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Magnetic Neutron Diffraction for Magnetic Structure
Determination.
Instruments and Methods
Juan Rodríguez-Carvajal
Institut Laue-Langevin, Grenoble, FranceE-mail: [email protected]
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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’
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'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
50
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.
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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