III. Analytical Aspects Summary Cheetham & Day, Chapters 2, 3

Chemical Characterization of Solid-State Materials

• Chemical Composition: Bulk, Surface, …

Inductively-Coupled Plasma-Mass SpectrometryX-ray Photoelectron Spectroscopy Scanning Electron MicroscopyElectron MicroprobeElectrochemistry

• Atomic Structure: Long-range order; short-range order; …

DiffractionNuclear Magnetic ResonanceTransmission Electron Microscopy

• Oxidation States of Atoms

Photoelectron Spectroscopy

III. Analytical Aspects Diffraction Cheetham & Day, Chapter 2

Reference:On-Line Tutorial: Proffen (LANL), Neder (Erlangen), Billange (Michigan St.)

Interatomic Distances in solids ca. 1010 m (= 1 Å = 0.1 nm = 100 pm).

2 22

2

Light:

1 3Particles:

2 2 2 2kinetic

hcE

p hE mv kT

m m

T (K) (Å) E (eV) v (m/sec)

Cu K --- 1.54178 8042 3 108

Mo K --- 0.71069 17450 3 108

Neutrons 298 1.4573 0.0385 2.72 103

Electrons 5.17 105 1.5000 66.9 4.85 106

Hand-Outs: 1

III. Analytical Aspects Diffraction: Producing X-rays and Neutrons Cheetham & Day, Chapter 2

X-Rays: Light created by (i) accelerating electrical charges(ii) inducing transitions between energy states

Conventional X-Ray Tube

• Isotropic emission (just a small fraction is used);

• Single wavelength;

• Water-cooled (high current will melt the anode);

• Rotating the anode allows more power (e beam travels over anode surface)

(Anode)

X-Rays

(W coil)

1.5 kWTube

Hand-Outs: 1

III. Analytical Aspects Diffraction: Producing X-rays and Neutrons Cheetham & Day, Chapter 2

When electrons hit the anode, they (i) collide with atoms & slow down - bremsstrahlung (ii) cause sharp transitions of core electrons through ionization and relaxation

L(2p) K(1s): K1(2p1/2), K2(2p3/2) M(3p) K(1s): K1(3p1/2), K2(3p3/2) Bremsstrahlung

MIN

E

To further control the wavelength, use a (i) monochromator: diffraction from an oriented single crystal (Ge, SiO2) (ii) filter: absorption – Beer’s law, I = I0exp( l)

~ a3 (Absorption Edge)

Filter for Element Zwould be Z1 or Z2;e.g., Ni is a filter for CuZr is a filter for Mo.

Hand-Outs: 2

III. Analytical Aspects Diffraction: Producing X-rays and Neutrons Cheetham & Day, Chapter 2

Synchrotron Radiation: accelerate electrons in circular motion

v = c

R

E

H (wavevector k)

Radiates with power

42 4 2

22 2 22

2 2

3 31

e c e c EP

R R mc

Energy lost per revolution is42

2

4

3

e EE

R mc

32

3

4

E

mcRMIN

Bremsstrahlung

Brightness = 1012-1015 photons/secmm2mrad2(X-ray tubes are 107; Rotating anodes are 108-109)

ADVANTAGES High intensity, tunable , Intrinsically collimated, pulsed, polarized

DISADVANTAGES Large facility, beam time is competitive, possible radiation damage and heating

Hand-Outs: 2

III. Analytical Aspects Diffraction: Producing X-rays and Neutrons Cheetham & Day, Chapter 2

Synchrotron Radiation: possible applications include

• Spectroscopy: EXAFS (Extended X-ray Absorption Fine Structure)XANES (X-ray Absorption Near-Edge Spectroscopy)

• Very fast crystallography: time-resolved phenomena (100 ps – 1 week)

• High resolution / diffuse scattering with small samples: intermediate length scales

• Magnetic scattering: Magnetic ordering in Gd

• Inelastic scattering: Vibrational modes and amplitudes.

Hand-Outs: 2

III. Analytical Aspects Diffraction: Producing X-rays and Neutrons Cheetham & Day, Chapter 2

Neutrons: produced in nuclear reactorscreate broad spectral distribution of wavelengths

2

2

2

2

2

1

22

1

t

Dm

m

hmvE

kinetic

Spallation Sources: bombarding metal targets with 800 MeV H+; generate pulsed neutron beams – analyzed by time-of-flight (TOF) methods.

t

Hand-Outs: 2

III. Analytical Aspects Diffraction: A Scattering Process Cheetham & Day, Chapter 2

X-Rays are scattered by electrons.

k = (2/) z

E = (oscillating) electric field

H = (oscillating) magnetic field

Plane Waves: E = E0 ei(t + kr) = [E0 ei(t + 2z/)] x; E = = c/

Scattering by a Single Electron:

ki

kf

2

e

Oscillating E, so electron will oscillate (accelerate) and emit radiation in all directions q

Elastic Scattering: | ki | = | kf | = 2/

Scattering Vector: q = kf ki

2 = scattering angle sin4

sin2 iif

q kkkq

NOTE: q and k have units m1

Hand-Outs: 3

III. Analytical Aspects Diffraction: A Scattering Process Cheetham & Day, Chapter 2

Scattering by a Single Electron:

ki

kf

2

e

Oscillating E, so electron will oscillate (accelerate) and emit radiation in all directions q = scattering vector

2 = scattering angle

4 sinq

Electron is accelerated:

Amplitude of scattered radiation (along kf direction):

Intensity of scattered radiation:

m

eEa 0

rmc

eE

rc

aeE

rmc

eE

rc

aeE

ss

1;

2cos2cos

2

2

0

2,2

2

0

2||,

4

,|| , 22 4 2

0 0

1 11 cos 2

2s ss

I II e

I I m c r

Polarization Factor

Hand-Outs: 3

III. Analytical Aspects Diffraction: A Scattering Process Cheetham & Day, Chapter 2

Scattering by an Atom: Continuous, spherical distributions of electron density

Two scattered rays show phasedifference () related to the pathdifference () traveled by the two rays

2

ki

kf

2

2

(r)dV

(r)dV

r(1)

(2)

D(2)

D(1)

coscossin4)2(2

sincos2

22cossin

)2( )1()2(

qrr

r

rr

DD

With respect to 1 electron: dVeE

dVeE

eE

dVE ii

s

s )()(

)1(

)(

0

0 rr

DifferentialScatteringAmplitude

Hand-Outs: 4

III. Analytical Aspects Diffraction: Atomic Scattering Factors Cheetham & Day, Chapter 2

Atomic Scattering Factor: shows constructive interference – atomic size ~ X-rays

2

0

00

sin( ) (r) 4 ( )

sin( )

iZ

Z Zq

qrf q e dV r r dr

qr

f q f Z

(NO Interference)

Size of Atom ~ (X-rays)Interference increases with 2

Both have 10 electrons;Si4+ more like point ion than O2

(1) Difficult to distinguish elementswith similar Z by diffraction alone(use interatomic distances);

(2) Light atoms next to heavy atomsare difficult to find (large scattering angles help…)

4 sinq

q

Scattering Vector: q(2)

Phase: = q r

Hand-Outs: 4