The plan
•
How does the neutron interact with magnetism?
•
The fundamental rule of neutron magnetic scattering
•
Elastic scattering, and how to understand it
•
Magnetic form factors
•
Inelastic scattering
INSTITUT MAX VON LAUE - PAUL LANGEVIN
How does the neutron interact with magnetism?
Neutrons have no charge, but they do have a magnetic moment.
The magnetic moment is given by the neutron’s spin angular momentum:
ˆ
N
where:
•
 is a constant (=1.913)
•
N is the nuclear magneton
•
ˆ is the quantum mechanical Pauli spin operator



Normally refer to it as a spin-1/2 particle
INSTITUT MAX VON LAUE - PAUL LANGEVIN
How does the neutron interact with magnetism?
Through the cross-section!
2
2
d 2
k mn 
ˆ
 
  p p  k , s,V r  k,s,  h  E  E 
d dE k 2h2   ,s  s  ,s

Probabilities of
initial target state
and neutron spin
Conservation of
energy
The matrix element, which contains all the physics.
Vˆ r is the pseudopotential,
which for magnetism is given by:

ˆ  Br
Vˆm r  N 
where B(r) is the magnetic induction.
G. L. Squires, Introduction to the theory of thermal neutron scattering, Dover Publications, New York, 1978
W. Marshall and S. W. Lovesey, Theory of thermal neutron scattering, Oxford University Press, Oxford, 1971
S. W. Lovesey, Theory of neutron scattering from condensed matter, Oxford University Press, Oxford, 1986

INSTITUT MAX VON LAUE - PAUL LANGEVIN
Elastic scattering
If the incident neutron energy = the final neutron energy, the scattering is elastic.
2
d  mn 
  2   ps k , sVˆ r  k,s
d 2h  s
2
Forget about the spins for the moment (unpolarized neutron scattering) and integrate over all r:
ˆ r  k   Vˆ r e iQr  dr
k
V


Momentum transfer Q = k – k´
The elastic cross-section is then directly proportional to the Fourier transform squared of the potential.

Neutron scattering
thus works in Fourier space, otherwise called reciprocal space.
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Evaluating kVˆm r k
(Squires, chapter 7)
Magnetism is caused by unpaired electrons or movement of charge.
spin, s
Momentum, p

k Vˆm ri  k 
Spin:
 
ˆ  s Q
ˆ
4 exp iQ ri  Q
i
Q
s
Movement / Orbital
4i
ˆ

hQ



ˆ  s Q
ˆ
Q
i
ˆ
sQ
Q

Perpendicular to Q
INSTITUT MAX VON LAUE - PAUL LANGEVIN

exp iQ ri  pi  Q
p
ˆ
pi  Q

The fundamental rule of neutron magnetic scattering
Neutrons only ever see the
components of the magnetization
that are perpendicular to the
scattering vector!
INSTITUT MAX VON LAUE - PAUL LANGEVIN
The fundamental rule of neutron magnetic scattering
Taking elastic scattering again:
2
d  mn 
  2  k Vˆ r  k
d 2h 
2
Neutron scattering therefore probes the components of the sample magnetization that are
perpendicular to the neutron’s momentum transfer,Q.

 Vm re iQr  dr  M  (Q)
and dmagnetic  M* (Q) M (Q)


d
Neutron scattering measures the correlations in magnetization,
i.e. the influence a magnetic moment has on its neighbours.

It is capable of doing this over all length scales, limited only by wavelength.
INSTITUT MAX VON LAUE - PAUL LANGEVIN
How to easily understand diffraction
Learn your Fourier transforms!
and
Learn and understand the
convolution theorem!
f (r )  g (r )   f ( x) g (r  x)  dx
 f r  F(q)
gr  G(q)
 f (r)  g(r)  F(q)  G(q)
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Elastic scattering
2
d  mn 
  2   ps k , sVˆ r  k,s
d 2  ,s


