Handout 6 Electrons in Periodic Potentials The Reciprocal Lattice ...
Chapter 2 Wave diffraction and the reciprocal lattice
Transcript of Chapter 2 Wave diffraction and the reciprocal lattice
Chapter 2 Wave diffraction and the
reciprocal lattice
1
How Can We Study Crystal Structure?
• The first Neutron diffraction experiment carried out in 1945
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Need probes that can penetrate into crystal
• X-ray: discovered by Roentgen in 1895
Laue condition in 1912 and Bragg Law in 1913
• Electron diffraction: 1927 by Davisson and Germer
Low Energy Electron Diffraction (LEED)
year Nobel Laureate Field Citation for
1914 Max von Laue Physics The discovery of the diffraction of x-ray by
crystals
1915 William Henry Bragg
and William Lawrence
Bragg
Physics The analysis of crystal structure by means of x-
ray
1937 Clinton Joseph
Davisson and George
Paget Thomson
Physics The experimental discovery of the diffraction of
electrons by crystals
1985 Herbert A. Hauptman
and Jerome Karle
Chemistry The outstanding achievements in the
development of direct methods for the
determination of crystal structures
1994 Bertram N.
Brockhouse and
Clifford G. Shull
Physics The development of neutron scattering and the
neutron diffraction techniques
year Nobel Laureate Field Citation for
1936 Peter Debye Physics The contributions to our knowledge of molecular structure
through his investigations on dipole moments and on the
diffraction of X-ray and electrons in gases
1962 Max F. Perutz and
John C. Kendrew
Chemistry The studies of the structures of globular proteins
1962 Francis Crick and
James Watson
Medicine The discoveries concerning the molecular structure of
nucleic acids and its significance for information transfer in
living material (DNA)
1964 Dorothy Crowfoot
Hodgkin
Chemistry The determinations by X-ray techniques of the structures of
important biochemical substances
1976 William Lipscomb Chemistry The studies on the structure of boranes illuminating
problems of chemical bonding
1982 Aaron Klug Chemistry The development of crystallographic electron microscopy
and his structural elucidation of biologically important
nucleic acid-protein complexes
1988 Johann Deisenhofer
and Robert Huber
Chemistry The determination of the three-dimensional structure of a
photosynthetic reaction centre
2009 Venkatraman
Ramakrishnan,
Thomas A. Steitz, and
Ada E. Yonath
Chemistry The studies of the structure and function of the ribosome
How Can We Study Crystal Structure?
• X-rays scatter from the electrons – intensity proportional to the
density n(r) – mainly the core electrons around the nucleus.
• Similarly for high energy electrons.
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• Neutrons scatter from the nuclei (and electron magnetic
moment) and can penetrate with almost no interaction with
most materials
• In all cases the scattering is periodic – that is it is the same in
each cell of the crystal.
• Diffraction is the constructive interference of the scattering
from the very large number of cells of the crystal.
The crystal can be viewed as being made up of different sets of planes
• Different sets of parallel planes
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Modern Physics for Scientists and Engineers by Thornton and Rex (2013).
Bragg Law
• Condition for constructive interference:
2d sin q = n l
• Maximum l = 2d
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Modern Physics for Scientists and Engineers by Thornton and Rex (2013).
• Only waves with l close to the atomic spacing can have Bragg scattering from a crystal.
Example of scattering
• Al is fcc with a = 0.405 nm.
• What is the minimum energy of the
x-ray that satisfies the Bragg condition?
8Solid-State Physics by Ibach and Lueth (2009)
• Higher energies are needed for all other planes. (d is smaller.)
– Maximum l is 2d = 0.468 nm
– Using E = hn = hc/l (hc = 1.240 x 103 nm), the minimum
energy of the x-ray for Bragg scattering is 2.65 keV.
– Largest distance between planes is for 111 planes:3
3
ad =
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Single crystal X-ray diffraction
The intensity of the diffracted beam is determined as a function of
scattering angle by rotating the crystal and the detector.
