Magnetic Neutron Diffraction
the basic formulas A.Daoud-aladine, (ISIS-RAL)
What’s a magnetic structure? Why study magnetic order?Example : Manganites
The magnetic structure factorUsing the k-vector Formalism (as required by the Fullprof input)(Alternative to Shubnikov symmetry, which are TABULATED crystallographic magnetic space groups : this is however restricted to commensurate structures)
Pitfalls
What’s a magnetic structure?
core
Ni2+
(atomic) magnetic moments (m) arise from quantum effects in atoms/ions with unpaired electrons
Intra-atomic electron correlationHund’s rule(maximum total S)+ Spin-orbit + CEF
m = gJ J (J=L+S 4f-rare earths)
m = gS S (3d-transition metals)
« Classical description»
« Quantum description»
What’s a magnetic structure?
Temperature (entropy) overcomes magnetic energy: Entropy essentially dominated by local magnetic momentfluctuations
Jij
Magnet: crystal containing magnetic atoms
S Sij ij i jE J
0Si
kT >> Jij,
paramagnetic (disordered) state
What’s a magnetic structure?
S Sij ij i jE J
Magnet: crystal contaning magnetic atoms
Jij0Si
Exemple here: Jij>0 Antiferromagnetic coupling
(AF)
kT < Jij,
Magnetic energy overcomes the entropy : Quasi-static configuration of magnetic momentwith small fluctuations that are made cooperative by the magnetic exchange (spin waves excitations )
Why study magnetic order?
• Fundamental properties of condensed matter. Exchange interactions related to the electronic structure.
• The first (necessary) step before determining the exchange interactions (generally, with inelastic neutron scattering)
• Permanent magnet industry. Chemical substitutions controlling single ion anisotropy, strength of effective interactions, canting angles, etc: NdFeB materials, SmCo5, hexaferrites, spinel ferrites.
• Spin electronics, thin films and mutilayers
core
kI=2/ uI kF=2/ uF
Q= kF - kI
r.rk de Fi . .rk IieIMF V kk
σμ NN
)(
)2
(4
.3
0
rB
Rσ
μ
S
BN R
s
N
Dipolar interaction term with one electron
rR
ri
)(rBS
core
Q
SQ
sQ.rQkk )
).(
).exp((.4
if
iVi
iiIMF
Total Sisi=> vector scattering amplitude for one atom
QSQQkk iIMF fV ).(.4
Magnetic scattering of a single atom
From Introduction to the Theory of Thermal Neutron Scattering, G SQUIRES – Cambridge University Press (1978)
Site A
Ideal cubic Perovskite
Pnma
[010]O
[001]C
[101]O
[010]C
La/Tb
n
n
aa
m
m
Mn O
LaMnO3 TN=150K
TbMnO3 TN=41K
Example: RMnO3 manganites
Mn-O-Mn
LaMnO3 TN=150K
Commensurate structure
AF
Example: RMnO3 manganites
T. Goto, et al. Phys. Rev. Lett. 92, 257201 (2004)
LaMnO3 TN=150K
Commensurate structure
AF
LaMnO3 : 50K and 150 K
Example: RMnO3 manganites
LaMnO3 TN=150K
Commensurate structure
AF
3d4
eg
t2g(e-
localized)
Crystalfield
HundJH
x2
dxy
dyz dxz
Jahn-TellerDistorsion
dz2
dx2
-y2
O
S=3/2
Electronic structure of Mn3+ (x=0)
Q
SQ
sQ.rQkk )
).(
).exp((.4
if
iVi
iiIMF scattering amplitude for one atom, tabulated f(Q)
Example: RMnO3 manganites
LaMnO3 TN=150K
Commensurate structure
AF
Example: RMnO3 manganites
T. Goto, et al. Phys. Rev. Lett. 92, 257201 (2004)
Example: RMnO3
manganites
T<TN2=23K Cycloid
TbMnO3 TN2<T<TN1=41K Sinusoidal LaMnO3 TN=150K
Commensurate structure
Incommensurate structures
AF
kk
2njnj
iexp k.RSm
What’s a magnetic structure? Why study magnetic order?Example : Manganites
