Magnetic Neutron Diffraction the basic formulas

$: Magnetic Neutron Diffraction the basic formulas$
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»


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


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


T. Goto, et al. Phys. Rev. Lett. 92, 257201 (2004)

LaMnO3 TN=150K


AF

LaMnO3 : 50K and 150 K


LaMnO3 TN=150K


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)


LaMnO3 TN=150K


AF


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


Incommensurate structures

AF

kk

2njnj

iexp k.RSm


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


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


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


Magnetic interaction vector for a crystal


).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


a1

a2

a3)000(

0r


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


b1mag = b1/2

a3mag = a3a1


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


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




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


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


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


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


The magnetic structure factor

Pitfalls


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):


h= H

k=(000)

a1

a2

a1pa2

p

Reciprocal space


Nuclear reflections

b1

nnj

iexp k.RSmkj

2Magnetic structure

b2

(110)(-110)

b2p b1

p

k=(100)


h= H + k

kp=(½½0)

AF order on a centred lattice

a1

a2

a1pa2

p

Reciprocal space



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


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

Magnetic Neutron Diffraction the basic formulas

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