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

PDF

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)