Magnetic Neutron Diffraction for Magnetic Structure

Determination.

Instruments and Methods

Juan Rodríguez-Carvajal

Institut Laue-Langevin, Grenoble, FranceE-mail: jrc@ill.eu

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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

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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"))

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Impact of Magnetic structures

Multiferroics

Methods and Computing Programs

Superconductors

5

Impact of Magnetic structures

Heavy Fermions

Nano particles

Multiferroics

Manganites, CO, OO

ComputingMethods

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Impact of Magnetic structures

Search with Topic = “Magnetic structure” or “magnetic structures”

Thin Films

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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.

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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

• …

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

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) 1670Sov. Phys. Solid State 4, 3461

Moon et al. (1969)Polarisation Analysis of Thermal Neutron scattering Phys. Rev. 181 (1969) 920

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History: Magnetic neutron scattering

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Magnetic Structures Pictures

How we get the information to construct these kind of pictures?

Thanks to Navid Qureshi for the pictures!

Magnetic scattering: Fermi’s golden rule

Differential neutron cross-section:

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’

222

'2

' '

'' ' ' ( )

' 2k kn

m

s s

md ks V s E E

d dE k

Vm = 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:

0

2 2

ˆ ˆ2

4

μ R p RB B B

j jBj jS jL

R R

13

O

Rj

sj

j

pj

R

R+Rj

σn

μn

Bj

Magnetic vector potential

of a dipolar field due to

electron spin moment

Biot-Savart law for a

single electron with

linear momentum p

Magnetic scattering: magnetic fields

Evaluating the spatial part of the transition matrix element for

electron j:

ˆ ˆ ˆ' exp( ) ( ) ( )k k QR Q s Q p Qj

m j j j

iV i

Q

( ')Q k k Where is the momentum transfer

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Summing for all unpaired electrons we obtain:

ˆ ˆ ˆ ˆ' ( ( ) ) ( ) ( ( ). ). ( )k k Q M Q Q M Q M Q Q Q M Qj

m

j

V

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.

M(Q) is the perpendicular component of the Fourier transform of

the magnetisation in the sample to the scattering vector.

Magnetic interaction vector

Magnetic structure factor:

Neutrons only see the component of the magnetisation that is

perpendicular to the scattering vector

M

M

Q=Q e

Magnetic scattering

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3( ) ( )expM Q M r Q·r ri d

*M MI

e M e M e (eM M)

1

( ) ( )exp(2 · )M H m H rmagN

m m m

m

p f H i

2 *

0( ) M Md

rd

Scattering by a collection of magnetic atoms

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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.

R R re lj je

( ) exp( · ) exp( · ) exp( · )M Q s Q R Q R Q r sj

e e lj je je

e lj e

i i i 3

3

( ) exp( · ) ( )exp( · )

( ) ( )exp( · ) ( )

F Q s Q ρ r Qr r

F Q m r Q r r m

j je je j

e

j j j j j

i r i d

i d f Q

( ) ( )exp( · )M Q m Q Rlj lj lj

lj

f Q i

Vector position of electron e:

The Fourier transform of the magnetization can be written in

discrete form as

Scattering by a collection of magnetic atoms

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3

3

( ) exp( · ) ( )exp( · )

( ) ( )exp( · ) ( )

F Q s Q ρ r Qr r

F Q m r Q r r m

j je je j

e

j j j j j

i r i d

i d f Q

If we use the common variable

s=sin/, then the expression of

the form factor is the following:

0,2,4,6

2 2

0

( ) ( )

( ) ( ) (4 )4

l l

l

l l

f s W j s

j s U r j sr r dr

2 2 2 2

2 2 2

0 0 0 0 0 0 0 0

( ) exp{ } exp{ } exp{ } 2,4,6

( ) exp{ } exp{ } exp{ }

l l l l l l l lj s s A a s B b s C c s D for l

j s A a s B b s C c s D

Elastic Magnetic Scattering by a crystal (1)

