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7/28/2019 Chapter 2_bruillion_Zone.pdf
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Chapter 2Reciprocal Lattice
Phys 175A
Dr. Ray Kwok
SJSU
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Crystal Lattice
• Periodic f(r + T) = f(r) for any observable
functions such as electronic density,electric potential….etc. which means theyare all periodic functions because of thetranslational properties of the latticevectors.
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Fourier Transform
Fourier series (1D): f(r) = ∑n An ei2πnr/a
where the period is a.
such that f(r + ua) = f(r); u is an integer.
Or write f(r) ≡ ∑G AG eiGr
where G = 2πn/a (for 1D lattice)
and AG = (1/a) ∫cell f(r) e –iGr
dr
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1D Translational Invariance
f(r + T) = ∑G AG eiG(r+T) = ∑G AG eiGr eiGT
= f(r) = ∑G AG eiGr
i.e. eiGT = 1
For example, if T = ua (1D),
then GT = 2πun = 2π·(integer).and eiGT = 1.
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3D Translational Invariance
For 3D lattice, eiG·T = 1
and T = u1a1 + u2a2 + u3a3 is the translationalvector or the lattice vectorDefine “reciprocal primitive vectors”
b1
= (2π /V) a2
x a3
b2 = (2π /V) a3 x a1
b3 = (2π /V) a1 x a2
where V is the volume of the primitive lattice cell
V = | a1 · a2 x a3|, such that bi· a j = 2πδij
and δij is the kronecker delta function.Note: {a}’s don’t have to be orthogonal.
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Reciprocal Lattice Vectors
Define G = v1b1 + v2b2 + v3b3
where the v’s are integersso that
G·T = 2π ( u1v1 + u2v2 + u3v3) = 2π·(integer)
[because bi· a j = 2πδij]and eiG·T = 1
In other words, because of the translational
invariance property of the crystal, there exist aset of vector G such that G·T = 2π·(integer).This set of vector defines another set of latticepoint in the Reciprocal Space.
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Reciprocal Vector Space
G has a unit of 1/length. Similar to the
wave-vector k in the plane waveexpression eik·r.
G has a meaning in Fourier transform, k-space, or momentum space.
It defines a set of lattice points in the k-
space.
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Direct and Reciprocal Lattice
• For every crystal, there is a set of space lattice
(crystal lattice) – location of lattice points in realspace where atoms and molecules are.
• There is also a set of reciprocal lattice in the
momentum space (k-space) – something we seein diffraction measurement. However, there isno physical object present at the reciprocal
lattice sites.
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Light Microscopy vs X-ray Crystallography
λ ~ 500 nm λ ~ 0.1 nm
see real space see reciprocal space
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Tunneling Microscope vs X-Ray Diffraction
Reciprocal Lattice
(volume)“see” Direct Lattice(surface only)
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d D
Diffraction maximum can be specified by the Miller indices [hkl]
Reciprocal nature of diffraction pattern
Bragg’s law: 2d sinθ = n λ d ∝ 1/sinθ
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What do we learn from x-ray?
•Lattice parameters(Space Group)
•Symmetry (Point Group)
•Miller Index (h,k.l) foreach point
•Intensity (square ofstructural factor) of each
reflection
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Properties of G – reciprocal lattice
each G is normal to a
lattice plane in realspace
G·T = 2πn. For a
fixed G and n, thereare many T vectorssatisfied this equation.
But they all lie on theplane perpendicular toG.
G
T
Crystal plane
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Distance between planes = 2ππππ/G• If G has no common
factor (prime #), then the
distance between crystalplane perpendicular tothe G is 2π /G
d = T cosθ = 2πn/G
d’ = T’cosθ’ = 2π(n+1)/G separation of planes
= d’ – d = 2π /G
and G must be thesmallest reciprocal latticefor a given direction in k-space (prime #).
G
T
Crystal planes
T’
d d’
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Small G dominates The larger the G, the closer the crystal plane, and less
atoms on the plane.