The magnetic structure factor
Pitfalls
core
kI=2/ uI kF=2/ uF
Q= kF - kI
r.rk de Fi . .rk Iie )(
)2
(4
.3
0
rB
Rσ
μ
S
BN R
s
N
IMF V kk
σμ NN
core
Q
SQ
sQ.rQkk )
).(
).exp((.4
if
iVi
iiIMF
Total Si
Dipolar interaction term with one electron
rR
ri
)(rBS
si=> vector scattering amplitude for one atom
QSQQkk iIMF fV ).(.4
Magnetic scattering of a single atom
From Introduction to the Theory of Thermal Neutron Scattering, G SQUIRES – Cambridge University Press (1978)
kI=2/ uI
kF=2/ uF
Q= kF - kI
n
Magnetisation density: M(r)=MS(r)+ML(r)
rQ.rrMQM di )exp()()(
).exp(.)()(
)()(
iii
i if rQSQQM
QQMQQM
=> vector scattering amplitude for one atom
Magnetic structure factor
QSQQkk iIMF fV ).(.4
Magnetic interaction vector for a crystal
In the magnetic cell
HQ
HQQMkk )().(.4 iMF V
Magnetic scattering from a magnet
=> vector scattering amplitude for one atom
Magnetic structure factor
QSQQkk iIMF fV ).(.4
Magnetic interaction vector for a crystal
In the magnetic cell
).exp(.)()(
)()(
iii
i if rQSQQM
QQMQQM
M(Q)Q=Q eOnly the perpendicularcomponent of M to Q=2h contributes to scattering
M(Q)M(Q)xQ
Magnetic structure factorsand interaction vectors
Reciprocal k-space
Nuclear reflections
H=h.b1+k. b2+l. b3
b2
b1
a1
a2
a3
R(101)
Crystal+AF mag structure
a1mag
Propagation vectors : Structure factor in crystal cell
k
)101(0r
a3mag = a3a1
mag = 2a1 a2mag = a2
Q=(hmagkmaglmag)
b1mag
b1mag = b1/2
)000(0r
In the magnetic cell
a1
a2
a3)000(
0r
Crystal+AF mag structure
a1mag
Propagation vectors : Structure factor in crystal cell
)101(0r
).exp(.)()( iii
i if rQSQQM )100(0r
),,()000(0 zyxr
),,2/1()100(0 zyx r
Atom 1
Atom 2
)1.()()( )(2 ihkzlyhxihkl eef SQQM
(2)(1)
))2/1((2
)(2
)()(kzlyxhi
kzlyhxi
hkle
ef
S
SQQM
the AF arrangement =>…hmag necessarily odd (type, 2h+1)
Q=(hmagkmaglmag)
Real space
Reciprocal space
R(101)
)101(0r
Reciprocal space
Q=hmag.b1mag+k. b2+l. b3
Magnetic reflections
Nuclear reflections
b2
b1
b1mag
Crystal+AF mag structure
b1mag = b1/2
a3mag = a3a1
mag = 2a1 a2mag = a2
Propagation vectors and Structure factor in crystal cell
the AF arrangement =>…hmag necessarily odd (type, 2h+1)
Q
a1
a2
a3)000(
0r
a1mag
)101(0r
)100(0r
(2)(1)
R(101)
)101(0r
Reciprocal space
Magnetic reflections
Nuclear reflections
b2
b1
b1mag
Q=(2h+1). b1/2+k. b2+l. b3
Q=h.b1+k. b2+l.b3 + b1/2
b1mag = b1/2
a3mag = a3a1
mag = 2a1 a2mag = a2
Crystal+AF mag structure
Propagation vectors and Structure factor in crystal cell
Q
Q= H + k
k=(½00)
« Propagation vector »
H
a1
a2
a3)000(
0r
a1mag
)101(0r
)100(0r
(2)(1)
R(101)
)101(0r
a1
Crystal+AF mag structureReciprocal space
Q= H + k
Magnetic reflections
Nuclear reflections
k=(½00) b2
b1
b1mag
Q=(2h+1). b1/2+k. b2+l. b3
Q=h.b1+k. b2+l.b3 + b1/2
because k.Rn=n1/2
m[n]i = Si. (-1)n1
Propagation vectors and Structure factor in crystal cell
QH
nini iexp k.RSm k 2
a1
a2
a3)000(
0r
a1mag
)101(0r
)100(0r
(2)(1)
R(101)
)101(0r
Crystal+AF mag structureReciprocal spacek=(½00) b2
b1
b1mag
because k.Rn=n1/2
i n
nni iQ.Rm 2exp
m[n]i = Si. (-1)n1
Propagation vectors and Structure factor in crystal cell
kHQ nuchkl
).exp(.)()( ihkli
ihkl ifi
rQSQQM k
HQ
nini iexp k.RSm k 2
a1
a2
a3)000(
0r
a1mag
)101(0r
)100(0r
(2)(1)
R(101)
)101(0r
Q
Reciprocal space
kk
