Diffraction: the Director’s Cut

Learning Outcomes

By the end of this section you should:• know the basic principles of neutron & electron

diffraction• be able to explain time-of-flight neutron diffraction

and make calculations relating tof to d-spacing• understand the uses of both techniques

Not just X-rays

There’s more to diffraction than X-rays, you know(but not much more)

(with apologies to the Smiths)

As we (Dr Gibson) stated previously (HO3)For diffraction from crystals:Interatomic distances 0.1 - 2 Åso = 0.1 - 2 ÅX-rays, electrons, neutrons suitable

Matter waves!

In 1901 the very first Nobel Prize in Physics was awarded

to Wilhelm Röntgen for his discovery of X-rays.

Wilhelm Röntgen

1845-1923

De Broglie

• Extended the idea of wave-particle duality• 1923 – particles can be wavelike• Idea that everything has a wavelength!

Louis de Broglie

1892-1987E = mc2 = (mc)c but momentum, p=mv

and for a photon v=c

mv

h

p

h

How to prove?

E = pc = p f but E=hf (Planck/Einstein)hf = p f

so

Using de Broglie

If we want a wavelength of 1.00 Å then…• h = 6.626 x 10-34 J s

• mN = 1.675 x 10-27 kg

How fast do the neutrons need to be travelling?

mv

h

p

h

1

ms4000m

hv

Neutron scattering

Neutron can be scattered by atoms by:• interaction with nucleus• interaction with spin of unpaired electrons - magnetic

interaction, magnetic scattering. This happens because the neutron has a magnetic moment. (later)

Also the interaction can be:• elastic (diffractometer) structural studies• inelastic (spectrometer)

loss of energy on scattering gives information on phonon dispersion (effect of vibrations in lattice) and stretching of bonds

Scattering from neutrons

X-rays: fj Z - can be calculated

Neutrons: small dependence of fj on Z but major part Z independent. fj must be determined experimentally

V

Good points/Bad points

Can detect light atoms Can often distinguish between adjacent atoms Can distinguish between isotopes Can accurately find atoms in presence of very high Z

atoms Covers a wide range of d-spacings - more hkl - BUT

Some atoms/isotopes good neutron absorbers (e.g. Cd, Gd (Gadolinium), 6Li (so use 7Li)

V has very low, ~0 scattering (but..) need neutron source VERY expensive (~£10,000 per DAY!)

Excellent complementary technique to XRD

Neutron source

• Need nuclear reactor (and accelerators, amongst other things)

• Very expensive to set up!

Clifford Schull

1915-2001

Bertram Brockhouse

1918-2003Nobel Prize 1994

"for the development of neutron spectroscopy"

“for the development of the neutron diffraction technique"

ISIS schematic

ISIS schematic

ILL, Grenoble, France

IPNS, Argonne, Chicago IL

Intense Pulsed Neutron Source

Other neutron sources are also available…

e.g.• Los Alamos Neutron Science Center (New Mexico, US)• Lucas Heights (Sydney Australia)• Oak Ridge (Tennessee, USA)• KENS (Tsukuba, Japan)• Chalk River (Ontario, Canada)• Risø (Roskilde, Denmark)

The experiment

At many sources (e.g. ILL at Grenoble) neutrons are produced by fission in a nuclear reactor and then selected by wavelength - but with neutrons there are no “characteristic” wavelengths:

..so by selecting a wavelength we lose neutrons and lose intensity

Alternative

UK neutron source at Rutherford Appleton Laboratory uses “time of flight” neutron diffraction

H- produced at source (pulsed)

Electrons stripped protons (~3 x 1013)

U or Ta - 25 neutrons per proton (i.e. 5 x 1014 per pulse)

Accelerator

Synchrotron

Target

Ion Source

Linear Accelerator

Extracted Proton Beam Line

Time-of-flight neutron diffraction

2dhklsin =

mv

h

where m,v = mass, velocity of neutron

L = length of flight path t = time of flight of neutron

t

Lv

• We are measuring d, so two variables, and • In lab X-ray powder diffraction, is constant, variable• In time-of-flight (t.o.f), is constant, variable

This takes advantage of the full “white” spectrum

• Two basic equations:

Time-of-flight equation

Combine: sind2

mL

ht

sindh

mL2t

L is a constant for the detector, h, m are constants so:

t d

d-spacings are discriminated by the time of arrival of the neutrons at the detector

Data

• e.g. from Polaris, ISIS (Medium resolution, high intensity diffractometer)

Range of d-spacings

• They thus interact with unpaired electrons in atoms• This leads to additional (magnetic) scattering

ExampleA neutron detector is located at a distance of 10m from the sample and at 145º. We measure a reflection with a tof of 14,200 s. What is its d-spacing?

