Magnetic Neutron Diffraction for Magnetic Structure Determination. - PowerPoint PPT Presentation

Magnetic Neutron Diffraction for Magnetic Structure Determination. Instruments and Methods Juan Rodríguez-Carvajal Institut Laue-Langevin, Grenoble, France E-mail: jrc@ill.eu 1

Outline: 1. Impact of magnetic structures in condensed matter physics and chemistry. 2. The basis of Neutron Diffraction for Magnetic structures. Instruments at ILL 3. Magnetic Crystallography: Shubnikov groups and representations, superspace groups 2

Impact of Magnetic structures Search in the Web of Science, TOPIC: (("magnetic structures" or "magnetic structure" or "spin configuration" or "magnetic ordering" or "spin ordering") and ("neutron" or "diffraction" or "refinement" or "determination")) 3

Impact of Magnetic structures Methods and Computing Programs Multiferroics Superconductors 4

Impact of Magnetic structures Nano particles Multiferroics Computing Methods Manganites, CO, OO 5 Heavy Fermions

Impact of Magnetic structures Search with Topic = “Magnetic structure” or “magnetic structures” Thin Films 6

Impact of Magnetic structures The impact of the magnetic structures as a field in physics and chemistry of condensed matter is strongly related to (1) the availability of computing tools for handling neutron diffraction data (2) adequate neutron instruments and to (3) the emergence of relevant topics in new materials. 7

Impact of Magnetic structures It is expected an increase of the impact of magnetic structures in the understanding of the electronic structure of materials with important functional properties. • Thin films • Multiferroics and magnetoelectrics • Superconductors • Thermoelectrics and magnetocalorics • Topological insulators • Frustrated magnets • … 8

Outline: 1. Impact of magnetic structures in condensed matter physics and chemistry. 2. The basis of Neutron Diffraction for Magnetic structures. Instruments at ILL 3. Magnetic Crystallography: Shubnikov groups and representations, superspace groups 9

History: Magnetic neutron scattering Historical milestones: Halpern et al. (1937-1941): First comprehensive theory of magnetic neutron scattering (Phys Rev 51 992; 52 52; 55 898; 59 960) Blume and Maleyev et al. (1963) Polarisation effects in the magnetic elastic scattering of slow neutrons. Phys. Rev. 130 (1963) 1670 Sov. Phys. Solid State 4 , 3461 Moon et al. (1969) Polarisation Analysis of Thermal Neutron scattering Phys. Rev. 181 (1969) 920 10

Magnetic Structures Pictures How we get the information to construct these kind of pictures? 11 Thanks to Navid Qureshi for the pictures!

Magnetic scattering: Fermi’s golden rule Differential neutron cross-section :   2    2 ' d k m 2          n   ' ' ' ( ) k s V k s E E     ' 2 m     ' 2 d dE k    ' ' s s This expression describes all processes in which: The state of the scatterer changes from  to  ’ - The wave vector of the neutron changes from k to k ’ where k ’ - lies within the solid angle d  The spin state of the neutron changes from s to s ’ - V m =  n . B is the potential felt by the neutron due to the magnetic field created by moving electrons. It has an orbital an spin part. 12

Magnetic scattering: magnetic fields Magnetic field due to spin and orbital moments of an electron: s j p j σ n Magnetic vector potential R of a dipolar field due to  j R j electron spin moment B j μ n R+R j   ˆ ˆ O      μ R  p R   2        j j 0   B B B B    j jS jL 2 2 4 R R       Biot-Savart law for a single electron with linear momentum p 13

Magnetic scattering: magnetic fields Evaluating the spatial part of the transition matrix element for electron j :   i ˆ ˆ ˆ      j   ' exp( ) ( ) ( ) k k QR Q s Q p Q V i m j j j   Q   Where is the momentum transfer ( ') Q k k Summing for all unpaired electrons we obtain:  ˆ ˆ ˆ ˆ       j ' ( ( ) ) ( ) ( ( ). ). ( ) k k Q M Q Q M Q M Q Q Q M Q V  m j M  ( Q ) is the perpendicular component of the Fourier transform of the magnetisation in the scattering object to the scattering vector. It includes the orbital and spin contributions. 14

