Chapter 4 Inelastic Neutron Scattering of Phonons Chapter 4 Inelastic Neutron Scattering of Phonons Phonon dispersion curves � Phonon dispersion curves are usually measured using inelastic neutron sca[.]
Trang 1Chapter 4: Inelastic Neutron
Scattering of Phonons
Trang 2Phonon dispersion curves
• Phonon dispersion curves are usually measured using
inelastic neutron scattering The energies of neutrons at root temperature are about the right value for the
average phonon (~ 1 meV)
• Dispersion relations can be measured in different
directions in the crystal (so we are really measuring
ω(K))
• Last time, we learned that the following relationship must
be held with scattering:
k + G = k’ +/- K
Incident neutron
Phonon wavevector (+ for phonon created,
- for phonon absorbed) Reciprocal lattice vector Scattered neutron
Trang 3(Cold, thermal, and hot refer to the temp of the moderator, which can
be cooled by cryogenic liquids or heated up high T (eg ~ 600 K))
Trang 4Inelastic neutron scattering
Triple-axis neutron spectrometer
G
K
Defines outgoing wavelength, which may
be different if the neutron lost a bit of energy to make a phonon
Defines incoming wavelength
Trang 5Triple Axis Spectrometer
Trang 6Geometry of experiments
• In reciprocal space, the geometry of the experiments looks a bit different
• For example: for measuring the dispersion along the (100) direction, we can do this in a few different ways Here is one way:
(010) (110) k
k’
K
G
Phonon created:
k + G = k’ + K
So
k – k’ = G + K
(but you can measure this in different
ways by varying G The energy transfer
is from changing the k and k’ vectors)
(210)
(000) (100) (200)
Trang 7Inelastic neutron scattering
• Using the conservation of energy, we can define the energy of the
phonon created by:
• So, you can measure the incoming neutron’s wavevector and
energy, and the outgoing neutron’s wavevector and energy, and
then solve for the the phonon’s energy The phonon’s wavevector is solved for using: k + G = k’ +/- K
ω
h
h
h2 22 = 2 22 ±
2
'
2 n M n
k M
k
Mass of neutron
Incoming neutron energy Scattered neutron energy
Phonon energy (sign changes if phonon
is created or destroyed)
Trang 8Phonon dispersion curves
La La
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c
a
• Phonon dispersion relations
can get very complicated!
• Eg La2CuO4 (material used to
make superconductors)
• The structure is tetragonal (a =
b ≠ c, all angles 90 degrees)
• There are two important
directions: the ab-plane, and
the c-direction
• We would expect the phonons
to act differently (have different
dispersion curves) in different
directions.
b
Blue Cu octahedra
Trang 9Phonon dispersion relations
ab-plane ab-plane
• This is the dispersion
relationship in different
directions, as measured by
neutron scattering.
• Why is this important?
• Phonons help some materials
become superconducting – the
lattice distortions play a role in
making the electrons stick
together to form Cooper pairs
(which conduct electricity with
zero resistance)
• We have to understand how
phonons work to figure out
how materials superconduct.
c-direction
Trang 10BCS Superconductivity
(Bardeen, Cooper, and Schrieffer, 1957, Nobel prize)
• Normally, electrons repel one
another But in a
superconductor, they ‘pair up’ to
form Cooper Pairs
• In BCS superconductivity, the
mechanism for this is phonons
Lattice distortions create
temporary clusters of positive
charge, which attract electrons
and enables them to pair up
• Many elements (eg Ti, V, etc.)
are superconducting at low
temperatures (< 1 K)
• The high-TC superconductors, on
the other hand, have much
higher TCs (up to 100 K or so)
• Do lattice vibrations still play a
role?
e
Cooper Pair
of electrons
Trang 11What else do phonons tell us?
• If a crystal structure changes, then
we often see something
happening to the lattice vibrations
first (the atoms want to move in a
slightly different way)
• This is called mode softening
• Eg : Nb3Sn has a phase transition
from a cubic to a tetragonal
structure at 43 K (the c-axis
shrinks, and ab-plane stretches
out a bit)
• We see something happening to
the phonon dispersion relation at
temperatures much higher than
this transition (from x-ray diffraction)
Trang 12Mode softening
• These are phonon
measurements in the ab-plane
(inelastic neutron scattering).
• Notice how you need less and
less energy to get the atoms to
move in the ab-plane as you
approach the transition temp.
• This is an example of mode
softening
• At 46 K, the mode almost
disappears at low enough
energies
(excitations in the ab-plane)
(K is called q here)
(Zone Boundary)
Trang 13Spin waves (magnons)
• Because neutrons have spin, they can also cause spin-wave
excitations in ferromagnets and antiferromagnets These are called magnons.
An ordered ferromagnet (aligned spins at low temperatures)
A spin wave (spin excitation)
(Top down view)
Trang 14Spin waves
• This is an example of a
ferromagnet, EuO, where
all the Eu spins align
below 69 K
• Neutrons can measure
the dispersion relations of
spin excitations.
• Notice how it looks like an
acoustic phonon (spins
moving in phase)
Trang 15Inelastic Neutron Scattering
This is an experiment done on CuGeO3, which has 1D chains of ordered spins
Notice how
it looks like
a sin curve: the theory for
spin waves
is similar
to lattice vibrations
Low Temp (below transition to ordered state) High Temp
(same as K)