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University of Stavangeruis.no

Magnetic ScatteringDiana Lucia Quintero Castro

Department of Mathematics and Natural Sciences

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 antiferromagnetnaked eye

Permanent magnet

Electron Configuration‐ Hund‘s Rules               back to modern physics

Magnetic Ions                  back to modern physics

Orbitalangularmomentum: Spinquantum number: Totalangularmomentum:

For an electron with l=1:   Lz=h

Bohr Magneton – used as a Unit

√

Quintero, PRB 2010

Total Magnetic moment

Magnetic Exchange InteractionAFM 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

.

12

∑ . . 4 ∑ . .

Neutrons only ever see the components of the magnetizationthat 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 cmSimilar 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 .

=

12

12

Spin Orbital

12

Fourier transform of the sample‘stotal magnetization

Axes

Scattering cross section – time dependence

Ω 212 . . 0 ′ .

For unpolarized neutrons, ↔  ‘ 

Ω 212 . 0

Squaredform factor

DW factor

Polarizationfactor

Fourier transform

Spin correlationfunction

Scattering cross section – Static

Ω 212

.

1

University of Stavangeruis.no

Magnetic Scattering IIDiana Lucia Quintero Castro

Department of Mathematics and Natural Sciences

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 212 . 0

Diffraction from a Paramagnet

Ω 212 . 0

0 213 1

Ω23 2 1

Diffuse scattering (continuosly distributed over all scattering directions)

Diffraction from a Ferromagnet

0

Proportional to the domain‘s magnetisation

.

∑ . =∑ .

Reciprocal lattice vector(magnetic)

Ω2

.

. .

. 2 .

Structure factor:

Nuclear Magnetic Nuclear‐MagneticIf:

4 1

0 1Polarized Beam!

Diffraction from a FerromagnetA

Diffraction from a Ferromagnet  IINi1.8Pt0.2MnGa

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

Diffraction from a simple cubic antiferromagnet I

Real SpaceReciprocal Space

am*bm*

Ω 212 . 0

A

B

Diffraction from a simple cubic antiferromagnet II

A

B

. . .

. 2

.

,

2

Sum overthe ions in thesublatticeA

Sum over theions in themagneticunit cell

1,  A

1, B

Ω2

1 . .

∑ .Magneticstructurefactor:

Diffraction from a simple cubic antiferromagnet III

. 2

.

+

. .

2,12 ,

12 ,

12

0, , ,

For a magnetic lattice: face centered cubic

Nuclear and magnetic Bragg scatter ocurr at different points in the reciprocal latticespace

Example: SrYb2O4

Example 2: SrYb2O4   IIRepresentation Analysis

Basireps ‐Fullprof

Example 2: SrYb2O4   IIIRietvel 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 methodsDiffractometers Triple axis spectrometers

Polarized diffractometers SANS