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( ) ( )exp( · )M Q m Q Rlj lj lj

lj

f Q i

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

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 Q r Q R m H rj j j l j j j

j l j

f Q i i f Q i

The lattice sum is only different from zero when Q=2H, 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

Elastic Magnetic Scattering by a crystal (2)

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( ) exp( 2 ) ( )exp(2 · )k

k

M h S kR h Rj l lj lj

lj

i f h i

For a general magnetic structure that can be described as a

Fourier series:

( ) ( )exp(2 · ) exp(2 ( )· )

( ) ( )exp(2 ( )· )

k

k

k

M h h r S h k R

M h S H k r

j j j l

j l

j j j

j

f h i i

f Q i

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 Skj instead of the magnetic

moments mj.

k

k kRSm ljlj iexp 2

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Diff. Patterns of magnetic structures

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

k is one of the propagation vectors of the magnetic structure

( k is reduced to the Brillouin zone)

Portion of reciprocal space

Magnetic reflections

Nuclear reflections

h = H+k

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Blume-Maleyev equations

Cross-section (Equation 1):

* *

* * *

h h h h h

h h h h h h

M M

M M P M M Pi i

I N N

N N i

Final polarisation (Equation 2):

* * * *

* *

* * *

h h h h h h h h h

h h h h

h h h h h h

P P M M P P M M P M M

M M P

M M M M

f i i i i

i

I N N

i N N

N N i

Polarized Neutrons

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Blume-Maleyev equations

Simplified notation for the Cross-section

* *

* * *

MM

M M P M M Pi i

I NN

N N i

Simplified notation for the Final polarisation

* * * *

* * * * *

P P M M P P M M P M M

M M P M M M M

f i i i i

i

I NN

i N N N N i

Most common case

Polarized Neutrons

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Blume-Maleyev equations

* *M MI NN

The simplest case: the major part of experiments

using neutron diffraction for determining magnetic

structures use non-polarised neutrons

The intensity of a Bragg reflection may contains

contribution from nuclear and magnetic scattering

Nuclear: proportional to the square of the structure

factor

Magnetic: proportional to the square of the magnetic

interaction vector

Powder diffractometers for magnetic

structure determination @ILL:

D20, D1B (CRG-CNRS-CSIC), D2B

Polarised cold neutrons, diffuse scattering: D7

D1B … the most productive

instrument at ILL

High Flux Powder diffractometers @ ILL

D20 … the most versatile

and highest flux

High Flux Powder diffractometers @ ILL

Single Crystal magnetic diffraction at ILL:

High resolution, magnetism:

D10, D23 (polarised neutrons, CRG-CEA)

High-Q, hot neutrons: D9, D3 (polarised neutrons)

Laue diffractometer: CYCLOPS

D10 …thermal neutrons,

triple axis single crystal

diffractometer.

Single Xtal diffractometers at ILL

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Single Crystal Diffractometers

D10

Projects of the diffraction group @ILL:

XtremeD: A diffractometer for extreme

conditions: small samples, high-pressure and

magnetic field, powders and single crystals.

Upgrade of D10, increase the flux (x 10) for

small samples and thin films

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D10: upgrade (single crystals and thin films)

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XtremeD: new diffractometer for magnetism

and extreme conditions

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XtremeD: new diffractometer for magnetism

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CryoPAD: instrument to exploit B-M equations

Polarised neutrons (D3)

Two configurations:

High field: Measurement of flipping ratios Output: magnetisation densities

Zero field: CryoPAD

3D-PolarimetryMeasurement of polarisation tensorOutput: Precise magnetic

structures

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Blume-Maleyev equations

Matrix form for the scattered polarization.

P P +Pf iP

* * *M M M M W T

P RN N i

I I

With

We have defined the vector W= 2 NM*= WR+i WI that we

shall call the nuclear-magnetic interference vector.

