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University of Stavangeruis.no
Magnetic ScatteringDiana Lucia Quintero Castro
Department of Mathematics and Natural Sciences
14/09/2017
1
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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
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Magnetic Materials
GdFe multilayer films
magnetic force microscope
Length Scale
Magnetic neutron diffraction
Kagome antiferromagnetnaked eye
Permanent magnet
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Electron Configuration‐ Hund‘s Rules back to modern physics
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Magnetic Ions back to modern physics
Orbitalangularmomentum: Spinquantum number: Totalangularmomentum:
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For an electron with l=1: Lz=h
Bohr Magneton – used as a Unit
√
Quintero, PRB 2010
Total Magnetic moment
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Magnetic Exchange InteractionAFM interaction FM interaction
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Static Magnetic Ordering
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Example: Manganosite (MnO)
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C. G. Shull & J. S. Smart, Phys. Rev. 76 (1949) 1256
Mn2+Electronic configuration:(3d5) S = 5/2, l=0,
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Partial differential cross section
′ ′
Dipole‐dipole interaction
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Magnetic Moment of Electron Systems back to electrodynamics
Orbital contribution:
2.0023
Spin contribution:
Bohr magneton:
By now—Only spin contribution
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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:
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Potential energy of a dipole in a field
Potential:
Torque:
Force:
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Generated Magnetic Field by one electron
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Generated magnetic field by multiple electrons
4 . electron j
neutron
Ω 2
Ω 2 4 .
Back to the partial differential cross section
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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
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https://www.ill.eu/sites/ccsl/ffacts/ffachtml.html
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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
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Generalization
r 2 .
=
12
12
Spin Orbital
12
Fourier transform of the sample‘stotal magnetization
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Axes
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Scattering cross section – time dependence
Ω 212 . . 0 ′ .
For unpolarized neutrons, ↔ ‘
Ω 212 . 0
Squaredform factor
DW factor
Polarizationfactor
Fourier transform
Spin correlationfunction
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Scattering cross section – Static
Ω 212
.
1
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University of Stavangeruis.no
Magnetic Scattering IIDiana Lucia Quintero Castro
Department of Mathematics and Natural Sciences
14/09/2017
1
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Contents‐ Second part
• Paramagnet• Ferromagnet• Antiferromagnet• Examples: MnO and SrYb2O4• Superconductors• Diffuse elastic magnetic scattering• 2D magnets• Parametric studies• Experimental methods
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Scattering cross section
Ω 212 . 0
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Diffraction from a Paramagnet
Ω 212 . 0
0 213 1
Ω23 2 1
Diffuse scattering (continuosly distributed over all scattering directions)
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Diffraction from a Ferromagnet
0
Proportional to the domain‘s magnetisation
.
∑ . =∑ .
Reciprocal lattice vector(magnetic)
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Ω2
.
. .
. 2 .
Structure factor:
Nuclear Magnetic Nuclear‐MagneticIf:
4 1
0 1Polarized Beam!
Diffraction from a FerromagnetA
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Diffraction from a Ferromagnet IINi1.8Pt0.2MnGa
Singh, Sanjay, et al. APPLIED PHYSICS LETTERS 171904 (2012)
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Diffraction from a simple cubic antiferromagnet I
Real SpaceReciprocal Space
am*bm*
Ω 212 . 0
A
B
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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:
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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
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Example: SrYb2O4
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Example 2: SrYb2O4 IIRepresentation Analysis
Basireps ‐Fullprof
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Example 2: SrYb2O4 IIIRietvel Refinement
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Example 2: SrYb2O4 IV
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Flux line lattices in Superconductors
Meissner effect
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Diffuse elastic magnetic scattering
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Short range magnetic order
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Short range magnetic order II
Petrenko, et al., Phys. Rev. B 78, 184410 (2008)Hayes, et al., Phys. Rev. B 84, 174435 (2011).
SrEr2O4
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Parametric studies
Zhao 2008 Toft-Petersen
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Experimental methodsDiffractometers Triple axis spectrometers
Polarized diffractometers SANS