Resonant magnetic x-ray scattering and Summary Resonant scattering –Why do it? –What is it? –How is it done? –Example(s) The Real World …. CaFe 2 As 2

Solving magnetic structures Determine the magnetic wavevector (what is the “magnetic unit cell”) Use the angular dependence of resonant and nonresonant scattering cross- sections to determine magnetic moment directions. Scattering amplitudes → magnitude of the ordered magnetic moment.

2p/d2p/d d Q 2d t = 2 p /2d tt Long-range order Bragg peaks I Q Bragg =2 p /d

Why Bother? Many of the technologically important RE compounds contain neutron opaque elements. Superior reciprocal space (Q) resolution allows more detailed study … reinvestigation of “solved” structures. Can be used for investigations of submillimeter-sized single crystals. Resonant magnetic scattering occurs at well-defined energies specific to elements of interest -- probe local magnetism. Studies of magnetic surfaces and interfaces.

X-ray Resonant Magnetic Scattering (XRMS) EFEF P 3/2 P 1/2 L 3 - edge Incoming photon outgoing photon (L 2, L 3 )-edge for rare-earths (6-10KeV) Electric multipole transition (dipole : 2p – 5d, quadrupole 2p – 4f) Dipole transition is dominant 4f : magnetic properties 5d : exchange splitting by 4f

TbNi 2 B 2 C

Non-resonant: d  /d  S 2 sin    outgoing  - pol.)  2sin 2  cos  (L 1 +S 1 ) + S 3 sin  }] 2 (outgoing  - pol.) Resonant (E1): d  /d  outgoing  - pol.)  M 1 cos  + M 3 sin    (outgoing  - pol.)  (k′· M) 2 Resonant (E2): Much more complicated, but can probe M 1, M 2, M 3 ^^ ^ analyzer sample

We can do this by plotting the q-dependence of integrated intensities (a la neutrons) Alternatively… Angular dependence of the scattering at (0 0 L ±  ) of GdCo 2 Ge 2 measured by resonant and nonresonant diffraction Gd L 3 edge

X-ray resonant magnetic scattering azimuth scans Intensity  |k' M| 2 Gd 5 Ge 4 Q k θfθf θiθi k' I  (Mk’) 2

Crystal structure Gd Ni Ge Magnetization measurement S.L. Bud’ko, Z. Islam, T.A. Wiener, I.R. Fisher, A.H. Lancerda, P.C. Canfield Journal of Magnetic Materials 205, 53 (1999) GdNi 2 Ge 2 – An Example TNTN TtTt

1 st harmonic 2 nd harmonic 3rd harmonic (magnetic) (charge)

So, what do you learn from diffraction? From peak positions –Lattice parameters and how they change with environmental conditions (e.g. temperature and pressure) From peak widths –Crystal quality (e.g. mosaic) –Presence of strain (e.g. longitudinal widths) From integrated intensities –Contents of the unit cell –Positions of atoms within the unit cell; magnetic ordering –Thermal parameters (thermal disorder)

CaFe 2 As 2 …an example

Peak positions: Temperature dependent studies

Peak Positions: Pressure dependent studies

Peak Widths – strain, crystallite size and mosaic Powder after grinding Single crystal mosaic

Integrated Intensities