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INSTITUT MAX VON LAUE - PAUL LANGEVIN
• 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
The plan
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Neutrons have no charge, but they do have a magnetic moment.
The magnetic moment is given by the neutron’s spin angular momentum:
where:
• is a constant (=1.913)
• N is the nuclear magneton
• is the quantum mechanical Pauli spin operator
Nˆ
ˆ
How does the neutron interact with magnetism?
Normally refer to it as a spin-1/2 particle
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Through the cross-section!
d2
d dE
k
k
mn
2h2
2
p ps
,s
k , s , ˆ V r k,s, , s
2
h E E
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
Conservation of
energy
Probabilities of
initial target state
and neutron spin
The matrix element, which contains all the physics.
How does the neutron interact with magnetism?
is the pseudopotential,
which for magnetism is given by:
where B(r) is the magnetic induction.
ˆ V m r Nˆ B r
ˆ V r
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Forget about the spins for the moment (unpolarized neutron scattering) and integrate over all r:
Momentum transfer Q = k – k´
Elastic scattering
k ̂V r k ̂V r eiQr dr
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.
If the incident neutron energy = the final neutron energy, the scattering is elastic.
d
d
mn
2h2
2
ps k , s ˆ V r k,ss
2
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Evaluating
k ˆ V m r k (Squires, chapter 7)
Magnetism is caused by unpaired electrons or movement of charge.
Spin: Movement / Orbital
spin, sMomentum, p
4 exp iQ ri ˆ Q si ˆ Q
k ˆ V m ri k
4i
hQexp iQ ri pi
ˆ Q s
p
pi ˆ Q
s ˆ Q
ˆ Q si ˆ Q
Perpendicular to Q
INSTITUT MAX VON LAUE - PAUL LANGEVIN
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
Neutron scattering therefore probes the components of the sample magnetization that are
perpendicular to the neutron’s momentum transfer,Q.
and
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.
Vm r eiQr dr M (Q)
dmagnetic
d M
* (Q) M (Q)
d
d
mn
2h2
2
k ˆ V r k2
The fundamental rule of neutron magnetic scattering
Taking elastic scattering again:
INSTITUT MAX VON LAUE - PAUL LANGEVIN
How to easily understand diffraction
Learn your Fourier transforms!and
Learn and understand the
convolution theorem!
dxxrgxfrgrf )()()()(
f r F(q)
g r G(q)
f (r) g(r) F(q) G(q)
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Elastic scattering
d
d
mn
2 2
2
psk , s ˆ V r k,s
, s
2
ˆ V 2
e iQr dr ˆ V 2 ˆ V 2
z(r) e iQr dr
The contribution from the average
structure of the sample:
Long-range order
The contribution from deviations from
the average structure:
Short-range order
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Magnetic structure determination
rrQ deV i
2
ˆd
d
Crystalline structures
f(r) F(Q)
The Fourier transform from a series of delta-functions
Bragg’s Law: 2dsinq=l
Leads to Magnetic Crystallography
a1/a
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Superconductivity
Normal state Superconducting state Flux line lattice
INSTITUT MAX VON LAUE - PAUL LANGEVIN
MgB2 is a superconductor below 39K, and expels all magnetic field lines (Meissner effect).
Above a critical field, flux lines penetrate the sample.
A simple example of magnetic elastic scattering
The momentum transfer, Q, is roughly
perpendicular to the flux lines, therefore all the
magnetization is seen.
(recall ))()( *QQ
MM
d
d magnetic
k
k´
Qk´
Scattering geometry
2q
INSTITUT MAX VON LAUE - PAUL LANGEVIN
A simple example of magnetic elastic scattering
~2
Cubitt et. al. Phys. Rev. Lett. 91 047002 (2003)
Cubitt et. al. Phys. Rev. Lett. 90 157002 (2003)
Via Bragg’s Law
2dsinq=l
l = 10 Å
d = 425Å
The reciprocal lattice has 60 rotational symmetry,
therefore the flux line lattice is hexagonal
MgB2 is a superconductor below 39K, and expels all magnetic field lines (Meissner effect).
Above a critical field, flux lines penetrate the sample.