2njnj
iexp k.RSm
b2
b1
-kk=(½00) nnj
iexp k.RSmkj
2In our example (AF):- k equivalent to –k(2k is a reciprocal lattice vector, or k at the border of the brillouin zone)
- Only k sufficient to index all the magnetic reflections, - And Skj must be real
Magnetic structure
b1mag
Propagation vectors and Structure factor in crystal cell : case 1
kHQ nuchkl
).exp(.)()( ihkli
ihkl ifi
rQSQQM k
H
Reciprocal space Magnetic structurek
b2
b1
- necessary condition for real
mnj : jj kk- SS
Formalism Required to describe Incommensurate structures
Propagation vectors and Structure factor in crystal cell : case 2
kk
2njnj
iexp k.RSm
In general k NOT equivalent to –k, so a set of {k}=k,-k is needed- Skj are complex vectors
kHQ nuchkl
).exp(.)()( ihkli
ihkl ifi
rQSQQM k
H-k
0
li l i r R r
1
2 .n
j jj
F b exp i
H H r
nuc ( ) *( )I F F H H H
Nuclear Phase:Scattering
vector h H k
i
k
2li lexp i km S kR
Structure Factor/
Intensity
*h h hM MI
Magnetic Phase:
h=HAtomic positionsStructural
model
Arrangement of the moments
h = ( )- ( ( )) with h h
M M h e e M h e
1
h 2 .n
jj k j
j
p f exp i
M h S h r
*
hh
*
hhhMM NNI
For non-polarised neutrons
Magnetic scattering and structure factors
What’s a magnetic structure? Why study magnetic order?Example : Manganites
The magnetic structure factor
Pitfalls
Magnetic reflections
Reciprocal space
b2
b1
h= H
k=(000) nnj
iexp k.RSmkj
2When k=(000) (primitive P):- mni=Skj whatever n: this only means that the cell of the the magnetic structure is the same as that of the nuclear structure (not necessarily ferromagnetic, if more than 2 atoms/cell)
Magnetic structure
Nuclear reflections
Pitfalls : k=(000)
LaMnO3 TN=150K
Commensurate AF structure
AF
4
23
1
Sk1= -Sk2= -Sk3= Sk4, all reals vectors
k=(000) mni=Skj
kk
2njnj
iexp k.RSm
Pitfalls : k=(000)
LaMnO3 : 50K and 150 K
Orthorhombic Pnma
I/F-centred lattice (2D):
Pitfalls : Centred cells
Nuclear reflectionsa1
nnj
iexp k.RSmkj
2Magnetic structure
a2
(110)(-110)
b2p b1
p
b1
b2
a1pa2
p
Reciprocal space
Nuclear reflections
b1
nnj
iexp k.RSmkj
2Magnetic structure
b2
(110)(-110)
b2p b1
p
I/F-centred lattice (2D):
Magnetic reflections
h= H
k=(000)
a1
a2
a1pa2
p
Reciprocal space
Pitfalls : Centred cells
Nuclear reflections
b1
nnj
iexp k.RSmkj
2Magnetic structure
b2
(110)(-110)
b2p b1
p
k=(100)
Magnetic reflections
h= H + k
kp=(½½0)
AF order on a centred lattice
a1
a2
a1pa2
p
Reciprocal space
Pitfalls : Centred cells
Magnetic reflections
Nuclear reflections
b1
h= H + k
k=(100)
nnj
iexp k.RSmkj
2I-centred lattice :- k equivalent to –k
(2k is a reciprocal lattice vector)
-This is because {Rn} contains type (½½½)… translations, - k vectors, which can have integer components, have in fact non-integer components in the primitive cell
Magnetic structure
b2
(110)(-110)
b2p b1
p
kp=(½½0)
Reciprocal space
Pitfalls : Centred cells
0
li l i r R r
1
2 .n
j jj
F b exp i
H H r
nuc ( ) *( )I F F H H H
Nuclear Phase:Scattering
vector h H k
i
k
2li lexp i km S kR
Structure Factor/
Intensity
*h h hM MI
Magnetic Phase:
h=HAtomic positionsStructural
model
Arrangement of the moments
h = ( )- ( ( )) with h h
M M h e e M h e
1
h 2 .n
jj k j
j
p f exp i
M h S h r
*
hh
*
hhhMM NNI
For non-polarised neutrons
Magnetic scattering and structure factors