B i theory
Basic
th
II
II: inelastic
i l ti neutron
t
scattering
tt i
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Zahra Yamani
Summer School, June 15-18 2009
NRC - Canadian Neutron Beam Centre, Chalk River, Canada
Canadian Neutron Beam Centre
Overview
 Neutron scattering
g & dynamics
y
 Lattice vibrations
 Magnetic excitations
 Double
D bl differential
diff
ti l cross
section.
 Dynamic structure factor
 Fluctuation dissipation theorem
 Detailed balance
 Scattering triangle: elastic vs. inelastic
 Triple-axis spectroscopy
 Phonon scattering
 Magnon scattering
Canadian Neutron Beam Centre
Neutron scattering
Nobel Prize 1994
“for the development
of the neutron
diff
diffraction
ti technique"
t h i
"
James Chadwick
1891-1974
Nobel
N
b lP
Prize
i 1935 for
f
“the discovery of the
neutron"
Clifford Schull
1915-2001
Where the atoms are?
Bertram Brockhouse
1918-2003
"for the development of
neutron spectroscopy"
p
py
What the atoms do?
Canadian Neutron Beam Centre
Why
y neutrons?
Remember what
you learned
about the
properties of
neutron from
Ian’s lecture:
http://neutrons.ornl.gov/aboutsns/importance.shtml#properties
Canadian Neutron Beam Centre
Neutron scattering
Neutron is scattered by matter via:
 interaction with nucleus
n
structural studies, this lecture
 interaction with spin of unpaired electrons, magnetic scattering
Dominic Ryan’s Lecture
These interactions can be:
 elastic
(diffraction)
structural studies
studies, yesterday lecture
 inelastic (spectroscopy) dynamical studies, today
analysis of the energy of scattered neutrons provides
information on excitations (lattice vibrations and magnetic
excitations)
Canadian Neutron Beam Centre
Neutron scattering
and dynamics
From Shirane et al, 2002
Canadian Neutron Beam Centre
Lattice vibrations &
why
h study
d them?
h ?
Can a rigid fixed model of lattice describe
properties of materials?
Atoms are not infinitely massive nor held in
place by infinitely strong forces → in classical
theory rigid model is valid only at T=0!
In quantum theory even at T=0, rigid model incorrect!
At T≠0, ions will have thermal energy: they have
mobility in vicinity of their equilibrium positions:
Animation from: http://wolf.ifj.edu.pl/phonon/animation/
Canadian Neutron Beam Centre
Lattice waves (phonons)
Many important features of crystal
dynamics is explained with atoms
coupled via spring-like forces!
Coupled atomic vibrations
generate a traveling wave with
displacements along the chain
(longitudinal) or perpendicular to it
Normal modes: all atoms vibrate
(transverse).
with same frequency.
d 2 ui
M 2  C (ui 1  ui )  C (ui 1  ui )
dt
4C
1
sin( qa )
 (q ) 
M
2
C=force constant between
nearest neighbours & can
be measured with neutrons
neutrons.
Quantum energy of lattice vibrations=phonon
Canadian Neutron Beam Centre
More on phonons:
acoustic vs. optic
Normal-mode vibrations of primitive bcc AB crystal, with atoms A and B at
positions (0,0,0) (blue) and (1/2,1/2,1/2) (red).
LA
TA
frequency
TO
LA
TO
TA
wavevector
Animations from: http://wolf.ifj.edu.pl/phonon/animation/
Canadian Neutron Beam Centre
More on phonons:
acoustic vs. optic
Normal-mode vibrations of primitive bcc AB crystal, with atoms A and B at
positions (0,0,0) (blue) and (1/2,1/2,1/2) (red).
LA
LA
frequency
TO
LA
LO
TA
wavevector
Animations from: http://wolf.ifj.edu.pl/phonon/animation/
Canadian Neutron Beam Centre
Magnetic excitations
A rigid model of magnetic moments fails
t describe
to
d
ib magnetic
ti materials
t i l ffully!
ll !
Normal excitations in a FM: spins of
magnetic atoms interact strongly. Ground
state: aligned spins. Excited state: a spin
flips (high energy). A small amount of
energy can create a much lower-energy
excited
it d state:
t t spins
i precess around
d
equilibrium direction: spin-wave!
This concept can be extended to AF and itinerant systems.
Canadian Neutron Beam Centre
Spin waves (magnons)
Magnetic materials: neighboring
spins are usually coupled.
When one spin changes direction,
it induces
i d
a wave-like
lik di
disturbance
t b
of all neighbouring spins!
Example: spins interact via
exchange interactions:
FM: Jij <0
H   J ij s i . s j
i, j
J can be measured with neutrons.
1
 (q)  2S  J (R ) sin ( q.R
q R)
2
R
2
Quantum energy of excitation=magnon
Canadian Neutron Beam Centre
How is it related to
experiment?
i
t?
Bertram Neville Brockhouse
E f, k f
2
Incident beam
Ei, ki
Conservation of momentum
Momentum transfer= ħQ
Scattering vector
Q = ki – kf
Energy transfer
E = Ei – Ef
Energy transfer= ħ
C
Conservation
ti off energy:
Ei + E= Ef + E →E =E
The number of scattered neutrons as a function of Q
and  is measured. The result is the scattering function
S(Q, ) depending only on the properties of the sample.
Canadian Neutron Beam Centre
Scattering cross
sections
Incident flux: 
=number
number of incident neutrons/cm2sec
Incident neutron beam directed along polar is
scattered by the sample along ().
Detector measures all the neutrons into
solid angle d in the direction of ().
Differential cross-section:
Partial differential cross
d number of neutrons scattered per second into d section (implies integration

