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

2. Neutron Scattering
Neutron 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

3. Atomic Nuclear Factor - Structure
Very strong interaction because it is the same force that bind the nucleus together
Approximate the scattering potential as a Fermi pseudo potential:
As r0 ~10-15 m the interaction is essentially point like

4. Atomic Nuclear Factor - Structure
The 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

5. X-ray and Neutron comparison:
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

6. Contrast Matching
Unlike x-rays the neutron cross section is isotope dependent.
Isotope b
1H, Hydrogen
(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.

7. 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

8. 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

9. Scattering in Reciprocal Space
Peak positions and intensity tell us about the structure:
Underlying translational symmetry
Periodicity within the sample
Peak Position
Peak Width
Extent of periodicity
Particle / Grain size
Peak Intensity
Atomic positions
Order / disorder

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

11. RuSrGdCuO Powder
Magnetic Diffraction Peaks

12. Conventional Wisdom
Everyone knows that x-rays are good for studying structure
X-rays
Energies typically keV
(Elastic scattering)
Scatters from electrons
(High Z materials)
Strongly absorbed
Good for imaging
High Flux
High resolution

13. Magnetic X-ray Scattering
Recall the atomic scattering factor:
Atomic scattering length:

14. From and e- to a magnetic atom
As electrons are orbiting a nucleus we need to include orbital as well as spin components in the magnetic term.
S(q) is the FT of time averaged spin density and L(q) the corresponding FT of the orbital density
A and B are vectors which contain the polarisation dependences….
M. Blume and D. Gibbs, Phys. Rev. B (1988)

15. Magnetic X-ray Scattering
Pre-factor relates reduces magnetic scattering intensity by ~106.
The maximum intensity is about 2counts / min above the background
Counts in 225 mins. or 3¾hrs
Data set takes 4 days to collect
M. Blume and D. Gibbs, Phys. Rev. B (1988) & F. de Bergevin PRL 39A(2) 1972

16. 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

17. Electronic resonances
A core electron is excited and creates a spin polarised photoelectron
Exchange 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

18. XMCD Examples at Resonant Edges
From Magnetism by J. Stöhr and H.C. Siegmann, Springer