INSTITUT MAX VON LAUE - PAUL LANGEVIN
C. G. Shull & J. S. Smart, Phys. Rev. 76 (1949) 1256
New Bragg peaks
Antiferromagnetism in MnO
80K (antiferromagnetic)
300K (paramagnetic)
Bragg peaks from crystal structure
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Antiferromagnetism in MnO
C. G. Shull et al., Phys. Rev. 83 (1951) 333
H. Shaked et al., Phys. Rev. B 38 (1988) 11901
C. G. Shull & J. S. Smart, Phys. Rev. 76 (1949) 1256
New magnetic Bragg peaks Magnetic structure Magnetic unit cell
a 2a
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Antiferromagnetism in MnO
The moments are said to lie in the (111) planeH. Shaked et al., Phys. Rev. B 38 (1988) 11901
(111)
fcc lattice
(i.e. h,k,l) all even or all odd
Real space
(220)
(311)
2
1,
2
1,
2
1
2
1,
2
1,
2
3
2
1,
2
3,
2
5
Reciprocal space
Moment direction
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Antiferromagnetism in Chromium
Reciprocal space Real space
a Spin Density wave
Chromium is an example of an itinerant antiferromagnet
E.Fawcett, Rev. Mod. Phys. 60 (1988) 209
a
1/a
The Fourier Transform for two Delta functions:
F(Q) f(r)
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 Z(Q), reciprocal space
A AZero probability?
a
A
1/a
AaGaussian probability?
Delta
function
Constant
Gaussian Gaussian
Any spherically
symmetric function?
e.g. a hollow sphere
Delta function at r 0
rQ
sin(Qr)/Qr
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Diffuse scattering from magnetic frustration, Gd2Ti2O7
magnetic scattering
magnetic scattering at 46mK
with fits
Structure
model 1
Structure
model 2
J. R. Stewart et al., J. Phys.: Condens. Matter 16 (2004) L321
Qr
Qrro
sin
d
dSS
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
f(Q) is the magnetic form factor
It arises from the spatial distribution of unpaired electrons around a magnetic atom
d
i
d
i
i
d
i
d
i
desQf
deSQf
deSdes
de
d
d
r
Q
rQ
RR=
RRr
rrMM
RQ
RQrQ
rQ
)()(
)(
)(
R
rd
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
INSTITUT MAX VON LAUE - PAUL LANGEVIN
f(Q) is the magnetic form factor
It arises from the spatial distribution of unpaired electrons around a magnetic atom
There is no form factor for nuclear scattering, as the nucleus can be considered as a point compared
to the neutron wavelength
RRR
MM
RRQdeSSQf
d
d
jii
ji
magnetic
*2
*
)(
Magnetic form factors
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)
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
Series2
Series3
Series4
...444 442201 QjCQjCQjCQf
<j0>
<j2>
<j4>
Q (Å1)
Form factors for iron
Am
pli
tude
For spin only
For orbital
contributions
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Magnetic electron density
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
d
i
d desQfd
rrQ
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Magnetic electron density
H. A. Mook., Phys. Rev. 148 (1966) 495
Nickel
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
A. Döni et al., J. Phys.: Condens. Matter 9 (1997) 5921
O. Moze., Handbook of magnetic materials vol. 11, 1998 Elsevier, Amsterdam, p.493
6,4,2 4,2
4400
l l
llll OBOBH O = Stevens parameters(K. W. Stevens, Proc. Phys. Soc A65 (1952) 209)
B = CF parameters,
measured by neutrons
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Quantum tunneling in Mn12-acetate
I. Mirebeau et al., Phys. Rev. Lett. 83 (1999) 628
6,4,2 4,2
4400
l l
llll OBOBH
Calculated energy terms
Neutron spectra
Fitted data with scattering from:
• energy levels
• elastic scattering
• incoherent background
INSTITUT MAX VON LAUE - PAUL LANGEVIN
Spin waves and magnons
A simple Hamiltonian for spin waves is:
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
ji
jiJH,
ss
INSTITUT MAX VON LAUE - PAUL LANGEVIN
2
2
2
1for
cos14
JSaD
qaDq
qaJS
0
1
2
0 1 2
q (/a)
(4JS
)
2/a
Reciprocal spaceSpin wave dispersion
Bragg
peaksBrillouin zone boundaries
Magnons and reciprocal space
A B
BA
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
G. Shirane et al., J. Appl. Phys. 39 (1968) 383
Magnons in crystalline iron
(000) (220)
(002)
INSTITUT MAX VON LAUE - PAUL LANGEVIN
T. Huberman et al., Phys. Rev. B 72 (2005) 014413
Spin-waves in Rb2MnF4
A quasi-two dimensional antiferromagnetic system
Gap in dispersion due to magnetic anisotropy
INSTITUT MAX VON LAUE - PAUL LANGEVIN
A. R. Wildes et al., JPCM 10 (1998) 6417
Spin-waves in MnPS3
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