over all energies or no
energy
analysis).
d
d
Double differential cross-section:
number
u be oof neutrons
eut o s scatte
scattered
ed pe
per seco
secondd into
to d aandd ddE f
d 2

ddE f
ddE f
Double differential cross section
(neutron flux into dΩ with final
energy between Ef and Ef +dEf)
Canadian Neutron Beam Centre
Total cross section
Partial differential cross
section (implies integration
over all energies or no energy
analysis)
Total number of scattered
neutrons in all directions  
(units: barn=10-24 cm2)
2
 d 
d

dEf
d 0 ddEf

In practice neutrons can also be
absorbed
abso
bed by tthe
e sa
sample,
p e, hence
e ce
the total cross-section for
neutrons (units: barn=10-24 cm2)
4
0
4
d
d  
0
d


0
d 2
dEf d
ddEf
 tot   scat   abs
Canadian Neutron Beam Centre
Scattering by many
nuclei
l i
Léon Van
Lé
V Hove
H
1924-1990
Measured scattering intensity is the sum of
scattering from each individual nucleus!
Pseudo-potential (Fermi): interaction between a
neutron and a nucleus is replaced by a much weaker
effective potential.
Perturbation approximation (Born): effective
potential
po
e a is
s weak
ea e
enough
oug to
o use pe
perturbation
u ba o in
calculating scattering!
Scattering law (Van Hove): probability of a neutron
wave ki, Ei being scattered by V(r
V(r,t)
t) into outgoing
wave of kf, Ef is:
iQ.r
2
V
(
r
,
t
)
e
d
r
 (E   )

Integration is over the volume of the sample. E is the
change in energy of the sample due to scattering.
Enrico Fermi
1901-1954
Nobel Prize 1938 for “his
work on induced
radioactivity”
Max Born
1882-1970
Nobel Prize 1954 for “his
fundamental research in
quantum mechanics”
Canadian Neutron Beam Centre
Scattering by many
nuclei
l i
2 2
V (r,,t ) 
b j (r - R j )
Fermi p
pseudo-potential
p
for an

m j
assembly of nuclei at positions Rj is: m is neutron mass,
 is Dirac delta function=1 at
position r and zero elsewhere, bj are scattering lengths.
k f 1  m 2
d 2


2 
ddE f
ki 2π  2 
kf
1

k i 2π



-
 V (r' , t )e
 b b  dt e
j
j,j'
R j (t )  e iHt /  R j e  iHt / 
j'
 i t
e
 Q.r'
2
dr ' e iEt e it dt
iQ . R j ( t )  i Q . R j ' ( 0 )
e
-
This is Heisenberg operator for the position of the jth nucleus, H
is the Hamiltonian of the scattering system. Classically it can be
regarded as the position of the jth nucleus.
Double sum over all of positions of nuclei in the sample. Angular brackets
mean a thermal average at the temperature of the scattering system.
Canadian Neutron Beam Centre
More on scattering
by many nuclei
kf 1
d 2

ddE f
k i 2π

 b b  dt e
j
j,j'
 i t
j'
-
e
iQ . R j ( t )  i Q . R j ' ( 0 )
e

kf

1

b j b j '  dt e it   (r  [R j ' (0)  R j (t )]) e iQ .r dr

ki 2π j,j'
j j'
-
-
d 2
N b2 k f
For bj=b
bk: ddE  2 k
f
i
Only considering
coherent scattering!
 