The vector M* M is purely imaginary, so that T = i(M* M)

is a real vector that we shall call the chiral vector.

* * * * * * * * *P P M M P P M M P M M M M P M M M Mf i i i i iI NN i N N N N i

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Blume-Maleyev equations

P P +Pf iP

The polarization “tensor” is the matrix

* * * * * *1M M M M + M M M M

D S AP NN i N N

I

* *

* *

* *

0 0 0 0

0 0 0 0

0 0 0 0

M M

M M

M M

N M

N M

N M

NN I I

D NN I I

NN I I

Diagonal

*

* * * *

*

x x

y x y z y x y z

z z

M M

S M M M M M M M M

M M

Symmetric

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Blume-Maleyev equations

P P +Pf iP

The polarization “tensor” is the matrix

Symmetric

* * * * *

* * * * *

* * * * *

2

2

2

x x x y x y x z x z

x y x y y y y z y z

x z x z y z y z z z

M M M M M M M M M M

S M M M M M M M M M M

M M M M M M M M M M

* * * * * *1M M M M + M M M M

D S AP NN i N N

I

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Blume-Maleyev equations

P P +Pf iP

The polarization “tensor” is the matrix

* * * * * *1M M M M + M M M M

D S AP NN i N N

I

Anti-Symmetric

* * * *

* * * *

* * * *

0 ( ) ( ) 0

( ) 0 ( ) 0

( ) ( ) 0 0

z z y y Iz Iy

z z x x Iz Ix

y y x x Iy Ix

i NM N M i NM N M W W

A i NM N M i NM N M W W

i NM N M i NM N M W W

Coming from the vectorial product: * *M M Pii N N

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Blume-Maleyev equations

In practice, the scattered polarisation is measured by putting the incident

polarisation successively along x, y and z and measuring the components

x, y, z of the scattered polarisation for each setting of Pi . The list of the

calculated polarisations, to be compared with observations, are gathered

in the columns of the following matrix:

( , , )P

y zIy xN M M x Iz x

x y z

y z

Iz Ry N M M Ry mix Ryx y z

f x y z

y zRz Iy mix Rz N M M Rz

x y z

W TI I I T W T

I I I

W W I I I W M W

I I I

W W M W I I I W

I I I

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Blume-Maleyev equations

The terms of the polarization matrix have to be

calculated taking into account all kind of possible

domains existing in the crystal.

( , , )P

y zIy xN M M x Iz x

x y z

y z

Iz Ry N M M Ry mix Ryx y z

f x y z

y zRz Iy mix Rz N M M Rz

x y z

W TI I I T W T

I I I

W W I I I W M W

I I I

W W M W I I I W

I I I

Diffractometers for magnetism in pulsed sources

Currently available:WISH @ ISISsuperHRPD @ J-Park

Project of Single crystal diffractometer:MAGIC @ ESS

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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

4. Computing tools

43

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1968: Describing 3-dimensional Periodic Magnetic Structures by Shubnikov GroupsKoptsik, V.A.Soviet Physics Crystallography, 12 (5) , 723 (1968)

2001: Daniel B. Litvin provides for the first time the full

description of all Shubnikov (Magnetic Space) Groups.

Acta Cryst. A57, 729-730

Magnetic Structure Description and Determination

The magnetic moment (shortly called “spin”) of an atom can be considered as an “axial vector”. It may be associated to a “current loop”. The behaviour of elementary current loops under symmetry operators can be deduced from the behaviour of the “velocity” vector that is a “polar” vector.

A new operator can be introduced and noted as 1′, it flips the magnetic moment. This operator is called “spin reversal” operator or classical “time reversal” operator

Time reversal = spin reversal

(change the sense of the current)1′

Magnetic moments as axial vectors

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' { | }r r t n r r t n=r aj j h j j h i gjg h h

' det( )m m mj j jg h h

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One obtains a total of 1651 Shubnikov groups or four different types.

