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Magnetic Scattering Diana Lucia Quintero Castro Department of Mathematics and Natural S ciences University of S tavanger uis.no 14/09/2017 1 Contents First part Introduction to Magnetism Example 1: MnO Partial differential


  1. Magnetic Scattering Diana Lucia Quintero Castro Department of Mathematics and Natural S ciences University of S tavanger uis.no 14/09/2017 1

  2. 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

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

  4. Electron Configuration‐ Hund‘s Rules back to modern physics

  5. Magnetic Ions back to modern physics Orbital angular momentum : Spin quantum number: Total angular momentum: � � � �

  6. Total Magnetic moment � � ��� For an electron with l=1: Lz=h � �� � Bohr Magneton – used as a Unit � ��� � �� � √��� � �� Quintero, PRB 2010

  7. Magnetic Exchange Interaction FM interaction AFM interaction �� �� � �

  8. Static Magnetic Ordering �� �� � � � �� � � � �� � � � �� � �

  9. Example: Manganosite (MnO)

  10. Mn2+ Electronic configuration: (3d5) S = 5/2, l=0, C. G. Shull & J. S. Smart, Phys. Rev. 76 (1949) 1256

  11. Partial differential cross section �′ �′ � � � �� � � � � � � � � � � � � � � � � Dipole‐dipole interaction

  12. Magnetic Moment of Electron Systems back to electrodynamics Orbital contribution: � � By now— Only spin contribution Spin contribution: � � � � � � 2.0023 Bohr magneton: � � � �� �

  13. 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: � � � � � � �

  14. Potential energy of a dipole in a field Potential: � � � � Torque: Force:

  15. Generated Magnetic Field by one electron � � � � � � � �� � � � ��� � � � � � �� �� � � � ��� � � � � �� � � � � � � � � � �

  16. Generated magnetic field by multiple electrons neutron � � � � �� � � � � � � � � � � � � � � � � � � � � 4� �� � � � �. � � � � � � � � � electron j � � � � � � Back to the partial differential cross section � � � � � � � � � � ��� � � � � � � ��� � � � � � � � � � � �� 2�� � ��� � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � 4� �� � � � �. � � ��� ��� � � � � � � ��� � ��� � � 2�� � � � � � � � �

  17. 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

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

  19. Scattering cross section � �� � � � � � r � 2 ��� �. � � � �� Where, r � is the classical electron radius: � � Similar to the bound coherence � � � � � 0.54 � 10 ��� cm r � � � 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

  20. Generalization � �� � � � � � r � 2 ��� �. � � � �� � = �� �� � Orbital Spin � � � � 1 � � � � � 1 � � �� � � � � �� � � ��� � 2� � 2� � � � � � � � � � � 1 ���� 2� � Fourier transform of the sample‘s total magnetization

  21. Axes

  22. Scattering cross section – time dependence � � � � � � � � � 1 2�� � �� � ���� � � ��.�� � � � � �. � � �0� �′ � � � ��� � ��� � � 2 � � � � � �. � � ��� � � �Ω�� � � � � For unpolarized neutrons, � ↔ � ‘ � � � � � � � � � 1 � ��� � ��� �� � � 2�� � �� � ���� � � ��.�� � �� � �� � � � � ��� � �0�� � � � 2 � � � � � � � � � � �Ω�� � � � �� ��� Squared DW Spin correlation Polarization Fourier form factor factor function factor transform

  23. Scattering cross section – Static 1 �Ω � � � � �� � � 1 � ��� � ��� �� � � � ��� � � � � � 2�� � � ��.�� �� �� � 2 � � � � � � � � � � � � � � � �� ���

  24. Magnetic Scattering II Diana Lucia Quintero Castro Department of Mathematics and Natural S ciences University of S tavanger uis.no 14/09/2017 1

  25. Contents‐ Second part • Paramagnet • Ferromagnet • Antiferromagnet • Examples: MnO and SrYb 2 O 4 • Superconductors • Diffuse elastic magnetic scattering • 2D magnets • Parametric studies • Experimental methods

  26. Scattering cross section � � � � � � � � � 1 � ��� � ��� �� � � 2�� � �� � ���� � � ��.�� �� �� � ��� � � � � ��� � �0�� � � � 2 � � � � � � � � � � ��� � � � �� ���

  27. Diffraction from a Paramagnet � � � � � � � � � 1 � ��� � ��� �� � � 2�� � �� � ���� � � ��.�� �� �� � ��� � � � � ��� � �0�� � � � 2 � � � � � � � � � � ��� � � � �� ��� � � 2 � 1 � � � �� �� � � ��� � � � � �0�� � � � � � � � � 3 � �� ��� � 1� � �٠� 2 �� � � � � ��� � ��� � 1� 3 � 2 � � � � Diffuse scattering (continuosly distributed over all scattering directions)

  28. Diffraction from a Ferromagnet � � 0 � � � � � � � � � � � � �̂ Proportional to the domain‘s magnetisation � � � ��� � � � � � � �� �.�� �� � � � � �� � ∑ � ��.�� � � � � ∑ ��� �. � � � = � � � Reciprocal lattice vector (magnetic)

  29. Diffraction from a Ferromagnet A 2� � �� � � � � ��� � � � � � �Ω � � � � � � � �. � � � � � � � � � � � � Structure factor: � � � � � � � � � �� � � �� � ��. ��� � � ��.� � ��.� � ��.� � � � � � � � �� � � �� � 2� � � � � � � �� �� � � � � Nuclear Magnetic Nuclear‐Magnetic If: 4 � � � ��� � � 1 � � � � � � � � �� Polarized Beam! � 0 ��� � � �1

  30. Diffraction from a Ferromagnet II Ni 1.8 Pt 0.2 MnGa Singh, Sanjay, et al. APPLIED PHYSICS LETTERS 171904 (2012)

  31. Diffraction from a simple cubic antiferromagnet I Reciprocal Space Real Space A � � � � B b m * a m * � � � � � � � � � 1 � ��� � ��� �� � � 2�� � �� � ���� � � ��.�� �� �� � ��� � � � � ��� � �0�� � � � 2 � � � � � � � � � � ��� � � � �� ���

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