X-rays – more bits and pieces

Learning Outcomes

By the end of this section you should:• be aware of Compton scattering• understand how Moseley’s law relates wavelength to

atomic number • understand the uses and implementation of the filter

and monochromator within an X-ray instrument• be aware of the uses of synchrotron (X-ray) radiation

and some of its uses

Classical vs quantum

• In the classical treatment, X-rays interact with electrons in an atom, causing them to oscillate with the X-ray beam.

• The electron then acts as a source of an electric field with the same frequency

Electrons scatter X-rays with no frequency shift

Compton Scattering

• Some radiation is also scattered, resulting in a loss of energy [and hence, E=h, shorter frequency and, c= , longer wavelength].

• The change in frequency/wavelength depends on the angle of scattering.

This effect is known as Compton scattering• It is a quantum effect - remember classically

there should be no frequency shift.

)cos1(cm

h

e

Arthur Compton

1892-1962

Implications?

Calculate the maximum wavelength shift predicted from the Compton scattering equation.

)cos1(cm

h

e

cm

h2

e

831

34

1031011.9

10626.62

= 4.85 x 10-12 m = 0.05Å

Moseley’s Law

1913 2)Z(C

C ~ 0.75 Rc

~ 1 for K

~ 7.4 for L

Henry Moseley

1887-1915

Periodic Table

Moseley corrected anomalies:

27Co 58.93 28Ni 58.71 29Cu 63.54

18Ar 39.95 19K 39.10 20Ca 40.08

52Te 127.6 53I 126.9 54Xe 131.3

Also identified a gap at Z=43 (Tc)

Coster & von Hevesy predicted for new element - Hf

Absorption

• X-ray photons absorbed when E is slightly greater than that required to cause a transition

- i.e. wavelength slightly shorter than K

Absorption

So, as well as characteristic emission spectra, elements have characteristic absorption wavelengths

e.g. copper

Absorption - example

Element At. No. K K Kedge

Ni 28 1.66 1.50 1.49

Cu 29 1.54 1.39 1.38

Zn 30 1.44 1.30 1.29

• Ni does not absorb its own lines

• Ni absorbs CuK - useful

• Ni absorbs Zn K and K strongly

Uses of absorption

We want to choose an element which absorbs K [and high energy/low white radiation] but transmits K

e.g. Ni K absorption edge = 1.45 Å

As a general rule use an element whose Z is one or two less than that of the emitting atom

Monochromator

Choose a crystal (quartz, germanium etc.) with a strong reflection from one set of lattice planes, then orient the crystal at the Bragg angle for K1

= 1.540 Å = 2dhklsin

Example

A monochromator is made using the (111) planes of germanium, which is cubic, a = 5.66 Å. Calculate the angle at which it must be oriented to give CuK1 radiation (1.540 Å)

22

222

2 )66.5(

3

a

lkh

d

1

)27.32(

540.1sin

d2sin 11

d=3.27Å

=2d sin

= 13.62°

Synchrotron X-rays

When charged particles are accelerated in an external magnetic field (according to Lorentz force), they will emit radiation (and lose energy)

Theory proposed initially by Ivanenko and Pomeranchuk, 1944. First observed in 1947. (Physics Today article)

Synchrotron X-rays

Acceleration in a circle…

• Electrons are kept in a narrow path by magnets• Emit e.m. radiation ahead• Large spectral range• Very focussed and intense X-rays produced (GeV)

(also applications in particle, medical physics amongst other things)

Schematic

(1) electron gun (2) linear accelerator (3) booster synchrotron (4) storage ring (5) beamlines (6) experiment stations.

(From: Australian Synchrotron, Illustrator: Michael Payne)

APS Argonne

Inside the synchrotron

LINAC: linear accelerator

• Electrons emitted from cathode ~1100° C. • Accelerated by high-voltage alternating electric fields in

linac. Accelerates the electrons to 450 MeV - relativistic

Inside the synchrotron

Bending magnet

• Electrons injected into booster synchrotron (a ring of electromagnets); accelerated to 7 GeV

Inside the synchrotron

• 7 GeV electrons injected into the 1 km storage ring• Circle of > 1,000 electromagnets etc.

Storage ring

ESRF, Grenoble

ESRF, Grenoble

Daresbury SRS, UK

• Will close in December 2008

Diamond, Oxfordshire - schematic

Diamond, Oxon

February 2004

April 2004

Sept 2004

July 2006

Photos courtesy Diamond Light Source Ltd.

Diamond + ISIS, OxonPhoto courtesy Diamond Light Source Ltd.

Synchrotron vs lab data

• Much higher count rates signal to noise better• Wavelengths are variable.• Incident beam is usually monochromatic and parallel.• Very sharp peaks (smaller instrumental contribution)

– FWHM can be 10 times narrower – better resolution

Comparison

Ru0.95Sn0.05Sr2GdCu2O8

A. C. Mclaughlin et al. J. Mat Chem (2000)

Synchrotron (ESRF)

= 0.325104 Å

Lab X-ray

= 1.54056 Å

Synchrotron Diffraction - Uses• High resolution X-ray powder diffraction • “Resonant” X-ray powder diffraction (can select

wavelength)• Analysis of strain (see later)• Sample environment (as with neutrons)• Surface XRD• Diffraction on very small single crystals (0.0001 mm3)

A-amylose crystals, ESRF highlights, 2006

Back to absorption

• X-ray absorption - generally in the range 2 – 100 keV

Photoelectron ejected with energy equal to that of the incoming photon minus the binding energy.

Characteristic of element.

The ejected photoelectron then interacts with the surrounding atoms

Absorption - equations

Beer’s law for X-rays

)xexp(I

Im

0

Also written as function of m (mass of element) and A (area

of beam)

m is the mass absorption coefficient

IIo

x

3

4

AE

Zm

Absorption energies

• Energies of K edges Z2

• Elements with Z>18 have either a K or L edge between 3 and 35 keV

Interference effects

The ejected photoelectron then interacts with the surrounding atoms

This gives information on the local environment round a particular element within the crystal structure

Interference effects

XAS

X-ray Absorption spectroscopy complements diffraction

• Diffraction gives you information on average 3d structure of crystalline solids

• XAS gives you localised environment in solids (including glasses), liquids, gases.

Info on bonds, coordination, valence.

XANES/EXAFS

• X-ray Absorption – near edge structure• Extended X-ray Absorption – Fine Structure

Thin wafer of Silicon

XANES

EXAFS

More detail

Copper compound

Processed + FT

Intensity vs R (radius from central atom)

Summary

The interaction of X-rays with matter produces a small wavelength shift (Compton scattering)

The wavelength of X-rays varies as a function of atomic number - Moseley’s law

Filters can be used to eliminate K radiation; monochromators are used to select K1 radiation.

Synchrotrons can produce high intensity beams of X-rays suitable for structural studies

Absorption can be exploited to give localised information on elements within a crystal structure.