As sindh

mLt2

so

sinmL

thd

2

d = 4.90 Å

sin14510101.6752

106.6261014200d

27

346

Polaris at ISIS

Errors

• The biggest error in the experiment is where the neutrons originate

• This gives an error in the flight path, L• typical value ~5cm

d

d

t

t

L

L

Hence as L increases, error in d is reduced - resolution of the instrument is improved

e.g. instrument at 10m compared to instrument at 100m

100m = HRPD, currently highest resolution in the world

Magnetic Diffraction

• Neutrons possess a magnetic dipole moment

Example:

MnO (also NiO, FeO)

Rock salt structure

Basics of magnetism

Ferromagnetic

Anti-ferromagnetic Ferrimagnetic

Paramagnetic

Magnetic transition

• Oxygen atoms missing for clarity• > 120 K

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 100 200 300T / K

Ma

gn

eti

c s

us

ce

pti

bil

ity

paraAF

Magnetic transition

• Oxygen atoms missing for clarity• < 120 K

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 100 200 300T / K

Ma

gn

eti

c s

us

ce

pti

bil

ity

paraAF

Magnetic transition

Schull, Strauser & Wollan, Phys Rev B 83 333 (1951)

3d view – magnetic transition in YMn2

With thanks to Prof Sue Kilcoyne:

R Cywinski, S H Kilcoyne and C A Scott, J. Phys C33 6473 (1991)

3d view – doping loses transition

With thanks to Prof Sue Kilcoyne:

R Cywinski, S H Kilcoyne and C A Scott, J. Phys C33 6473 (1991)

More Complex Structures

Heavy equipment

• Furnaces, cryostats, pressure cells, magnets, humidity chambers, etc.

Review of sample environments

CryostatCryomag

High Pressure cell “Paris-Edinburgh”

Heavy equipment

• Furnaces, cryostats, pressure cells, magnets, humidity chambers, etc.

Review of sample environments

Humidity chamber

High Pressure for low angle work

Electron Diffraction

Similar principle – matter waves, but me = 9.109 x 10-31 kg

• Also applied “accelerating potential” V such that:

em

eVv

2

Typical values 10-200 kV, so v up to 2.65 x 108 ms-1

Relativistic speeds!

Calculate v for an accelerating voltage of 10 kV. What is ? (Question sheet)

G. P. Thomson

Experiments performed at Marischal College in the late 1920's

(also Lester Werner and Clinton Davisson at Bell labs in New York)

100 keV electrons.

His father had won the Nobel prize for proving electrons were particles. G. P. won the prize for proving that they were waves…

George Paget Thomson

1892-1975

Electron Diffraction

Picture of diffraction taken by Thomson

Bragg’s Law redux

• Since is very small, is also very small, so we can rewrite Bragg’s law as:

= 2d

D

L

2

As previously, we can derive:

d ~ L/D

Instrument

See applet at: Matter.org.uk

Schematic

Unlike X-ray diffraction we can refocus to produce an image, as well as producing a diffraction pattern.

HREM

Refocussed image

Uses

• Can be used to look at individual crystallites: must be thin (why?)

• Useful to help determine unit cell parameters; need many orientations (see animation here)

• Shape of spots: streaking can give information on crystal size and shape

• Can identify packing defects (see later)

Added extra: EDX for elemental analysis:• Electrons knock out inner shell electrons• Characteristic X-rays emitted as outer

shell electron drops down to fill gap

Conclusions

• Both neutron and electron diffraction are very useful complementary techniques to X-ray diffraction

• Neutron diffraction has a number of advantages over X-ray diffraction – but cost is a major disadvantage!

• Both fission and spallation sources are used• Magnetic diffraction is possible due to the dipole

present with neutrons• Electrons can be focussed, allowing high resolution

imaging as well as diffraction• Information on defects and unit cells