a slit size.
• The width of the beam is constant: 12mm for the Rigaku RU300.
185mm Radius Goniometer, XRPD
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
0 20 40 60 80 100
2Theta (deg)
Irradiated
Length
(
mm
)
2°DS
1°DS
0.5°DS
0.15°DS
43
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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
• limited in size, expensive – solid state real-time multiple semiconductor strips
• high speed with high resolution, robust
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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 error in most data– A systematic error:
• 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– Can be corrected by using an internal calibration standard – Can be analyzed and compensated for with many data analysis
algorithms• For sample ID, simply remember that your peak positions may be shifted a
little bit
– Can be eliminated by using parallel-beam optics
)(cos2
2 radiansinR
s
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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– The sample might be textured or have preferred orientation
• Creates a systematic error in peak intensities• Some peaks might be entirely absent
– The sample might have large grain sizes• Produces ‘random’ peak intensities and/or spotty diffraction peaks
• http://www.gly.uga.edu/schroeder/geol6550/XRD.html
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
• Can be eliminated by using parallel-beam optics or reduced by using a thin sample
R
2
2sin2
is the linear mass absorption coefficient for a specific sample
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)
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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