 ir. Q  i t
G
(r
,
t
)
e
e
drdt

- -
1
G (r,t )    (r  ( R j ' (0)  R j (t )))
N j, j'
Fourier transform
of time-dependent
pair correlation
function G(r,t)
Intensity is proportional to Fourier transform of time-dependent pair
correlation function (probability of finding two atoms being a certain distance
apart at a certain time). Scattering gives information as how correlations
between pairs of nuclei evolves with time.
Canadian Neutron Beam Centre
Dynamic structure
f t S(Q,)
factor
S(Q )
d 2
N b2 k f

2 ki
ddE f
N
kf
ki
 
ir.
r Q it
G
(
(r
,
t
)
e
e drdt
d

- -
b 2 S (Q,  )
S(Q,)
S(Q
) contains all the physics of system: neutron scattering probes dynamical
processes over a length scale ~1/Q & over a time scale ~1/. S(Q,) can be
calculated and compared with measurements to test theories.
Ph
Phonon
dispersion measured
in the hexagonal
phase of ice:
http://www.isis.rl.ac.u
k/isis97/excite.htm
Magnetic
excitation of
URu2Si2:
Nature Physics
2007.
Canadian Neutron Beam Centre
Fluctuation dissipation
theorem
th
1
S(Q,)  / kBT ''(Q,)
1e
For  both positive and negative.
Induced fluctuations due to
an external perturbation
Spontaneous fluctuations in
thermodynamic equilibrium
’’(Q,): imaginary part of dynamical susceptibility: basic excitation not
complicated by thermal population of states, often calculated in theoretical
modeling. M(Q,)=(Q,) H(Q,): linear response due to magnetic
perturbation varying in space and time.
What does it really mean?
Evolution of an externally induced perturbation is similar to that of a
spontaneous fluctuation! Neutrons interact weakly with the system (a small
perturbation) causing a linear response:
fluctuations in unperturbed system are observed!
Canadian Neutron Beam Centre
Detailed balance
kf 2
d 2
N
b S (Q ,  )
ki
ddE f
S (Q, )  e
Neutron energy gain


k BT
S (Q,  )
Neutron energy loss
 is
i assumed
d tto b
be positive.
iti
Probability of a transition in
sample depends on statistical
weight factor of the initial
state: always lower for
annihiling an excitation than
creating one!
Canadian Neutron Beam Centre
Scattering triangle:
elastic
l ti vs. inelastic
i l ti
Elastic: ki=kf
Neutron Energy
gy
loss: kf<ki
Neutron Energy
gy
gain: kf>ki
Kinematic range that can be covered in a scattering event:
Q  k k  G
q
i
f
hkl
2 2
Q  Ei  E f  2 Ei E f cos 2
2
2
2
Q

k

k
 2ki k f cos 2
2m
ki
2

E  Ei  E f 
Q 2  2 E f  E  2 E f ( E f  E ) cos 2
2m
i
kf
2
f
For a fixed final energy experiment
Q
Ghkl
q
Canadian Neutron Beam Centre
Triple-axis spectroscopy
NRU
neutron source
Elastic: Ei=Ef
Inelastic: Ei≠Ef
Detector
Sample
This techniques allows performing
measurements point by point in
momentum- energy space!
Monochromator
Analyzer
C5 spectrometer
Canadian Neutron Beam Centre
Triple-axis experiment

Planning the experiment:
1. Elastic or inelastic scattering?
2. Magnetic scattering?
3 Energy (E) & momentum transfer (Q) range?
3.
4. What resolution? (uncertainties in Q & E
determine the instrument resolution)
Final energy

Monochromator/analyzer
mosaic
collimations
What needs to be done:
1. Determine the wavelengths of neutrons (wavelength resolution)
2 Beam
2.
B
collimations
lli ti
((angular
l resolution)
l ti )
3. Detect neutrons (statistical process with uncertainty proportional to
square root of the counts)
Canadian Neutron Beam Centre
Phonon scattering
Sums are over all reciprocal lattice
vectors and all the phonon modes.
k f (2 ) 3
2
d 2