For example, in sc, separation of (100) is a,of (110) is a/ √2 = 0.71a
(110)
(100)
a1
a2
(210)
(100)
(110)
a 0.71a
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Volume of the reciprocal cell
If the volume of a unit cell of the direct lattice is V, thenthe volume of a unit cell of the reciprocal lattice is (2π)D/Vwhere D is the dimension of the lattice, usually 3.
3D:
Vg = | b1 · b2 x b3|= (2π /V)3 {(a2 x a3) · (a3 x a1) x (a1 x a2)}
= (2π /V)3 {(a2 x a3) · [a3·(a1 x a2)] a1 - [a1·(a1 x a2)] a3}
= (2π /V)3 {(a2
x a3) · [a
3·(a
1x a
2)] a
1}
= (2π /V)3 {(a2 x a3) · V a1 }
= (2π)3 /V2 {(a2 x a3) · a1 }
= (2π)3 /V
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Primitive Reciprocal Lattice Cell
The Wigner-Seitz cell of the reciprocal lattice
is called the Brillouin Zone
Reciprocal Lattice
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Cubic – reciprocal lattice
=
=
=
za
ya
xa
3
2
1
a
a
a
r
r
r
( )( )
( )
+=
+=+=
xza
zyayˆ
xˆ
a
21
3
21
2
2
1
1
a
a
a
r
r
r
( )( )
( )
+−=
++−=
−+=
zyxa
zyxa
zyxa
21
3
21
2
21
1
a
a
a
r
r
r
( )
( )
( )
=
=
=
z / 2b
y / 2b
x / 2b
3
2
1
a
a
a
π
π
π
r
r
r
( )( )
( )
+=
+=+=
xzb
zyb
yxb
23
22
21
a
a
a
π
π
π
r
r
r
( )( )
( )
+−=
++−=−+=
zyxb
zyxbzyxb
23
22
2
1
a
a
a
π
π
π
r
r
r
( )3 / 2 aπ
( )3 / 24 aπ
( )3 / 22 aπ
SC
FCC
BCC
Direct Lattice Reciprocal Lattice Volume (k-space)
SC
BCC
FCC
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bcc
Direct LatticePrimitive Cell
Wigner Cell
Reciprocal Lattice(fcc)
Brillouin Zone
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fcc
Direct LatticePrimitive Cell
Wigner Cell
Reciprocal Lattice(bcc)
Brillouin Zone
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1D - Reciprocal Lattice
a
Direct Lattice
bi·a j = 2πδij means b·a = 2π
so b = 2π /a along the same direction
2ππππ/a
First BZ
b=2π /a
Reciprocal Lattice
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Diffraction in quantum mechanic Diffraction in quantum mechanic description: (Born
approx)
The transitional amplitude to scatter from k-state to k’-state = Mkk’ = <k’|V(r)|k> = ∫Ψ*(r) V(r) Ψ(r) dr
Ψ(r) = eik·r is the plane wave (think of the Bragg
scattering plane incident & diffracted waves) In Fourier series, V(r) = ∑ VG eiG·r (with the
translational property of the lattice)
Mkk’= ∑ VG ∫ ei(G + k - k’)·r dr = ∑ VG δ(G – ∆k)
= VG if G = ∆k , Mkk’= 0 otherwise. So the condition for Bragg scattering becomes
G = ∆k = k’ - k
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Bragg Scattering For elastic scattering, frequency ω = ck is
unchanged. Speed of light is the same before &
after scattering off the target, so |k| = |k’|.write k + G = k’ and square both sides gives
2 k·G + G2 = 0,
k
k’
θ
∆k=G
θ
θ
k αβ
2k·G = 2kG cosα = -G2
-2k cosβ = -G2k sinθ = G
2(2π/λ) sinθ = 2πn/d2d sinθ = nλwhich is the Bragg condition.
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X-ray data
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Bragg Conditions
2 / G|Gk |
0GGk 2 2
=⋅
=+⋅r
r r
k
k’ ∆k=G
B r a g g P l a
n e
k’
ko
Can use different k to probethe same G. Different circles,
different k and angles.
