FullProf for magnetic structures. New features Juan Rodríguez-Carvajal Institut Laue-Langevin, Grenoble, France E-mail: jrc@ill.eu 1
Programs for handling Magnetic Structures Physica B 192, 55 (1993) 2
Programs for handling Magnetic Structures J. Rodriguez-Carvajal, Physica B 192, 55 (1993) Summary of the k -vector formalism for magnetic structure description In FullProf the sign of the phase was changed later… 3
Programs for handling Magnetic Structures J. Rodriguez-Carvajal, Physica B 192, 55 (1993) First description of some features of the program FullProf Description of the program MagSan for determining commensurate magnetic structures using Simulated Annealing (later included in FullProf for general structures) 4
Programs for handling Magnetic Structures J. Rodriguez-Carvajal, Physica B 192, 55 (1993) Introduction of the method C2-b for treating magnetic structures (commensurate and incommensurate) The phase conventions where changed in subsequent versions of the program FullProf Anisotropic broadening due to size and strain effects was already present for both commensurate and incommensurate structures 5
Outline: 1. The formalism of propagation vectors in FullProf 2. Representation Analysis and Magnetic Structures 3. Different options for treating magnetic structures in FullProf 6
Description of magnetic structures: k-formalism The position of atom j in unit-cell l is given by: m lj R lj = R l + r j r j R l where R l is a pure lattice translation 7
Formalism of propagation vectors Whatever kind of magnetic structure in a crystal can be described mathematically by using a Fourier series 2 m S kR exp i lj k j l m lj k r j R l R R r l a l b l c x a y b z c 1 2 3 lj l j j j j Necessary condition for real m lj S S - k j k j 8
Formalism of propagation vectors Another convention (Used in Superspace formalism) 2 2 ( ) m M exp i kR M exp i k R r lj k j lj k j l j k k 1 ( cos sin ) M M i M k j k j k j 2 For a single pair (k,-k) and its harmonics: sin cos sin(2 ) cos(2 ) m M n kR M n kR lj n j k lj n j k lj n sin cos ( ) sin(2 ) cos(2 ) m m x M nx M nx 4 4 4 lj n j k n j k n 9
Formalism of propagation vectors A magnetic structure is fully described by: i) Wave-vector(s) or propagation vector(s) { k } . ii) Fourier components S k j for each magnetic atom j and wave-vector k , S k j is a complex vector (6 components) !!! 10
Formalism of k-vectors: a general formula { 2 } m S exp i kR ljs k js l k l : index of a direct lattice point (origin of an arbitrary unit cell) j : index for a Wyckoff site (orbit) s : index of a sublattice of the j site Necessary condition for real moments m ljs S S - k js k js General expression of the Fourier coefficients (complex vectors) for an arbitrary site (drop of js indices ) when k and – k are not equivalent: 1 ( )exp{ 2 } S R i I i k k k k 2 Only six parameters are independent. The writing above is convenient when relations between the vectors R and I are established (e.g. when | R |=| I |, or R . I =0) 11
Single propagation vector: k=(0,0,0) The propagation vector k =(0,0,0) is at the centre of the Brillouin Zone. 2 m S exp{ i kR } S lj k j l k j k • The magnetic structure may be described within the crystallographic unit cell • Magnetic symmetry: conventional crystallography plus spin reversal operator: crystallographic magnetic groups 12
Single propagation vector: k=1/2H The propagation vector is a special point of the Brillouin Zone surface and k = ½ H , where H is a reciprocal lattice vector. exp{ 2 } exp{ } m S i k R S i HR lj k j l k j l k ( ) n l m S -1 lj k j REAL Fourier coefficients magnetic moments The magnetic symmetry can be described using crystallographic magnetic space groups 13
Fourier coefficients of sinusoidal structures - k interior of the Brillouin zone (IBZ) (pair k , - k ) - Real S k , or imaginary component in the same direction as the real one exp( 2 ) exp(2 ) m S i kR S i kR lj k j l -k j l 1 ( 2 ) S m u exp i k j j j k j 2 2 cos ( ) m m u kR lj j j l k j 14
