1. Magnetic Neutron Diffraction the basic formulas
A.Daoud-aladine, (ISIS-RAL)

2. What’s a magnetic structure?
Why study magnetic order? Example : Manganites
The magnetic structure factor Using 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

3. What’s a magnetic structure?
(atomic) magnetic moments (m) arise from quantum effects in atoms/ions with unpaired electrons
Ni2+
Intra-atomic electron correlation
Hund’s rule(maximum total S)
+ Spin-orbit + CEF
core
« Classical description»
« Quantum description»
m = gJ J (J=L+S 4f-rare earths)
m = gS S (3d-transition metals)

4. What’s a magnetic structure?
Magnet: crystal containing magnetic atoms
kT >> Jij,
Jij
paramagnetic
(disordered) state
Temperature (entropy) overcomes magnetic energy:
Entropy essentially dominated by local
magnetic moment
fluctuations

5. What’s a magnetic structure?
Magnet: crystal contaning magnetic atoms
kT < Jij,
Jij
Exemple here: Jij>0
Antiferromagnetic coupling
(AF)
Magnetic energy overcomes the entropy :
 Quasi-static configuration of magnetic moment
with small fluctuations that are made cooperative
by the magnetic exchange (spin waves excitations )

6. 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

7. Magnetic scattering of a single atom
From Introduction to the Theory of Thermal Neutron Scattering, G SQUIRES –
Cambridge University Press (1978)
Q= kF - kI r R
kI=2/ uI
core
kF=2/ uF ri
Dipolar interaction term with one electron
core
Total Si si
=> vector scattering amplitude for one atom

8. Example: RMnO3 manganites
LaMnO3 TN=150K
Ideal cubic Perovskite
TbMnO3 TN=41K O Mn
Pnma
La/Tb
Site A m
[010]O
[001]C
[101]O
[010]C n
Mn-O-Mn m n a a

9. Example: RMnO3 manganites
LaMnO3 TN=150K AF
Commensurate
structure
T. Goto, et al. Phys. Rev. Lett. 92, (2004)

10. Example: RMnO3 manganites
LaMnO3 TN=150K
LaMnO3 : 50K and 150 K AF
Commensurate
structure

11. Example: RMnO3 manganites
LaMnO3 TN=150K
Electronic structure of Mn3+ (x=0)
dxy
dyz dxz
Jahn-Teller
Distorsion
dz2
dx2-y2 O
Hund JH
Crystal
field x2 eg
3d4 AF
S=3/2
Commensurate
structure
t2g
(e- localized)
scattering amplitude
for one atom, tabulated f(Q)

12. Example: RMnO3 manganites
LaMnO3 TN=150K AF
Commensurate
structure
T. Goto, et al. Phys. Rev. Lett. 92, (2004)

13. Example: RMnO3 manganites
TbMnO3 TN2<T<TN1=41K Sinusoidal
LaMnO3 TN=150K
T<TN2=23K Cycloid AF
Commensurate
structure
Incommensurate structures

14. What’s a magnetic structure? The magnetic structure factor Pitfalls
Why study magnetic order? Example : Manganites
The magnetic structure factor
Pitfalls

15. Magnetic scattering of a single atom
From Introduction to the Theory of Thermal Neutron Scattering, G SQUIRES –
Cambridge University Press (1978)
Q= kF - kI r R
kI=2/ uI
core
kF=2/ uF ri
Dipolar interaction term with one electron
Total Si si
core
=> vector scattering amplitude for one atom

16. Magnetic scattering from a magnet
Magnetisation density: M(r)=MS(r)+ML(r)
kI=2/ uI
kF=2/ uF n
Q= kF - kI
Magnetic interaction vector for a crystal
Magnetic structure factor
=> vector scattering amplitude for one atom
In the magnetic cell

17. Magnetic structure factors and interaction vectors
Q=Q e
M(Q)
Only the perpendicular
component of M to Q=2h
contributes to scattering q q
M(Q)xQ
M(Q)
Magnetic interaction vector for a crystal
Magnetic structure factor
=> vector scattering amplitude for one atom
In the magnetic cell

18. Propagation vectors : Structure factor in crystal cell
Reciprocal k-space
Crystal+AF mag structure
a3mag = a3
a1mag = 2a1
a2mag = a2
Q=(hmagkmaglmag)
b1mag
b1mag = b1/2 b2 k b1
R(101) a3
Nuclear reflections
a1mag a2 a1
H=h.b1+k. b2+l. b3

