Magnetic Scattering Diana Lucia Quintero Castro Department of - - PowerPoint PPT Presentation

University of S tavanger

uis.no

Magnetic Scattering

Diana Lucia Quintero Castro

Department of Mathematics and Natural S ciences

14/09/2017 1

Contents‐ First part

• Introduction to Magnetism
• Example 1: MnO
• Partial differential cross section
• Electron and Neutron dipolar interaction
• Magnetic matrix element
• Time independent scattering cross section –

Magnetic diffraction

Ch 7 ‐ 8

Magnetic Materials

GdFe multilayer films magnetic force microscope Length Scale Magnetic neutron diffraction Kagome antiferromagnet naked eye Permanent magnet

Electron Configuration‐ Hund‘s Rules back to modern physics

Magnetic Ions back to modern physics

Orbital angular momentum:

• Spin quantum number:
• Total angular momentum:

For an electron with l=1: Lz=h

• Bohr Magneton – used as a Unit

√

Quintero, PRB 2010

Total Magnetic moment

Magnetic Exchange Interaction

• AFM interaction

FM interaction

Static Magnetic Ordering

Example: Manganosite (MnO)

• C. G. Shull & J. S. Smart, Phys. Rev. 76 (1949) 1256

Mn2+ Electronic configuration: (3d5) S = 5/2, l=0,

Partial differential cross section

• ′ ′

Dipole‐dipole interaction

Magnetic Moment of Electron Systems

back to electrodynamics

Orbital contribution:

• 2.0023

Spin contribution: Bohr magneton:

• By now—

Only spin contribution

Neutron‘s magnetic properties

The magnetic moment is given by the neutron‘s spin angular momentum

• Gyromagnetic ratio, 1.97
• : Pauli spin operator, eigenvalues 1
• And for the electron:

Potential energy of a dipole in a field

Potential: Torque: Force:

Generated Magnetic Field by one electron

Generated magnetic field by multiple electrons

• 4

.

• electron j

neutron

• Ω
• 2
• Ω
• 2
• 4

.

• Back to the partial differential cross section

The magnetic matrix element

• .
• 1

2 ∑ . .

• 4 ∑ .
• .
• Neutrons only ever see the components of the magnetization

that are perpendicular to the scattering vector!

r

• 2

. .

• Magnetic form factor:

Spatial extend of the spin density

https://www.ill.eu/sites/ccsl/ffacts/ffachtml.html

Scattering cross section

r

• 2

.

Where, r is the classical electron radius:

r

• 0.54 10 cm

Similar to the bound coherence scattering length for many nuclei

• We can only measure spin components perpendicular to the transfered momentum
• The strenght of the magnetic scattering is close to the nuclear scattering
• The magnetic scattering depends on the spatial distribution of the spin density of

the sample

• The magnetic scattering strength falls off at high wave vector transfers

Generalization

r

• 2

.

=

• 1

2

• 1

2

• Spin

Orbital

1 2

• Fourier transform of the sample‘s

total magnetization

Axes

Scattering cross section – time dependence

• Ω
• 2
• 1

2 .

• . 0 ′

.

• For unpolarized neutrons, ↔ ‘
• Ω
• 2
• 1

2 .

• Squared

form factor DW factor Polarization factor Fourier transform Spin correlation function

Scattering cross section – Static

• Ω
• 2
• 1

2 .

• 1

University of S tavanger

uis.no

Magnetic Scattering II

Diana Lucia Quintero Castro

Department of Mathematics and Natural S ciences

14/09/2017 1

Contents‐ Second part

• Paramagnet
• Ferromagnet
• Antiferromagnet
• Examples: MnO and SrYb2O4
• Superconductors
• Diffuse elastic magnetic scattering
• 2D magnets
• Parametric studies
• Experimental methods

Scattering cross section

• Ω
• 2
• 1

2 .

Diffraction from a Paramagnet

• Ω
• 2
• 1

2 .

• 2 1

3 1

• Ω 2

3

• 2
• 1

Diffuse scattering (continuosly distributed over all scattering directions)

Diffraction from a Ferromagnet

̂

• Proportional to the domain‘s magnetisation
• .
• ∑ .
• =

∑

• .
• Reciprocal lattice vector

(magnetic)

• Ω
• 2

.

• .
• .
• .
• 2
• .

Structure factor: Nuclear Magnetic Nuclear‐Magnetic If:

• 4 1

0 1 Polarized Beam!

Diffraction from a Ferromagnet

A

Diffraction from a Ferromagnet II

Ni1.8Pt0.2MnGa

Singh, Sanjay, et al. APPLIED PHYSICS LETTERS 171904 (2012)

Diffraction from a simple cubic antiferromagnet I

Real Space Reciprocal Space am* bm*

• Ω
• 2
• 1

2 .

• A

B

Diffraction from a simple cubic antiferromagnet II

A B

• .
• .
• .
• . 2
• .
• ,

2

Sum over the ions in the sublattice A Sum over the ions in the magnetic unit cell

1, A 1, B

• Ω
• 2
• 1

. ̂

• .
• ∑ .
• Magnetic

structure factor:

Diffraction from a simple cubic antiferromagnet III

. 2

• .
• +

.

.

• 2, 1

2 , 1 2 , 1 2 0, , , For a magnetic lattice: face centered cubic Nuclear and magnetic Bragg scatter ocurr at different points in the reciprocal lattice space

Example: SrYb2O4

Example 2: SrYb2O4 II

Representation Analysis Basireps ‐Fullprof

Example 2: SrYb2O4 III

Rietvel Refinement

Example 2: SrYb2O4 IV

Flux line lattices in Superconductors

Meissner effect

Diffuse elastic magnetic scattering

Short range magnetic order

Short range magnetic order II

Petrenko, et al., Phys. Rev. B 78, 184410 (2008) Hayes, et al., Phys. Rev. B 84, 174435 (2011).

SrEr2O4

Parametric studies

Zhao 2008 Toft-Petersen

Experimental methods

Diffractometers Triple axis spectrometers Polarized diffractometers SANS