Magnetic scattering M  ( Q ) is the perpendicular component of the Fourier transform of the magnetisation in the sample to the scattering vector.     3 Magnetic interaction vector ( ) ( )exp M Q M r i Q·r d r        M e M e M e (e M) Q= Q e M     d  2 *   ( ) M M r    0   d  * I M M M    N   mag  Magnetic structure factor: ( ) ( )exp(2 · ) M H m H r p f H i m m m  1 m Neutrons only see the component of the magnetisation that is perpendicular to the scattering vector 15

Scattering by a collection of magnetic atoms We will consider in the following only elastic scattering. We suppose the magnetic matter made of atoms with unpaired electrons that remain close to the nuclei.   Vector position of electron e : R R r e lj je The Fourier transform of the magnetization can be written in discrete form as      ( ) exp( · ) exp( · ) exp( · ) M Q s i Q R i Q R i Qr s e e lj je je e lj e j     ρ r Qr 3 r ( ) exp( · ) ( )exp( · ) F Q s i Q r i d j je je j e     3 ( ) ( )exp( · ) ( ) F Q m r Qr r m i d f Q j j j j j   ( ) ( )exp( · ) M Q m f Q i Q R lj lj lj lj 16

Scattering by a collection of magnetic atoms     ρ r Qr r 3 ( ) exp( · ) ( )exp( · ) F Q s Q i r i d j je je j e     3 ( ) ( )exp( · ) ( ) F Q m r i Qr d r m f Q j j j j j If we use the common variable s=sin  /  , then the expression of the form factor is the following:   ( ) ( ) f s W j s l l  0,2,4,6 l      2 2 ( ) ( ) (4 )4 j s U r j sr r dr l l 0           2 2 2 2 ( ) exp{ } exp{ } exp{ } 2,4,6 j s s A a s B b s C c s D for l l l l l l l l l        2 2 2 ( ) exp{ } exp{ } exp{ } j s A a s B b s C c s D 0 0 0 0 0 0 0 0 17

Elastic Magnetic Scattering by a crystal (1) The Fourier transform of the magnetization of atomic discrete objects can be written in terms of atomic magnetic moments and a form factor for taking into account the spread of the density around the atoms   ( ) ( )exp( · ) M Q m f Q i Q R lj lj lj lj For a crystal with a commensurate magnetic structure the content of all unit cell is identical, so the expression above becomes factorised as:       ( ) ( )exp( · ) exp( · ) ( )exp(2 · ) M Q m f Q i Qr i Q R m f Q i Hr j j j l j j j j l j The lattice sum is only different from zero when Q =2  H , where H is a reciprocal lattice vector of the magnetic lattice. The vector M is then proportional to the magnetic structure factor of the magnetic cell 18

Elastic Magnetic Scattering by a crystal (2) For a general magnetic structure that can be described as a Fourier series:       2 m S exp i kR lj k j l   k      ( ) exp( 2 ) ( )exp(2 · ) M h S i kR f h i h R k j l lj lj lj k        ( ) ( )exp(2 · ) exp(2 ( )· ) M h h r S h k R f h i i j j k j l j k l     ( ) ( )exp(2 ( )· ) M h S f Q i H k r k j j j j The lattice sum is only different from zero when h-k is a reciprocal lattice vector H of the crystallographic lattice. The vector M is then proportional to the magnetic structure factor of the unit cell that now contains the Fourier coefficients S k j instead of the magnetic 19 moments m j .

Diff. Patterns of magnetic structures Portion of reciprocal space Magnetic reflections Nuclear reflections h = H+k Magnetic reflections : indexed by a set of propagation vectors { k } h is the scattering vector indexing a magnetic reflection H is a reciprocal vector of the crystallographic structure is one of the propagation vectors of the magnetic structure k ( k is reduced to the Brillouin zone) 20

Blume-Maleyev equations Polarized Neutrons Cross-section (Equation 1):    * * I N N M M   h h h h h         * * * N M N M P i M M P     h h h h i h h i Final polarisation (Equation 2):            * * * * P I N N P M M P P M M P M M       f h h h i h h i i h h i h h       * * i N M N M P   h h h h i       * * * N M N M i M M     h h h h h h 21