Modern Physics for Scientists and Engineers by Thornton and Rex (2013).
Fermi-Golden Rule approach
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'
2
' '2( , ) ( )
kkk k k V k E E
= −
transition rate per time
'
' 3
3 3( )
i k r i k re e
k V k d r V rL L
−
=
( ) ( )V x R V x+ ='
' '
( ) 3
3
( ) ( ) 3
3
1( )
1( )
i k k r
i k k R i k k r
R unitcell
e V r d rL
e e V r d rL
− −
− − − −
=
=
k
incident
scatteredk’
k
unscattered
sample
Fermi-Golden Rule approach
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k
incident
scatteredk’
k
unscattered
sample
' '' ( ) ( ) 3
3
1( )
i k k R i k k r
R unitcell
k V k e e V r d rL
− − − − =
'3
( ) 3 '(2 )(( ) )
i k k R
R G
e k k GV
− − = − −
reciprocal lattice
'k k G− = Laue condition
Reciprocal lattice and
Fourier analysis in 1-d
( ) ( ),
:integers.
n
n r r an
n
= −
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a
[ ( )] ( )ikr
F n r e n r dr=
ikan
n
e=
( )ikr
n
e r an dr= −
n=30
2
ikan
n
e
Condensed Matter Physics by Marder(2000)
Reciprocal lattice and
Fourier analysis in 1-d
• The set of all (p·b) is the reciprocal lattice.
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• In 1-d, b = 2/a.
b
a
[ ( )] ( ) ( )ikr ikr
n
F n r e n r dr e r an dr= = − 2 2
( )ikan
n m
me k
a a
= = −
Reciprocal Lattice: definition
• Consider a set of Bravais lattice points represented by and a
plane wave of wave number satisfying
the set of points is the reciprocal lattice of this
Bravais lattice.
1 1 2 2 3 3T n a n a n a= + +
for any , r( )iG r T iG re e
+ =
iG re
G
G
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2 31
1 2 3
3 12
1 2 3
1 23
1 2 3
2( )
2( )
2(
)
a ab
a a a
a ab
a a a
a ab
a a a
=
=
=
• If {ai} are primitive vectors of the crystal lattice, then
{bi} are primitive vectors of the reciprocal lattice.
• Unit of bi : [1/L].
2i j ij
a b =
Reciprocal lattice vectors
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• The reciprocal of the reciprocal lattice is the real lattice itself.
Reciprocal lattice vectors - continuous
1 1 2 2 3 3
1 1 2 2 3 3
1 1 2 2 3 3
2 ( )
2 ( )
1i n k n k n kiG T
G k b k b k b
G T n k n k n k
e e
+ +
= + +
= + +
= =
: lattice vectorT
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(r) (r )n n T= +
( )( ) ( )
i G r G TiG r T iG r
G G GG G G
n r T n e n e n e n r − + − + − + = = = =
• The only information about the actual basis of atoms is in the
quantitative values of the Fourier components nG in the Fourier
analysis
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cell
1( )
iG r
G
G
n d r n r eV
=
Three Dimensional Lattices
Simplest examples
• Long lengths in real space imply short lengths in reciprocal space and vice versa.
Simple Orthorhombic Bravais
Lattice with a3 > a2 > a1
Reciprocal Lattice
Note: b1 > b2 > b3
a
1
a1
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Three Dimensional Lattices Simplest examples
• Reciprocal lattices is also hexagonal, but rotated.
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Reciprocal lattices Hexagonal Bravais Lattices
Brillouin Zone
Brillouin Zone : Winger-Seitz Cell
for Reciprocal Lattice
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Each Brillouin zone has
exactly the same total area,
since there is a one-to one
mapping of points in each
Brillouin zone to the first
one.The Oxford Solid State Basics by S. H. Simon (2017).