2
2

ˆ
V e iQ r  dr   Vˆ 2  Vˆ

2

 z(r)e iQ r  dr


The contribution from deviations from
the average structure:
Short-range order
The contribution from the average
structure of the sample:
Long-range order
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Magnetic structure determination
Crystalline structures
2
d
ˆ
  V eiQr  dr
d
The Fourier transform from a series of delta-functions
f(r)
F(Q)
1/a
a
Bragg’s Law: 2dsinq=l
Leads to Magnetic Crystallography
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Superconductivity
Normal state
Superconducting state
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Flux line lattice
A simple example of magnetic elastic scattering
MgB2 is a superconductor below 39K, and expels all magnetic field lines (Meissner effect).
Above a critical field, flux lines penetrate the sample.
Scattering geometry
k
Q
k´
2q
k´
The momentum transfer, Q, is roughly
perpendicular to the flux lines, therefore all the
magnetization is seen.
(recall
d magnetic
d
 M * (Q) M  (Q) )
INSTITUT MAX VON LAUE - PAUL LANGEVIN
A simple example of magnetic elastic scattering
MgB2 is a superconductor below 39K, and expels all magnetic field lines (Meissner effect).
Above a critical field, flux lines penetrate the sample.
Via Bragg’s Law
2dsinq=l
l = 10 Å
d = 425Å
~2
The reciprocal lattice has 60 rotational symmetry,
therefore the flux line lattice is hexagonal
Cubitt et. al. Phys. Rev. Lett. 91 047002 (2003)
Cubitt et. al. Phys. Rev. Lett. 90 157002 (2003)
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Antiferromagnetism in MnO
Bragg peaks from crystal structure
80K (antiferromagnetic)
300K (paramagnetic)
New Bragg peaks
C. G. Shull & J. S. Smart, Phys. Rev. 76 (1949) 1256
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Antiferromagnetism in MnO
New magnetic Bragg peaks
Magnetic structure
C. G. Shull & J. S. Smart, Phys. Rev. 76 (1949) 1256
a
C. G. Shull et al., Phys. Rev. 83 (1951) 333
H. Shaked et al., Phys. Rev. B 38 (1988) 11901
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Magnetic unit cell
2a
Antiferromagnetism in MnO
The moments are said to lie in the (111) plane
H. Shaked et al., Phys. Rev. B 38 (1988) 11901
Real space
Reciprocal space
(111)
(311)
1 1 1
 , , 
2 2 2
3 1 1
 , , 
2 2 2
5 3 1
 , , 
2 2 2
(220)
Moment direction
fcc lattice
(i.e. h,k,l) all even or all odd
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Antiferromagnetism in Chromium
The Fourier Transform for two Delta functions:
F(Q)
f(r)
a
E.Fawcett, Rev. Mod. Phys. 60 (1988) 209
1/a
Chromium is an example of an itinerant antiferromagnet
Reciprocal space
Real space
a Spin Density wave
INSTITUT MAX VON LAUE - PAUL LANGEVIN
The physical meaning of z(r)
Given an atom at position r1,
what’s the probability of finding a similar atom at position r2?
z(r), real space
Zero probability?
Gaussian probability?
Z(Q), reciprocal space
Constant
Delta
function
A
A
Gaussian
Gaussian
a
Any spherically
symmetric function?
e.g. a hollow sphere
Aa
A
1/a
Delta function at r  0
sin(Qr)/Qr
r
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Q
Diffuse scattering from magnetic frustration, Gd2Ti2O7
d
sin Qr 
  S oS r
d
Qr
Structure
model 1
Structure
model 2
magnetic scattering
magnetic scattering at 46mK
with fits
J. R. Stewart et al., J. Phys.: Condens. Matter 16 (2004) L321
INSTITUT MAX VON LAUE - PAUL LANGEVIN
The ‘Family Tree’ of Magnetism
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Magnetic form factors
A magnetic moment is spread out in space
Unpaired electrons can be found in orbitals, or in band structures
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Magnetic form factors
A magnetic moment is spread out in space
Unpaired electrons can be found in orbitals, or in band structures
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Magnetic form factors
A magnetic moment is spread out in space
Unpaired electrons can be found in orbitals, or in band structures
e.g.
Take a magnetic ion with total spin S at position R
The (normalized) density of the spin is sd(r)
around the equilibrium position
rd
R
M (Q)   M r e iQr  dr
  sd e iQrd  drd  S (R )e iQR  dR
= f (Q )  S (R )e iQR  dR
f Q    sd e
iQrd
 drd
f(Q) is the magnetic form factor
It arises from the spatial distribution of unpaired electrons around a magnetic atom
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Magnetic form factors
f(Q) is the magnetic form factor
It arises from the spatial distribution of unpaired electrons around a magnetic atom
d magnetic
d
 M* (Q) M  (Q)
 f 2 Q  S  (R i ) S * R j e