19. Propagation vectors : Structure factor in crystal cell
In the magnetic cell
Crystal+AF mag structure
Reciprocal
space
Q=(hmagkmaglmag)
Atom 1
Real
space
Atom 2
R(101) a3
(1)
(2)
a1mag a2 a1
the AF arrangement =>…
hmag necessarily odd (type, 2h+1)

20. Propagation vectors and Structure factor in crystal cell
Reciprocal space
Crystal+AF mag structure b2 Q b1
b1mag
R(101) a3
(1)
(2)
Nuclear reflections
a1mag a2 a1
Magnetic reflections
a1mag = 2a1
a2mag = a2
a3mag = a3
Q=hmag.b1mag+k. b2+l. b3
b1mag = b1/2
the AF arrangement =>…
hmag necessarily odd (type, 2h+1)

21. Propagation vectors and Structure factor in crystal cell
Reciprocal space
Crystal+AF mag structure
Q= H + k
k=(½00)
« Propagation vector » H b2 Q b1
b1mag
R(101) a3
(1)
(2)
Nuclear reflections
a1mag a2 a1
Magnetic reflections
a1mag = 2a1
a2mag = a2
a3mag = a3
Q=(2h+1). b1/2+k. b2+l. b3
b1mag = b1/2
Q=h.b1+k. b2+l.b3 + b1/2

22. Propagation vectors and Structure factor in crystal cell
Reciprocal space
Crystal+AF mag structure
k=(½00) b2 Q H b1
b1mag
R(101) a3
(1)
(2)
Nuclear reflections
a1mag a2 a1 a1
Magnetic reflections
m[n]i = Si. (-1)n1
Q=(2h+1). b1/2+k. b2+l. b3
Q=h.b1+k. b2+l.b3 + b1/2
because k.Rn=n1/2
Q= H + k

23. Propagation vectors and Structure factor in crystal cell
Reciprocal space
Crystal+AF mag structure
k=(½00) b2 Q H b1
b1mag
R(101) a3
(1)
(2)
a1mag a2 a1
m[n]i = Si. (-1)n1
because k.Rn=n1/2

24. Propagation vectors and Structure factor in crystal cell : case 1
Reciprocal space
Magnetic structure
k=(½00) -k b2 Q H
In 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 b1
b1mag

25. Propagation vectors and Structure factor in crystal cell : case 2
Reciprocal space
Magnetic structure k b2 -k H
In general k NOT equivalent
to –k, so a set of {k}=k,-k
is needed
- Skj are complex vectors b1
- necessary condition for real mnj :
Formalism Required to describe Incommensurate structures

26. Magnetic scattering and structure factors
For non-polarised
neutrons
Magnetic Phase:
Nuclear Phase:
Scattering
vector
h=H
Arrangement of the moments
Atomic positions
Structural
model
Structure
Factor/
Intensity

27. What’s a magnetic structure? The magnetic structure factor Pitfalls
Why study magnetic order? Example : Manganites
The magnetic structure factor
Pitfalls

28. Pitfalls : k=(000) Reciprocal space Magnetic structureb2
When 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) b1
Nuclear reflections
Magnetic reflections
h= H

29. Pitfalls : k=(000) mni=Skj Sk1= -Sk2= -Sk3= Sk4, all reals vectors
LaMnO3 TN=150K
Orthorhombic Pnma
k=(000)
mni=Skj
Sk1= -Sk2= -Sk3= Sk4,
all reals vectors 3 2 4
LaMnO3 : 50K and 150 K 1 AF
Commensurate
AF structure

30. Pitfalls : Centred cells
Reciprocal space
Magnetic structure
(110)
(-110) b2
b1p
b2p
I/F-centred lattice (2D): b1 a2
a2p
a1p a1
Nuclear reflections

31. Pitfalls : Centred cells
Reciprocal space
Magnetic structure
(110)
(-110) b2
b1p
b2p
I/F-centred lattice (2D): b1 a2
a2p
a1p
k=(000) a1
Nuclear reflections
Magnetic reflections
h= H

32. Pitfalls : Centred cells
Reciprocal space
Magnetic structure
(110)
(-110) b2
k=(100)
Magnetic reflections
h= H + k
kp=(½½0)
b1p
b2p
AF order on a centred lattice b1 a2
a2p
a1p a1
Nuclear reflections

33. Pitfalls : Centred cells
Reciprocal space
Magnetic structure
(110)
(-110) b2
b1p
b2p
I-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 b1
k=(100)
kp=(½½0)
Nuclear reflections
Magnetic reflections
h= H + k

34. Magnetic scattering and structure factors
For non-polarised
neutrons
Magnetic Phase:
Nuclear Phase:
Scattering
vector
h=H
Arrangement of the moments
Atomic positions
Structural
model
Structure
Factor/
Intensity