The Muppet’s Guide to: The Structure and Dynamics of Solids Neutron Diffraction and Magnetism.

The Muppet’s Guide to:The Structure and Dynamics of Solids

Neutron Diffraction and Magnetism

∂

Neutron ScatteringNeutron scattering is more complicated than x-rays because neutrons are defined both by their wavevector, k, and spin, s.

They can be polarised - ±S

Scattering cross section will depend on both the neutron-nuclear (structure) and the nuclear spin interaction with any magnetic moments

∂

Atomic Nuclear Factor - StructureVery strong interaction because it is the same force that

bind the nucleus together

Approximate the scattering potential as a Fermi pseudo potential:

22

V b rm

As r0 ~10-15 m the interaction is essentially point like

∂

Atomic Nuclear Factor - StructureThe spatial density distribution of the nucleus is an infinitesimal point with respect to the neutron wavelength.

Thus, nuclear scattering factor for neutrons is a constant and independent of q.

No sinq/l correction

∂

X-ray and Neutron comparison:

q exp q r exp q T N j jj T

A b i i

T

q q exp q r exp q Tj jj

A fi i

Neutrons:

X-rays:

Neutrons: Point like, isotope dependent, sensitive to light elements. Contrast varied by isotope substitution.

X-rays: Extended scatters, depends only on number of electrons – can’t ‘see’ light elements. Contrast change through anomalous scattering

∂

Contrast MatchingUnlike x-rays the neutron cross section is isotope dependent.

Isotope b1H, Hydrogen -3.7406(11)2H, Deuterium 6.671(4)3H, Tritium 4.792(27)

Controlled mixing of 1H and 2H allows contrast to be changed. Very powerful technique for soft condensed matter.

∂

Magnetic Interactions…The magnetic moment of the neutron interacts with the magnetic moment of any unpaired electrons within a crystal - and this probes the magnetic structure.

energy associated with the neutron magnetic moment, mN in the internal field of the ion, H

SPIN ORBIT

V curlp

n ni B ir Hr

r

r

r2 2

2

∂

magnetic scattering amplitude for an ion is related to the Fourier Transform of the total magnetisation density, M(r)::

As the magnetism arises from unpaired electrons in outer shells and not the nucleus there is a

dependence on intensity, similar to the sin( ) q / l

used for x-rays

3M q M r exp q ri d r

∂

Scattering in Reciprocal Space

T

q q exp q r exp q Tj jj

A f i i Peak positions and intensity tell us about the structure:

Peak PositionPeriodicity within the

sample

Peak Width Extent of periodicity

Peak Intensity Atomic positions

Underlying translational symmetry

Particle / Grain size

Order / disorder

T

q exp q r exp q Tj jj

A b i i

∂

Magnetic Super-Structures

Ferromagnetic:Magnetic and charge have the same unit cell

aa

c

c

Anti-Ferromagnetic: Magnetic and charge have the different unit cells. Magnetic cell double nuclear cell.

∂

RuSrGdCuO Powder

Magnetic Diffraction Peaks

∂

Conventional Wisdom

X-raysEnergies typically keV

(Elastic scattering)Scatters from electrons

(High Z materials)Strongly absorbed

Good for imagingHigh FluxHigh resolution

Everyone knows that x-rays are good for studying structure

∂

Magnetic X-ray Scattering

e spinf r R q f

arg exp[ ]Ch eV

f R q r iq r dr Z

argCh e spinff f

Recall the atomic scattering factor:

Atomic scattering length:

arg ( ) ( )ik r ik r iQ rch e fi

V V

f k V k e V r e V r e

∂

From and e- to a magnetic atom

M. Blume and D. Gibbs, Phys. Rev. B 37 1779-1789 (1988)

S(q) is the FT of time averaged spin density and L(q) the corresponding FT of the orbital density

2Total ef r R q i L q A S q B

mc

ˆ ˆ ˆ ˆ ˆ ˆB q q q q q q

A and B are vectors which contain the polarisation dependences….

ˆ ˆA q q

As electrons are orbiting a nucleus we need to include orbital as well as spin components in the magnetic term.

∂

Magnetic X-ray Scattering

M. Blume and D. Gibbs, Phys. Rev. B 37 1779-1789 (1988) & F. de Bergevin PRL 39A(2) 1972

2Total ef r R q i L q A S q B

mc

The maximum intensity is about 2counts / min above the background

Counts in 225 mins. or 3¾hrs Data set takes 4 days to collect

Pre-factor relates reduces magnetic scattering intensity by ~106.

∂

Realistic Magnetic X-ray scattering

Make electronic interactions sensitive to the magnetic moment.

X-ray Magnetic Circular Dichroism (XMCD)2 step process that couples circular polarised x-rays to a the absorption processes in a magnetic material

X-ray analogue to the Kerr effect

2Total ef r R q i L q A S q B

mc

∂

Electronic resonancesA core electron is excited and creates a spin polarised photoelectronExchange split final states act as a filter of the spin

Magnetic sensitivity comes through the spin-orbit coupling and exchange and has strong polarisation dependence (MOKE)

Courtesy W. Kuch, Freie Universität Berlin

∂

XMCD Examples at Resonant Edges

From Magnetism by J. Stöhr and H.C. Siegmann, Springer