Fourier coefficients of helical structures - k interior of the Brillouin zone - Real component of S k perpendicular to the imaginary component 1[ ] ( 2 ) S m u im v exp i k j uj j vj j k j 2 cos ( ) sin ( ) m m u 2 kR m v 2 kR lj uj j l k j vj j l k j 15
How to play with magnetic structures and the k-vector formalism { 2 } m S kR exp i ljs k js l { } k The program FullProf Studio performs the above sum and represents graphically the magnetic structure. This program can help to learn about this formalism because the user can write manually the Fourier coefficients and see what is the corresponding magnetic structure immediately. Web site: http://www.ill.eu/sites/fullprof 16
Outline: 1. The formalism of propagation vectors in FullProf 2. Representation Analysis and Magnetic Structures 3. Different options for treating magnetic structures in FullProf 17
Magnetic Bragg Scattering Intensity (non-polarised neutrons) * * M M I N N h h h h h Magnetic interaction vector M e M(h) e M(h) e (e M(h)) h h h H k e Scattering vector h 18
Magnetic structure factor: Shubnikov groups The use of Shubnikov groups implies the use of the magnetic unit cell for indexing the Bragg reflections * I M M M e M e M e (e M) N mag ( ) ( )exp(2 · ) Magnetic structure factor: M H p m f H i H r m m m 1 m n independent magnetic sites labelled with the index j The index s labels the representative symmetry operators of the Shubnikov group: is the magnetic moment det( ) m h h m js s s s j of the atom sited at the sublattice s of site j . n ( ) det( ) {2 [( { } ]} M H p O f H T h h m exp i H h t r j j j s s s j s j 1 j s The maximum number of parameters n p is, in general, equal to 3 n magnetic moment components. Special positions make n p < 3 n . 19
Magnetic Structure Factor: k-vectors n ( ) ( ) {2 [( ){ } ]} M h p O f h T S exp i H k S t r j j j k js s j 1 j s j : index running for all magnetic atom sites in the magnetic asymmetric unit ( j =1,… n ) s : index running for all atoms of the orbit corresponding to the magnetic site j ( s =1,… p j ) . Total number of atoms: N = Σ p j { } S t s Symmetry operators of the propagation vector group or a subgroup If no symmetry constraints are applied to S k , the maximum number of parameters for a general incommensurate structure is 6N (In practice 6N-1, because a global phase factor is irrelevant) 20
The magnetic representation as direct product of permutation and axial representations An inspection to the explicit expression for the magnetic representation for the propagation vector k , the Wyckoff position j , with sublattices indexed by ( s , q ), shows that it may be considered as the direct product of the permutation representation, of dimension p j p j and explicit matrices: Permutation j 2 i ka k j j ( ) representation P g e gs , Perm qs q gs by the axial (or in general “vector”) representation, of dimension 3, constituted by the rotational part of the G k operators multiplied by -1 when the operator g ={ h | t h } corresponds to an improper rotation. Axial representation ( ) det( ) V g h h Axial Magnetic j 2 i ka k j j ( ) det( ) g e h h gs representation , , Mag q s q gs 21
Basis functions of the Irreps of G k The magnetic representation, hereafter called M irrespective of the indices, can be decomposed in irreducible representations of G k . We can calculate a priori the number of possible basis functions of the Irreps of G k describing the possible magnetic structures. This number is equal to the number of times the representation is contained in M times the dimension of . The projection operators provide the explicit expression of the basis vectors of the Irreps of G k 1 * ε k k j ( ) ( ) ( ) ( 1,... ) j g O g l [ ] s ( ) G n g G 0k 0k 1 * ε k j j k j ( ) ( ) exp(2 )det( ) j g i k a h h [ ] , gs s gq q ( ) n G g G q 0k 0k 22
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