Elastic x-ray scattering
Peak
amplitude:
(slide from last time)
A(2θ ) ∝ ∑∑
n
j
! !
Nδ ( q − K )
Peak positions
(Bravais lattice)
(
! ! !
f j (2θ ) exp − iq ⋅ ( Rn + d j )
= ∑e
! !
−iq ⋅ Rn
n
∑f
j
)
(2θ )e
! !
− iq ⋅ d j
j
S K! ≡ ∑
j
(
! !
f j (2θ ) exp − iK ⋅ d j
)
Note, background may be due to:
• Impurities, defects in crystal.
• Phonons
• Inelastic core excitations (fluorescence)
• Etc.
Thermal displacement parameters
displacement ellipsoids for
PbBi2Nb2O9 ferroelectric;
Ismunandar et al. Solid State
Ionics 112, 281 (1998)
• 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
Bloch Theorem and Electron Bands
v Assume: noninteracting electrons, simple product states.
E = ∑εi
Ψ = ψ 1 (r1 )ψ 2 (r2 )ψ 3 (r3 ) !
i
v Also Born-­‐Oppenheimer approximation, and assume crystal potential includes average of all other electron interactions. (Hartree approximation)
v Justification comes later (Landau theory, ch. 17).
v Then can show, Bloch theorem form of wavefunction.
ψ i = ∑α ke

ik⋅r
! !
!
iK ⋅ r
U (r ) = ∑ U K! e
K
k
∑e
k
""
ik ⋅ r
U with crystal symmetry; ψ with
periodic boundary conditions.
⎧⎛ ! 2 k 2
⎫
⎞
− ε ⎟⎟α k + ∑ U K α k − K ⎬ = 0
⎨⎜⎜
K
⎠
⎩⎝ 2m
⎭
!!
! !
! ik!⋅r! ⎛
iK ⋅ r ⎞ ik ⋅ r
ψ i = u ( r ) e = ⎜ ∑ α k , K e ⎟e
⎝ K
⎠
!
!
!
u (r ) = u (r + R)
Schrödinger equation
Bloch form (general
form of wavefunction)
Bloch States and Electron Bands
v Can always re-­‐define states by replacing k by k ± K.
v Consequences: k overdetermined modulo K; shall see, k
conserved only within single Brillouin Zone; k conservation rather than overall momentum. [k = “Crystal momentum”]
Free-­‐electron states
K
• Note, each state is displayed multiple times in the figure.
• States appear once in each Brillouin Zone. [Twice with spin.]
Bloch States and Electron Bands
v Can always re-­‐define states by replacing k by k ± K.
v Consequences: k overdetermined modulo K; shall see, k
conserved only within single Brillouin Zone; k conservation rather than overall momentum. [k = “Crystal momentum”]
Free-­‐electron states
Free-­‐electron states folded into 1BZ.
K
• Note, each state is displayed multiple times in the figure.
• States appear once in each Brillouin Zone. [Twice with spin.]
Electrons with a crystal potential:
› Electron energies (and wavefunctions) no longer same as simple plane wave states; strongest effects at zone boundaries.
› Shown is case of “nearly free electron model”.
Free-­‐electron states folded into 1BZ.
Brillouin Zones:
•
•
•
Mth zone ≡ region having origin as Mth nearest K point.
Equivalent definition: Region reached from origin by crossing (M - 1)
perpendicular bisector planes.
Each zone contains N “allowed k points” (e.g. as defined with periodic boundary
conditions), where N = number of Bravais-lattice cells in crystal.
! ! !
! ! !
b1 ⋅ b2 × b3 = (2π )3 / a1 ⋅ a2 × a3
•
All zones have same total volume; can “fold” zones into 1st zone by translation
through K vectors.
2D case