The first Brillouin zone is
connected, while higher
Brillouin zones typically
are made of disconnected.
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Any point from 0 without
crossing a perpendicular
bisector is in the first
Brillouin zone. If one
crosses only one bisector,
it is in the second
Brillouin zone.
The Oxford Solid State Basics by S. H. Simon (2017).
The boundaries of the
Brillouin zone are in
parallel pairs symmetric
around the central point
and are separated by a
reciprocal lattice vector.
Brillouin Zone
Primitive vectors and the
conventional cell of bcc lattice
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Introduction to Solid State Physics by Kittel (2005)
Reciprocal lattice is Body
Centered Cubic
• Note if the conventional cell is bcc,
then the corresponding reciprocal
lattice is fcc.
Body Centered Cubic
Winger-Seitz Cell for Body
Centered Cubic Lattice
Brillouin Zone = Winger-Seitz
Cell for Reciprocal Lattice
1
2
3
( )2
( )2
( )2
aa y z x
aa x y z
aa x y z
= + −
= − +
= + −
1
2
3
3
3
(2 )1st
2( )
2( ),
2
( )
2
,
BZ/
b y za
ab x z
a
b x ya
= +
= +
=
=
+
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1 2 3
3
2
) (V a a a
a
=
=
Face Centered Cubic
Winger-Seitz Cell for Face
Centered Cubic Lattice
Brillouin Zone = Winger-Seitz
Cell for Reciprocal Lattice
1
2 1 2 3
3
3
( )2
( ),2
( ),2
V ( )
4
a
aa y z
aa x z
a
a
a
a
ya
x
= +
= +
= +
=
=
1
2
3
3
3
(2 )1st BZ
2( )
2( ),
2
/
( ),
4
b y z xa
b x y za
b x y za
a
= + −
= − +
= + −
=
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Ewald construction
Procedure:
• Draw the reciprocal lattice,
• Draw kin with arrow head coinciding with a lattice point,
• Draw a sphere of radius |k| centered at the other end of k,
• Any reciprocal lattice point intercepted by the sphere satisfies the
diffraction condition.
Laue condition:
' + k k G=
Solid State Physics by Schmool (2017)
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Laue and Bragg Conditions
• From last slide,
since G 2 = |G|2 :
2 sinG k q=
Solid State Physics by Schmool (2017)
2 4sin
n
d
q
l=
• But |k| = 2π/λ, and |G| = n(2π/d), where
d = spacing between planes
Bragg condition 2d sinq = n λ
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Laue and Bragg Conditions
• The Laue condition and the Bragg condition are equivalent.
⚫ It is equivalent to say that interference is constructive (asBragg indicates) or to say that crystal momentum isconserved (as Laue indicates).
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Structure factor
• Electron density
( ')
''cell cell
( ) iG r i G G r
GG
N dV e n r n N dV e− − − =
3
' 3
3
(2 )( ) ( )
iG r
unitcell
Nk n r k e n r d r
V L
− =
( ')
''
'
''
( )
( )
iG r i G
iG r
G
r
G
G
GGV V
dV e n r n dV e
n r n e
− −
−
=
=
periodicperiodic
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( ') cell
cell
, '
0, '
i G G r V G GdV e
G G
− − =
=
cell
cell
( ),iG r
G GS V n dV e n r
− =
The structure factor SG is defined as
determined by the charge distribution in a unit cell.
