Inelastic Neutron scattering:
•
Neutron energy vs. k comparable to phonon (&
magnetic) excitations
•
Observe elastic and inelastic scattering.
•
Crystal momentum, energy conserved
sample
!
k′
!
k
Reactor
(or beam source)
Analyzer
(or time of flight
system)
Triple-axis: analyze for momentum/energy of outgoing
neutrons (NMI3 website)
Example, scanned
energy transfer,
giving phonon
dispersion curves.
Inelastic Neutron scattering:
Probability of scatter includes zero and one-phonon
terms (Fermi golden rule). (Also higher-order.)
P ∝ ψ H o + H1 + ... ψ
2
H o ⇒ bragg scattering
H1 ∝ uˆ (or Qq )
Recall ~ sum of a and a†
(One phonon term.)
gives isolated peaks.
H2: gives broad background. (also
incoherent atom displacements or
impurities give broad signal.)
Example, scanned
energy transfer,
giving phonon
dispersion curves.
Quantized Modes (phonons) recall from before:
⎛ pˆ i2
⎞
⎛ pˆ i2
⎞
H = ∑i , j ⎜
+ uˆi ⋅ Dij ⋅ uˆ j ⎟ ⇒ ∑i , j ⎜
+ K (uˆi − uˆ j ) 2 ⎟
2
⎝ 2M
⎠
⎝ 2M
⎠ (1D)
Canonical operators
N atoms
pˆ i uˆi = −i! + uˆi pˆ i
Convert to sum over N wave-vectors k (appendix L)
1
Qk =
N
Now we are using
q for the phonon.
∑ uˆ e
i
−ikxi
1
Pk =
N
mω
1
ak =
Qk + i
Pk
2!
2!mω
∑ pˆ e
i
−ikxi
(note N k vectors in 1st BZ make complete set)
mω t
1
t
ak =
Qk − i
Pk
2!
2!mω
t
• Analogs of lowering, raising operators; “remove/add a phonon”
• ω(k) is classical lattice solution
Inelastic Neutron scattering:
Probability of scatter includes zero and one-phonon
terms (Fermi golden rule). (Also higher-order.)
P ∝ ψ H o + H1 + ... ψ
2
H o ⇒ bragg scattering
H1 ∝ uˆ (or Qq )
Recall ~ sum of a and a†
(One phonon term.)
gives isolated peaks.
H2: gives broad background. (also
incoherent atom displacements or
impurities give broad signal.)
Example, scanned
energy transfer,
giving phonon
dispersion curves.
Inelastic Neutron scattering:
Probability of scatter includes zero and one-phonon
terms (Fermi golden rule). (Also higher-order.)
P ∝ ψ H o + H1 + ... ψ
Selection rules:
! !
!
Δk = K ± ∑ q
Δε = ± ∑ !ωq
2
Example, scanned
energy transfer,
giving phonon
dispersion curves.
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
Bragg scatter intensity: Debye-­‐Waller
H ∝ uˆ (or q)
Displaced atom
Contribution of phonons:
I ∝ u2
Intensity = square amplitude:
u
So,
∝ ψ′ aψ
2
etc.
∝ sum of n(T )
Elastic scattering attenuation
vs. T – appendix N.
2
n=
1
e
!ω
kT
−1
∝ T , high T
Debye-Waller Factor: phonon scattering increases linear in T, high T;
elastic peaks decrease initially linear in T (full solution is exponential drop).
(Appendix N of A&M)
Thermal displacement parameters
displacement ellipsoids for
PbBi2Nb2O9 ferroelectric;
Ismunandar et al. Solid State
Ionics 112, 281 (1998)
I showed this slide
earlier in semester
• Generally static + dynamic disorder
both reduce x-ray peaks.
• Dynamics (phonons) see appendix N
(Debye-Waller factor, discuss later)
displacement parameter (thermoelectric material)
shows T 0 disorder. Falmbigl et al., Dalton Trans.
41, 8839 (2012)
Xray peaks -> average
displacement; (can’t resolve
coherent vs. random this way.)
u2