Notes:
• Reminder,)Exam:))Tues,)Oct)13.
!
!
!
In)class.
Covers)through)ch. 5
Open)book)=)Kittel)only.
• HW)4:))Can)turn)in)Thursday,)Oct.)8.)(I)will)have)solutions)
ready.)
• Note)on)HW)grades:))
averages)38/50)(HW1),)31/40)(HW2),)37/45)(HW3).))
Important)to)understand)what)you)missed!))
Neutron)scattering:
Recall with periodic crystal:
• Crystal momentum conserved
(!k; text uses !K).
• also energy conserved
(phonon:))ΔE = !ω)
•
Neutron energy vs. k comparable to phonon (& magnetic) excitations
•
Used to observe both elastic and inelastic scattering.
Neutron)scattering)(also)xRray):
Atomic displacements
Elastic scattering attenuation
measured this way – This
vs. T – appendix A;
example ReO3
Debye-Waller Factor.
(Rodriguez et al., J Appl Phys 2009)
Neutron)scattering)(also)xRray):
Displaced atom
H ∝ uˆ (or q)
Contribution of phonons:
Intensity = square amplitude:
I∝ u
u 2 ∝ ψ at a ψ
∝ sum of n(T )
1
n = !ω
∝ T , high T
e kT − 1
So,
Elastic scattering attenuation
vs. T – appendix A.
Debye-Waller Factor: phonon scattering increases linear in T, high T;
elastic peaks decrease initially linear in T (actual solution, exponential).
2
Inelastic)Neutron)scattering:
Triple-axis: analyze for momentum/energy of outgoing
neutrons (NMI3 website; & see figure 4.12 in text)
Example, scanned
energy transfer,
giving phonon
dispersion curves.
Inelastic)Neutron)scattering:
H ∝ uˆ (or q)
Recall ~ sum of a and a†
Probability of scatter includes zero and one-phonon
terms (Fermi golden rule). (Also higher-order.)
P ∝ ψ H o + H1 + ... ψ
Example, scanned
energy transfer,
giving phonon
dispersion curves.
Inelastic)Neutron)scattering:
(B.)Gaulin)
Inelastic)Neutron)scattering)other)examples:
Quasicrystal and related phonons
(de Boissieu et al. Nature Materials
6, 977 (2007))
Magnetic excitations
(pyrochlore “spin ice”).
Kimura et al. Nature Communicatons 2013
Thermal)conductivity)of)phonons:
General case for thermal conductivity by a “gas”:
κ = 13 C!v
Heat
capacity
(or an appropriate average of these quantities)
velocity
Mean
free
path
(v g )
Q! = −κA dT
dx
!
!
jq = −κA∇T
Thermal)conductivity)of)phonons:
General case for thermal conductivity by a “gas”:
κ = 13 C!v
Heat
capacity
(or an appropriate average of these quantities)
Mean free path
(schematic)
→ const.
const.
boundary
Umklapp
1/ T
T3
0
T
Umklapp =)“flipRover”
Thermal)conductivity)of)phonons:
κ = 13 C!v
→ const.
Mean free path
const.
boundary
T
3
Heat
capacity
Umklapp
1/ T
T
0
T3
1/ T
Thermal)conductivity)of)phonons:
T3
Scattering from point
defects analogous to
Rayleigh scattering
(long-wave case)
Negligible at low T
κ = 13 C!v
1/ T
Boron nitride
isotope effect
S. Barman,
Europhys. Letters
2011