Or similarly, all k end on theperpendicular bisecting plane
of G would give Braggdiffraction. (BZ !!)
oGG/2
k
G/2
k’
BZ
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Ewald Sphere
k
k’2θ
G
A sphere with radius k. Any point lands on the surface ofthe sphere is an allowed G value, i.e. diffraction occurs atthat particular angle & k.In principle, it can map out the whole reciprocal latticeusing different k.
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(0,0)
RECIPROCAL LATTICE
(0,1)(1,1)
(2,1)(3,1)
REAL LATTICE
a1
a2
b1
b2
ExampleNote:a1·b2 = 0
a2·b1 = 0here {a} are the directlattice vectors & {b}are the reciprocallattice vectors
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(0,1) planes(0,0)
RECIPROCAL LATTICE
(0,1)(1,1)
(2,1)(3,1)
length=1/d0,1
REAL LATTICE
a1
a2
b1
b2
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(1,1) planes(0,0)
RECIPROCAL LATTICE
(0,1)(1,1)
(2,1)(3,1)
length=1/d1,1
Note:
length is longerthan (0,1) sincespacing between(1,1) planes issmaller.
REAL LATTICE
a1
a2
b1
b2
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(0,0)
(2,1) planes
RECIPROCAL LATTICE
(0,1)(1,1)
(2,1)(3,1)
length=1/d2,1
REAL LATTICE
a1
a2
b1
b2
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(0,0)
(3,1) planes
RECIPROCAL LATTICE
(0,1)(1,1)
(2,1)(3,1)
length=1/d3,1
REAL LATTICE
a1
a2
b1
b2
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(0,1) planes
(1,1) planes(0,0)
(2,1) planes
(3,1) planes
RECIPROCAL LATTICE
(0,1)(1,1)
(2,1)(3,1)
REAL LATTICE
a1
a2
b1
b2
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(0,2) planes(0,0)
RECIPROCAL LATTICE
(0,1)(1,1)
(2,1)(3,1)
length=1/d0,2
(0,2)
REAL LATTICE
a1
a2
b1
b2
with basis,for example
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(1,2) planes(0,0)
RECIPROCAL LATTICE
(0,1)(1,1)
(2,1)(3,1)
length=1/d1,2
(1,2)(0,2)
REAL LATTICE
a1
a2
b1
b2
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(2,2) planes(0,0)
RECIPROCAL LATTICE
(0,1)(1,1)
(2,1)(3,1)
length=1/d2,2
(1,2)(0,2)
(2,2)
REAL LATTICE
a1
a2
b1
b2
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(0,0)
(3,2) planes
RECIPROCAL LATTICE
(0,1)(1,1)
(2,1)(3,1)
(1,2)(0,2)
(2,2)(3,2)
length=1/d3,2
REAL LATTICE
a1
a2
b1
b2
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(0,1) planes
(1,1) planes(0,0)
(2,1) planes
(3,1) planes
RECIPROCAL LATTICE
(0,1)(1,1)
(2,1)(3,1)
(0,0)(0,0)(0,0)(0,0)
(1,2)
(2,2)(3,2)
(0,2)
(0,2) planes
(1,2) planes
(2,2) planes
(3,2) planes
REAL LATTICE
a1
a2
b1
b2
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(0,1) planes(0,0)
RECIPROCAL LATTICE
(0,1)(1,1)
(2,1)(3,1)
length=1/d0,1
How do weorient thecrystal to
observediffraction fromthe (0,1)reflection?
REAL LATTICE
a1
a2
b1
b2
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(0,0 )
(0,1) planes
nλ=2dsinθ
θ
Bragg condition-- upper beamhas to be an integral number ofwavelengths from the lower
beam forconstructive interference.