iQ  R i  R j

 dR
There is no form factor for nuclear scattering, as the nucleus can be considered as a point compared
to the neutron wavelength
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Magnetic form factors
Approximations for the form factors are tabulated
(P. J. Brown, International Tables of Crystallography, Volume C, section 4.4.5)
f Q   C1 j0 Q 4   C2 j2 Q 4   C4 j4 Q 4   ...
Form factors for iron
For spin only
Amplitude
1
Series2
<j
0>
<j
Series3
2>
<j
4>
Series4
0.8
For orbital
contributions
0.6
0.4
0.2
0
0
5
10
Q (Å1)
15
INSTITUT MAX VON LAUE - PAUL LANGEVIN
20
Magnetic electron density
f Q    sd e
iQrd
 drd
UPtAl
P. Javorsky et al., Phys. Rev. B 67 (2003) 224429
From the form factors, the magnetic moment density in the unit cell can be derived
crystal structure
magnetic moment density
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Magnetic electron density
Nickel
H. A. Mook., Phys. Rev. 148 (1966) 495
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Neutron inelastic scattering
Magnetic fluctuations are governed by a wave equation:
H=E 
The Hamiltonian is given by the physics of the material.
Given a Hamiltonian, H, the energies E can be calculated.
(this is sometimes very difficult)
Neutrons measure the energy, E, of the magnetic
fluctuations, therefore probing the free parameters in the
Hamiltonian.
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Crystal fields in NdPd2Al3
H

l  2, 4, 6
Bl0Ol0 
B O
l  2, 4
4
l
4
l
O = Stevens parameters
(K. W. Stevens, Proc. Phys. Soc A65 (1952) 209)
B = CF parameters,
measured by neutrons
A. Döni et al., J. Phys.: Condens. Matter 9 (1997) 5921
O. Moze., Handbook of magnetic materials vol. 11, 1998 Elsevier, Amsterdam, p.493
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Quantum tunneling in Mn12-acetate
H

l  2, 4, 6
Bl0Ol0 
B O
l  2, 4
4
l
4
l
Fitted data with scattering from:
• energy levels
• elastic scattering
• incoherent background
Calculated energy terms
Neutron spectra
INSTITUT MAX VON LAUE - PAUL LANGEVIN
I. Mirebeau et al., Phys. Rev. Lett. 83 (1999) 628
Spin waves and magnons
A simple Hamiltonian for spin waves is:
H   J  sis j
i, j
J is the magnetic exchange integral, which can be measured with neutrons.
Take a simple ferromagnet:
The spin waves might look like this:
Spin waves have a frequency () and a wavevector (q)
The frequency and wavevector of the waves are directly measurable with neutrons
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Magnons and reciprocal space
Reciprocal space
Spin wave dispersion
 (4JS)
2
A
B
1
0
2/a
0
1
2
A
q (/a)
B
  4 JS 1  cos qa
 Dq 2 for qa  1
Brillouin zone boundaries
Bragg
peaks
D  2 JSa2
C. Kittel, Introduction to Solid State Physics, 1996, Wiley, New York
F. Keffer, Handbuch der Physik vol 18II, 1966 Springer-Verlag, Berlin
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Magnons in crystalline iron
(002)
(000)
(220)
G. Shirane et al., J. Appl. Phys. 39 (1968) 383
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Spin-waves in Rb2MnF4
A quasi-two dimensional antiferromagnetic system
Gap in dispersion due to magnetic anisotropy
T. Huberman et al., Phys. Rev. B 72 (2005) 014413
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Spin-waves in MnPS3
A. R. Wildes et al., JPCM 10 (1998) 6417
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Other magnetic Hamiltonians
Model magnetic systems (one, two and three dimensions)
Superconductivity
Giant and colossal magnetoresistance
Quantum magnetic fluctuations
Heavy fermion materials
Overdamped excitations in amorphous materials
Multiple magnon scattering
Slow relaxation in spin glasses
Fluctuations in Fractals and percolation theory
etc. etc. etc.
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Conclusions:
•
Neutrons only ever see the components of the
magnetization,M, that are perpendicular to the scattering
vector, Q
•
Magnetization is distributed in space, therefore the
magnetic scattering has a form factor
INSTITUT MAX VON LAUE - PAUL LANGEVIN