x-ray, neutron and e-beam scattering - iowa state...
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X-ray, Neutron and e-beam scattering
•
Introduction–
Why scattering?
–
Diffraction basics–
Neutrons and x-rays
•
Techniques–
Direct and reciprocal space
–
Single crystals–
Powders
•
CaFe2
As2
–
an example
•
atoms pack in periodic, 3D arrays
•
typical of:
26
Crystalline
materials...
-metals-many ceramics-some polymers
•
atoms have no “regular”
packing
•
occurs for:
Noncrystalline
materials...
-complex structures-rapid cooling
Si Oxygen
crystalline SiO2
noncrystalline SiO2"Amorphous" = NoncrystallineAdapted from Fig. 3.18(b),Callister 6e.
Adapted from Fig. 3.18(a),Callister 6e.
What is the atomic scale structure?
Distance between atoms ~ Å
(10-9 m)
d(hkl)
d(h’k’l’)
Visible light
X-rays
Resolution ~ wavelength
So, 10-9
m resolution requiresλ
~ 10-9 m
How do we study crystal structures?
Interference of two waves Double slit diffraction
2 slits 2 slits and 5 slits
You can also do this with light (as well as neutrons and electrons)!
Diffraction
Diffraction from periodic structures
2θBragg
d
2θ
I
Diffraction
nλ
= 2dsinθ
θ θ
Neutrons and x-rays•
Neutrons–
λ
~ interatomic
spacings–
E ~ elementary excitations–
Penetrates bulk matter (neutral particle)
–
Strong contrasts possible (H/D)–
Scattered strongly by magnetic moments
–
Low brilliance of sources•
Large samples–
Strong absorption for some elements (e.g. Cd, Gd, B)
•
X-rays–
λ
~ interatomic
spacings–
E >> elementary excitations–
Strong absorption for low energy photons
•
Not for high energies (e.g. 100keV)
–
Weak scattering for light elements, magnetic moments and little contrast for hydrocarbons
–
High brilliance of x-ray sources•
High resolution; small samples; high degree of coherence
•
Lets us look at “weak scattering processes”
Techniques•
Single crystal diffraction
•
Powder diffraction•
Anomalous (resonant) x-ray scattering
•
Small angle scattering•
Reflectivity and surface scattering
T x-ray sourceC detectorS specimen
The linguistics of scattering from periodic crystals
7 crystal systems
(cubic, tetragonal, orthorhombic, monoclinic, trigonal, hexagonal) 14 Bravais
lattices (above + centering (body, base, face))230 periodic space groups (14 Bravais
lattices + 32 crystallographic point groups)
Lost in reciprocal spaceFor an infinite 3D lattice defined by primitive vectors (a1
, a2
, a3
) we can define areciprocal lattice generated by:
For R
= m1
a1
+ m2
a2
+ m3
a3 andG
= m1
b1
+ m2
b2
+ m3
b3
eiG·R
= 1; G·R = 2π
x integer
G is normal to sets of planes of atoms
Each point (hkl) in the reciprocal lattice corresponds to a set of planes (hkl) in the real space lattice.
(H 0 0)(0 K
0)
Miller indices and reciprocal space
(100) Reflection = diffraction from planes of atoms spaced 2π/a apart(200) Reflection = diffraction from planes of atoms spaced 2π/2a apart
Reciprocal space, angle space and diffraction
nλ
= 2dhkl
sinθ
↔ k –
k0
= G; k = k0
= 2π/λ
2θ
Ewald
sphere; Radius = 2π/λ
For single crystal diffraction –both the detector angle and the sample orientation matter
Things to keep in mind about single crystal measurements
•
Because “low-energy”
x-rays are strongly absorbed by most materials: we are looking at only the first few microns from the surface.–
Surface quality matters quite a bit.
•
Surface oxidation/contamination–
Is this characteristic of the bulk?
–
High energy x-rays level the playing field. •
What are the consequences of using high enegy
x-rays?•
Extinction effects
•
Multiple scattering effects•
We are probing only a small volume of reciprocal space …. What else is out there?
Powder diffraction
The intensity at discrete points in reciprocal space are now distributed over a sphereOf radius G or 2π/dhkl
(sample angle is irrelevant)
Q ( )Å5 10 15
Inte
nsity
Q10 12 14
Rietveld
Refinement
Point detection
2D area detectors
PDF measurements
Raw data
Structure function
QrdQQSQrG sin]1)([2)(0∫∞
−=π
What happens if your sample is made of disordered dodecahedra?
•Sit on an atom and look at your neighborhood
•G(r) gives the probability of finding a neighbor at a distance r
•PDF is experimentally accessible
•PDF gives instantaneous structure.
Things to keep in mind about powder diffraction measurements
•
Easy to do, but make sure you have a good powder!!!
•
Powder diffraction is excellent for getting the “big picture” but since intensities are spread over a sphere, small (but
perhaps important) details are missed.•
Lose information about anisotropy
CaFe2
As2
…an example
Peak positions: Temperature dependent studies
Peak Positions: Pressure dependent studies
Peak Widths
–
strain, crystallite size and mosaic
20 30 40 50 60 70 80
100
1000
10000
Sn tracesSi standard
CaFe2As2
311
301
310
224
222
206
21500
8
007
116
21120
2
00611
4
112
103
004
101
Inte
nsity
(cou
nts)
2θ (deg)45.60 45.65 45.70
0
200
400
Fig. X1
Rockingcurve
FWHM =0.017 deg
Cou
nts
/ s
θ (deg)
(1 1 10)
Powder after grinding Single crystal mosaic
Integrated Intensities
What do you learn?•
From peak positions–
Lattice parameters and how they change with environmental conditions (e.g. temperature and pressure)
•
From peak widths–
Crystal quality (e.g. mosaic)–
Presence of strain (e.g. longitudinal widths)
•
From integrated intensities–
Contents of the unit cell–
Positions of atoms within the unit cell–
Thermal parameters (thermal disorder)
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