Upload
others
View
0
Download
0
Embed Size (px)
Citation preview
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