T1: 230 are of the form M0=G (“monochrome” groups)

T2: 230 of the form P=G+G1′ (“paramagnetic” or “grey” groups)

1191 of the form M= H + (G H)1′ (“black-white”, BW, groups).

T3: Among the BW group there are 674 in which the subgroup H G is an equi-translation group: H has the same translation group as G (first kind, BW1). Magnetic cell = Xtal cell

T4: The rest of black-white groups, 517, are equi-class group (second kind, BW2). In this last family the translation subgroup contains “anti-translations” => Magnetic cell bigger than Xtal cell

Shubnikov Magnetic Space Groups

Magnetic structure factor:

Magnetic structure factor

47

*M MI

1

( ) ( ) exp(2 · )M H m H rmagN

m m m

m

p f H i

M e M e M e (e M)

1

( ) det( ) {2 [( { } ]}M H m H t rn

j j j s s s j s j

j s

p O f H T h h exp i h

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

of the atom sited at the sublattice s of site j.det( )m mjs s s s jh h

The use of Shubnikov groups implies the use of the

magnetic unit cell for indexing the Bragg reflections

48

Magnetic Structure Description and Determination

Only recently the magnetic space groups are available in a

computer database

Magnetic Space Groups

Compiled by Harold T. Stokes and Branton J. Campbell

Brigham Young University, Provo, Utah, USA

June 2010

These data are based on data from:

Daniel Litvin, Magnetic Space Group Types,

Acta Cryst. A57 (2001) 729-730.

http://www.bk.psu.edu/faculty/litvin/Download.html

49

Magnetic Structure Description and Determination

Bertaut is the principal developer of the method base in irreps

Representation analysis of magnetic structuresE.F. Bertaut, Acta Cryst. (1968). A24, 217-231

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Magnetic Structure Description and Determination

Symmetry Analysis in Neutron Diffraction Studies of Magnetic Structures, JMMM 1979-1980

Yurii Alexandrovich Izyumov (1933-2010)

He published a series of 5 articles in Journal of Magnetism and Magnetic Materials on representation analysis and magnetic structure description and determination, giving explicit and general formulae for deducing the basis functions of irreps.

51

Formalism of propagation vectors

A magnetic structure is fully described by:

i) Wave-vector(s) or propagation vector(s) {k}.

ii) Fourier components Skj for each magnetic atom j and wave-vector k, Skj is a complex vector (6 components) !!!

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{ 2 }k

k

m S kRljs js lexp i

jsjs kk- SSNecessary condition for real moments mljs

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

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 }

2k k k kS R Ii i

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)

Formalism of propagation vectors

53

Magnetic Structure Factor

1

( ) ( ) {2 [( ){ } ]}k

M h h S H k t rn

j j j js s j

j s

p O f T exp i 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,… pj). Total number of atoms: N = Σ pj

{ }t sS Symmetry operators of the propagation vector

group or a subgroup

If no symmetry constraints are applied to Sk, the maximum number of

parameters for a general incommensurate structure is 6N (In practice

6N-1, because a global phase factor is irrelevant)

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Fourier coefficients and basis functions

k

kS Sjs n n

n

C js

1

2kM h h S h r

n

j j j n n s j

j n s

p O f T C js exp i

The fundamental hypothesis of the Representation Analysis of magnetic

structures is that the Fourier coefficients of a magnetic structure are

linear combinations of the basis functions of the irreducible

representation of the propagation vector group Gk , the “extended little

group” Gk,-k or the full space group G.

Basis vectorsFourier coeff.

55

Seminal papers on superspace approach

Wolff, P. M. de (1974). Acta Cryst. A30, 777-785.

Wolff, P. M. de (1977). Acta Cryst. A33, 493-497.

Wolff, P. M. de, Janssen, T. & Janner, A. (1981). Acta Cryst. A37, 625-636.