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(0,0 )
( 0, 1 ) ( 1, 1 )
( 2, 1 )
( 3, 1 )
( 0, 0 )
(0,1) planes
nλ=2dsinθ
θ 2θ
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(1,1) planes
( 0, 0 )
( 0 , 1 ) ( 1 ,
1 )
( 2 , 1 )
( 3 , 1 )
( 0 , 0 )
3D
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( 0 , 0 )
(2,1) planes
( 0 , 1 )
( 1 , 1 )
( 2 , 1 )
( 3 , 1 )
( 0 , 0 )
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Diffraction by a monatomic lattice
Bragg condition:In phase to get constructive interference (+ + + +)
Zero reflection otherwise (+ - + - + -)monatomic lattice
k k’
+
+
primitive cell
d
From here we got:
G = ∆k // d
G=2π /dBragg diffraction
Laude Equations
Construction of reciprocal lattice from EACH diffraction observedi.e. each reciprocal lattice point gives a spot on the screen
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Diffraction by a lattice with a Basis
diatomic lattice
k k’+
+
d
conventional cell
d1
+
+
For diatomic lattice (assume identical atoms)ALL in phase to get max amplitude (++ ++)
Zero amplitude if the reflection from the basis
cancel (−+ −+).Likewise, if the basis reflections are in phase,
but the lattice reflections are out-of-phase,
amplitude still cancels (++ −−)i.e. to get max Bragg scattering, both latticereflections and basis reflections should be inphase. Amplitude is less if they are partially inphase or out-of-phase.
With multiple atoms, or non-identical atoms, scattering amplitudewould not be uniform. There are partial cancellations.
The resulting reciprocal lattice of the direct lattice would be modified.
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Fourier Analysis of a Basis Scattering amplitude
Mkk’ = <k’|V(r)|k> =∫Ψ*(r) V(r) Ψ(r) dr
where Ψ(r) = eik·r
V(r) = ∑ j V(r-r j) = ∑G,j VG eiG·(r-r j)
r j is the position of the jth atom in the unit cell. Mkk’= ∑ VG ∫ ei(G + k - k’)·(r-r
j) dr
= ∑G ∑ j e-iG·r j [VG ∫ ei(G + k - k’)·r dr]
Define SG = ∑ j e-iG·r j f j the Structure Factor and f jis the Atomic Form Factor such that Mkk’∝ SG
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Now if we specify G and rj as
we get:
Note that S G can be complex, because the scatteringintensity involves the magnitude squared of S G.
The evaluation of atomic form factor is complicated but forspherically-symmetric electron density, it can be written as
(Kittel)
aaar bbbG 321332211 and j j j j z y xvvv ++=++=
( )3212
1
v zv yv xis
j jG
j j je f S ++π−
=∑=
( )∫π=∞
0
2 sin)(4 dr
Gr
Gr r r n f j j
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For a BCC lattice, r j = (000) and (½ ½ ½). The structurefactor is then: ( )
f eS
vvvi
G321
1
++π−
+=
bcc structure factor
The structure factor is maximum SG = 2f when the sumof the indices is even, i.e. v1+v2+v3=2n .
The structure factor is SG = 0 when the sum of theindices is odd, i.e. v1+v2+v3=2n+1 .
(0,0,0)
The only G show up in diffraction
pattern are the ones fit the fcc latticepoints.
Same physical measurementregardless how the cell was defined.
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Example – fcc
primitive cellfccV=a3 /4
reciprocal cell
bccV =(4π /a)3 /2
* see primitive vectors
•lattice cell sc•w/ 4-pts basis
•V=a3
(0,0,0)
•reciprocal lattice sc
•V= (2π /a)3 [8x smaller
than the cube (4π /a)3
]•extra lattice points•extra diffraction spots?
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For a FCC lattice, we have four atoms per init cell located at(000), (0 ½ ½), (½ 0 ½) and (½ ½ 0). The structure factor isthen:
fcc structure factor
(0,0,0)
The only G show up in diffractionpattern are the ones fit the bcc latticepoints.
Same physical measurement
regardless how the cell was defined.
( ) ( ) ( ) f eeeS
vvivvivvi
G
3231211+π−+π−+π− +++=
When all indices are even or odd, then SG = 4f.When the indices are partially even and partially odd, thenSG = 0.
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