Fj (q, Q) 

ddE f
k i v c G jq


n (q) (   (q)) (Q  q  G)  (n (q) 1) (   (q)) (Q  q  G))
j



j j  j
 Phonon annihilation

Phonon creation
Phonon dynamic structure factor:
Fj (q , Q )  
l
1
2 ml  j (q )
Intensityy increases
as Q2, measure at
higher zones!
bl [Q.e j (q )] e iQ.rl e Wl ( Q )
Debye-Waller factor to
describe attenuation due to
thermal motion.
Selection rule: separate
transverse from longitudinal
Delta-functions mean
that we only observe
peaks only when:
ħ ± ħphh
ħQ ħqph ±G)
Canadian Neutron Beam Centre
Phonon scattering:
examples
[0 0 1]
[0,0,1]
Axe, Shirane, s-wave superconductor: Nb3Sn
Phys. Rev. Lett. 30 (1973) 214.
Yildirim et al, Superconductivity in MgB2, a special type
of vibration in the MgB2 lattice is responsible for high Tc.
http://www.ncnr.nist.gov/staff/taner/mgb2/
Canadian Neutron Beam Centre
Magnon scattering
Magnetic
M
ti diff
diffraction:
ti
L
Lecture
t
by Dominic tomorrow.


2
k
d 
2  2W Q 

f  g
ˆ Q
ˆ S  Q,



Q
Q 
  r0  F Q  e




ddE f
ki  2


|F(Q|: Magnetic form factor due to
Magnetic scattering
i t
intra-atomic
t i iinterference,
t f
Fourier
F i
function: time and spatial
transform of spin density distribution
Fourier transform of spinon atom decreases with increasing Q.
p correlation function.
spin
2
For a FM:
S (Q,  )  S 
G,q


n (q) (   (q)) (Q  q  G )  (n (q)  1) (   (q)) (Q  q  G ))
j



 j j  j


How to separate from phonons?
Use differences from phonon
scattering: intensity decreases with Q
and no Q2 dependence! Usually also
decreases with T unlike phonons.
 



ˆ Q
ˆ S  Q,  
Q


Neutrons scatter from m
perpendicular component of atomic
magnetic moment to Q
Canadian Neutron Beam Centre
Magnons in MnF2
Tetragonal (P42/mnm): a=4.873,
c=3.130 Å
Mn: (2a) (0, 0, 0) ; (1/2,1/2,1/2)
F: (4f) (x,x,0); (-x,-x,0); (1/2+x,1/2x,1/2); (1/2-x,1/2+x,1/2); x=0.3
NN coupling, J1, much
weaker & FM
NNN coupling, J2, through
fluorine ligands
Single-ion anisotropy
1
1
H  J 2  S r . S r  d 2  J 1  S r . S r  d 1  D  (S r, z ) 2
2 r ,d 2
2 r ,d 1
r
q  2 (1   q ) 2   q2 ]
1
D  21 sin 2 ( q z c )
2
q 
with  i  2 Sz i J i (z 1  2, z 2  8)
2  2
1
2
1
2
1
2
 q  cos( q x a ) cos( q y a ) cos( q z c)
Canadian Neutron Beam Centre
Observation of magnons
N5 spectrometer
McGill Condensed Matter
Graduate Course with Prof. Ryan
and students, March 2009.
MnF2 dispersion
Student’s Report.
Canadian Neutron Beam Centre
More on inelastic
scattering
tt i
… look for lots more
inelastic examples in talks
b Maikel
by
M ik l Rh
Rheinstaedter
i t dt
& Bruce Gaulin.
Canadian Neutron Beam Centre
Final word!
Hope tto h
H
have convinced
i
d you ((and
d more so b
by th
the end
d off
Summer School) that neutron scattering is a powerful probe:
 used to directly study fundamental
structural & magnetic correlations (both
static and dynamic) in condensed matter!
and hence to enhance our understanding of
microscopic origin of physical properties of
materials.
materials
Canadian Neutron Beam Centre
References and further
readings
 Squires,
q
, Introduction to the theory
y of thermal Neutron Scattering.
g Dover.
 Shirane, Shapiro, Tranquada, Neutron scattering with a triple-axis
spectrometer, basic techniques. Cambridge University Press.
 Willis and Carlile, Experimental neutron scattering. Oxford University
Press.
 Lovesey, Theory of neutron scattering from condensed matter, Oxford
University Press.
 Pynn,
P
N t
Neutron
scattering:
tt i
a primer.
i
L Al
Los
Alamos N
Neutron
t
S
Science
i
C
Centre.
t
 Warren, X-ray Diffraction. Dover.
 Kittel, Introduction to solid state physics. Wiley.
 Ashcroft and Mermin,
Mermin Solid state physics.
physics Saunders College.
College
 http://www.ncnr.nist.gov/resources/n-lengths/.
 http://neutrons.ornl.gov/science/index.shtml.
http://neutrons ornl gov/science/index shtml
 http://www.neutron.anl.gov/reference.html.
 http://neutron.nrc-cnrc.gc.ca/home_e.html.