'
2
' '2( , ) ( ) ( )
kkk k k n r k E E
−
• The intensity of an x-ray reflection |SG|2
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Lattice with basis
: within the cell
( ) ( )j j
j
n r n r r= −
jr r −
cell
cell
( )
( )j
iG r
j j
j
iG r iG
j
G
j
S
e
dV n r r e
dV n e
−
− −
= −
=
cell
( )iG
jj dV n ef −
fj : atomic form factorjiG r
j
j
f e−
=
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Example: bcc Bravais lattice 1/2
• fcc in reciprocal lattice
• Simple cubic with basis at (0,0,0) and a/2 (1,1,1)
( )
n 1
0 odd
2 ev
e if
i h k l
ff e h k l
− + + = + = + + =
f0 = f1, for identical atoms
1 2 3j j j jr x a y a z a= + +
2 ( )j j j j
G r hx ky lz = + +
1 2 3G hb kb lb= + +
( )
0 1
jiG r i h k l
j
j
S f e f f e− − + += = +
•Structure factor:
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Example: bcc Bravais lattice 2/2
• The diffraction pattern does not contain reflections: (100), (300),
(111), (221) …
but with reflections: (110), (200), (222), …
There are additional planes of atoms half-way between the (100)
planes which then cause perfect destructive interference.
Introduction to Solid State Physics by Kittel (2005)
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Example: fcc Bravais lattice
• Cubic cell with atoms at:
(0,0,0), a(0,½ , ½ ), a(½ ,0, ½ ), and a(½ , ½ ,0)
2 ( )j j j j
G r hx ky lz = + +
( ) ( ) ( )1
4 h, k, an
d l are all odd or even
0 otherwise
i k l i h l i h kS f e e e
f
− + − + − + = + + +
=
• Structure factor:
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a face-centered cubic
K: 000; ½ ½ 0; ½ 0 ½ ; 0 ½ ½
Cl: ½ ½ ½ ; 0 0 ½ ; 0 ½ 0; ½ 00
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Equivalent to monatomic sc
lattice of lattice constant a/2
(331), (311), (111) missing.
Scattering amplitude
(K )f+ -
(Cl )f~
Introduction to Solid State Physics by Kittel (2005)
Both KCl and KBr have an fcc lattice
differs significantly
from .
(Br )f−
(K )f+
All reflections of the fcc
lattice are present.
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X-ray powder diffraction of PrO2
λ=0.123 nm
The Oxford Solid State Basics by Steven H. Simon (2013).
36
X-ray powder diffraction of PrO2
2sin
dl
q=
2θ
a 22.7°
b 26.3°
c 37.7°
d 44.3°
e 46.2°
f 54.2°
2 2 2N h k l= + +
2 2 2d*a h k l= + +
d
0.313 nm
0.270 nm
0.190 nm
0.163 nm
0.157 nm
0.135 nm
a
0.542 nm
0.540 nm
0.537 nm
0.541 nm
0.544 nm
0.540 nm
3da2/d2 N
3 3
3.99 4
8.07 8
11.01 11
11.91 12
16.05 16
{hkl}
111
200
220
311
222
400
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Powder x-ray diffraction: LaCu3Fe4O12
Nature 458, 60 (2009)
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Powder x-ray diffraction: LaCu3Fe4O12
Nature 458, 60 (2009)
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Powder x-ray diffraction: example
New J. of Phys. 12 (2010) 063029
40
From the talk of Christian Grünzweig,
Paul Scherrer Institut summer school 2014.
41
From the talk of Christian Grünzweig,
Paul Scherrer Institut summer school 2014.
42
From the talk of Christian Grünzweig,
Paul Scherrer Institut summer school 2014.
43
From the talk of Christian Grünzweig,
Paul Scherrer Institut summer school 2014.
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X-ray vs Neutron
X-ray Neutron
interact electron nuclei
scattering intensities higher lower
scatter strength atomic number erratically
form factor reciprocal lattice
vector
nuclear scattering-
length
spin no yes
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Neutron
Energy (meV) Temperature (K) Wavelength (nm)
Cold 1-10 20-120 0.3-0.7
Thermal 10-100 120-1000 0.1-0.3
Hot 100-500 1000-6000 0.04-0.1
46Introduction to Solid State Physics by Kittel (2005)
47
BaFe2As2
PRL 101, 257003 (2008)
48
BaFe2As2
PRL 101, 257003 (2008)