chapter 2 wave diffraction and the reciprocal lattice

$: Chapter 2 Wave diffraction and the reciprocal lattice$
Chapter 2 Wave diffraction and the

reciprocal lattice

1

How Can We Study Crystal Structure?

• The first Neutron diffraction experiment carried out in 1945

2

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.

5

• 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

6

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

7


• 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 =

9

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.


Fermi-Golden Rule approach

10

'

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

11

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

= −

12

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.

13

• 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

14

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

15

• 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

16

(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

3

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

17

Three Dimensional Lattices Simplest examples

• Reciprocal lattices is also hexagonal, but rotated.

18

Reciprocal lattices Hexagonal Bravais Lattices

Brillouin Zone

Brillouin Zone : Winger-Seitz Cell

for Reciprocal Lattice

19

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.

20

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

21

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

= +

= +

=

=

+

22

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

= + −

= − +

= + −

=

23

24

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)

25

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

26

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).

27

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

28

( ') 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

29

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−

=

30

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:

31

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.


32

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:

33

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

34

Equivalent to monatomic sc

lattice of lattice constant a/2

(331), (311), (111) missing.

Scattering amplitude

(K )f+ -

(Cl )f~


Both KCl and KBr have an fcc lattice

differs significantly

from .

(Br )f−

(K )f+

All reflections of the fcc

lattice are present.

35

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

37

Powder x-ray diffraction: LaCu3Fe4O12

Nature 458, 60 (2009)

38

Powder x-ray diffraction: LaCu3Fe4O12

Nature 458, 60 (2009)

39

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



42



43



44

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

45

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)

chapter 2 wave diffraction and the reciprocal lattice

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