Lattice Planes
Parallel equal-­‐spaced planes, intersect all Bravais lattice points, equal density of points.
(Existence proof we’ll see in terms of Reciprocal lattice) Indexing: • place origin on an atom at corner of cell, edges = lattice vectors. • choose plane nearest origin. • indices are integer divisors of the lengths of cell edges intercepted by plane. • Usual notation is h, k, ℓ.
d = distance between planes
(200) planes, simple cubic.
Note, each plane intersects equivalent number of atoms, provided indices have no common denominators.
Lattice Planes
(100) planes, FCC
Lattice planes intersect Bravais lattice
points (not necessarily all atoms)
FCC and BCC are always indexed as if they were simple cubic with a basis • h, k, ℓ called Miller indices.
• Also note, Bragg planes are the “mirror planes” of x-ray scattering.
Reciprocal Lattice:
Vectors form k-space Fourier components of Bravais (direct-space) lattice.
• Plane waves have symmetry of lattice:
!
!
!
!
• To construct: K = hb + kb + "b
1
2
3
& find:
! !
!
a2 × a3
b1 = 2π ! ! !
a1 ⋅ a2 × a3
e
! ! !
iK ⋅( r + R )
with
=e
! !
iK ⋅ r
! !
ai ⋅ b j = 2πδ ij
! !
!
2π a × a
b2 = ! !3 !1
a1 ⋅ a2 × a3
! 2π a! × a!
b3 = ! !1 !2
a1 ⋅ a2 × a3
(3D cases)
• Then can show: Wavefronts of
e
! !
iK ⋅ R
are Bragg planes;
! 2π
e.g. K =
nˆ must be a K vector;
d
• Reciprocal lattice is a Bravais lattice; “dual” to direct lattice
(uniquely associated with it);
• Direct lattice: reciprocal of reciprocal lattice.
K
! !
!
!
n( r ) = ∑
nK exp[iK ⋅ r ]
!
Reciprocal Lattice:
K
Vectors form k-space Fourier components of Bravais (direct-space) lattice.
• Plane waves have symmetry of lattice:
!
!
!
!
• To construct: K = hb + kb + "b
1
2
3
& find:
! !
!
a2 × a3
b1 = 2π ! ! !
a1 ⋅ a2 × a3
e
! ! !
iK ⋅( r + R )
with
=e
! !
iK ⋅ r
! !
ai ⋅ b j = 2πδ ij
! !
!
2π a × a
b2 = ! !3 !1
a1 ⋅ a2 × a3
! 2π a! × a!
b3 = ! !1 !2
a1 ⋅ a2 × a3
(3D cases)
• Then can show: Wavefronts of
e
! !
iK ⋅ R
are Bragg planes;
! 2π
e.g. K =
nˆ must be a K vector;
d
• Reciprocal lattice is a Bravais lattice; “dual” to direct lattice
(uniquely associated with it);
• Direct lattice: reciprocal of reciprocal lattice.
K
n̂
• Examples
• Fourier expansion: see more remarks, Appendix D
! !
!
n( r ) = ∑
nK! exp[iK ⋅ r ]
!
K
1
nK! =
Vcell
! !
∫ d r n(r ) exp[−iK ⋅ r ]
3
cell