Superspace Symmetry

56

Magnetic superspace groups

The future of Magnetic Crystallography is clearly an unified approach of symmetry invariance and representations

Journal of Physics: Condensed Matter 24, 163201 (2012)

57

Superspace operations

4

4 ,0 , 4 , 4( ) [ sin(2 ) cos(2 )]

( , | ) 1( ) 1( )

r r kr

M M M M

t

l l

ns nc

n

b

x

x nx nx

time reversal otherwise

l

R

The application of (R, | t) operation to the modulated structure

change the structure to another one with the modulation functions

changed by a translation in the fourth coordinate

The original structure can be recovered by a translation in the

internal space and one can introduce a symmetry operator

'

4( )M M x

( , | , )t R

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k k HIR RR

HR is a reciprocal lattice vector of the basic structure and is

different of zero only if k contains a commensurate component.

If in the basic structure their atomic

modulation functions are not independent and should verify

( | )t r r R l

4 0 4

4 0 4

0

( ) det( ) ( )

( ) ( )

M H r M

u H r u

kt

I

I

R x x

R x x

R

R

R R

R

If belongs to the (3+1)-dim superspace group of an

incommensurate magnetic phase, the action of R on its propagation

vector k necessarily transforms this vector into a vector equivalent to

either k (RI = +1) or -k (RI = -1).

( , | , )t R

Superspace operations

59

( , 1| , )t k t 1The operators of the form: constitute the lattice

of the (3+1)-dim superspace group

Using the above basis the we can define 3+1 symmetry

operators of the form:

( , 1|100, ), ( , 1| 010, ), ( , 1| 001, ), ( , 1| 000,1)x y zk k k 1 1 1 1

The basic lattice translations are:

1 2 3 0

11 12 13

21 22 23

1 2 3 0

31 32 33

1 2 3

( , | , ) ( , | ) ( , , )

0

0( , , , )

0

t t t kt

t

S S

S S

R R R I

t t t

R R R

R R Rt t t

R R R

H H H R

R R

R

The operators of the magnetic superspace groups are the same a those

of superspace groups just extended with the time inversion label

Superspace operations

60

Simplified Seitz symbols for 3+1 symmetry operators

1 2 3 0 0

1 2 3 0 1 2 3 0 1 2 3 0

( , | , ) ( , | ) ( , , , )

( , | , ) { , | } { | } { ' | }

1 1 1 1 1( , | 00 , ) { , | 00 } *

2 2 2 2 2

t t t kt

t

k c

S S S t t t

t t t t t t or t t t

with

R R

R R R R

R R

Simple example

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

11'( )0

{1| 0000} , , , , 1

{1| 0000} , , , , 1

{1' | 0001/ 2} , , , 1/ 2, 1

{1' | 0001/ 2} , , , 1/ 2, 1

P s

x x x x

x x x x

x x x x

x x x x

Superspace operations

61

4

4 ,0 , 4 , 4( ) [ sin(2 ) cos(2 )]

r r kr

M M M M

l l

ns nc

n

x

x nx nx

l

, ,

1

( ) | | ( ) {2 }2

M MF H k H k H k Hr

nmc ms

mag

im p f m T m exp i

Simplified magnetic structure factor in the superspace

description with the notation used in JANA-2006

Magnetic Structure Factor

62

{ 2 } { 2 }k k

k k

M S k M krl lexp i exp i l

, ,

1 1( )exp{ 2 } ( )exp{ 2 }

2 2k k k k k kS R I M M krc si i i i

1

, ,

1

( ) {2 ( ) }

1( ) ( ) {2 }

2

k

k k

M h S H k r

M h M M Hr

n

n

c s

p f h T exp i

p f h T i exp i

Simplified magnetic structure factor in the

description used in FullProf (with notations adapted

to those used in superspace approach)

Magnetic Structure Factor

63

Magnetic Structure Factor

Superspace formalism computing modules are being

prepared to be used within FullProf

Thank you for your attention