Reciprocal Lattice (1)

7/27/2019 Reciprocal Lattice (1)

1/31

Alternative description, not atomic positions but lattice

planes: the reciprocal lattice

Miller indexes


2/31

Crystal projectionsThe stereogram


3/31

The Reciprocal lattice

A family of crystal planes (hkl) is characterized by:

a) The normal to the planesb) The interplanar spacing dhkl

A more practical way to characterize the crystal planes is to

define the reciprocal lattice formed by the vectors

hklhklhkl /d2G n ClBkAhGhkl

cba

cb2A

B and C are given by cyclic permutation

With the property:


4/31

Thus G is perpendicular to twovectors spanning a plane i.e. it

is perpendicular to the plane

Each G

coincides

with a setof crystal

planes

only every third (300) plane

goes through lattice points


5/31

http://www.doitpoms.ac.uk/tlplib/miller_indices/lattice.php

For orthogonal axis:222

2

1

a

l

a

k

a

h

dhkl

See also:


6/31

Determination of lattice parameters by diffraction

Nowadays crystal structures can

be observed by microscopy but aprecise determination of lattice

constants is still based on

diffraction methods

To this scope one has several

probe particles at disposition

which differ with respect to the

energy dependence of wavelength

and of penetration depth inside

the solid.

De Broglie wavelength: =h/mv

m mass of the particle

v velocity

h Planck constant


7/31

The reciprocal lattice has the dimension 1/[L], like the wavevector k

and is therefore related to electron momentum, velocity and

energy. The concept is needed to describe therefore the properties

of valence electrons in the crystal.

The electron wavefunction is thereby assumed as a plane wave:

The Reciprocal lattice

t)rki(

A(r)

ewith k, wavevector related to the particle momentum

//2 pk

The reciprocal space defines the acceptable k values for the valence

electrons in the solid and provides thus its description with respect to

electron velocities and energies. It plays also a pivotal role to describe

probe particle diffraction from crystal lattices


8/31


)''(

'

trki

A

Aer

f

))'(''( kktrki

B

er

f

At the detector, placed at r

the wave scattered from

lattice site A

The wave scattered

from lattice site B

has a phase delay (is

retarded)

Wave scattered

from all atoms of

the solid (difference

between different rj

is negligible

j

)''( ktrki

j

jer

f

1-P1-N1M

0

)(

j

kcpbnami

j

e

r

f

amplitude of incident wave, f atomic form factor

Solid consisting of M ,N and P atoms along x, y and z, respectively


9/31

)(

)(

1

1/2kai/2kai-/2kai

/2kaMi/2kaM-i/2kaMi

kai

kaMi1M

0

kami

eee

eee

e

ee

m

ka2

1sin

kaM2

1sin

Since we measure intensities we do not

care about the phase factor and find:

0ka21sin

which has maxima whenever:

i.e. for Ank1

with n1 integer andcba

cb2A

There are M-2 subsidiary maxima between adjacent principalmaxima due to the numerator, which become insignificant for

large M.CBAklkhGhkl In general:

Whereby each diffracted beam is associated to one particular

family of lattice planes. The ensemble of G is the reciprocal lattice


10/31

First order diffraction

Second and higher order

diffractions correspond to

Miller planes with

reducible indexes (e.g.

(300) instead of (100)


11/31


Ewald construction

Only events corresponding

to vectors on the Ewald

sphere satisfy energy and

momentum conservation


12/31

The other factor in the scattering formula is the form factor f

which convolutes the scattered amplitudes .

For a Bravais lattice with more than one kind of atoms we

have to consider the scattering power of the basis rather than

from the isolated atom

hkl

j

j

j

j Sefef hkljjj Giki

structure factor

Atom factor for

X ray scattering

Angular dependence for

X ray scattering arises

from dephasing of the

different trajectories for

angles different from zero


13/31

Simple cubic lattice

(one atom per unit cell)


14/31

Body centered cubic lattice (2 atoms per unit cell)

The (100) beam, as well as the (111) beam would thus be absent in the

diffraction pattern


15/31

Simple cubic structure with two atom basis


16/31


17/31

Reciprocal lattice of bcc structure

)(2

1

1zyxaKi

KeS

)(2

321 znynxn

a

K

)()( 321321 )1(11 nnnnnniK

eS

2 n1+n2+n3 even0 n1+n2+n3 odd

))(2/(

0

2

1

zyxad

d

face centered cube


18/31

)(2

1 xzya

b

)(22 yxza

b

)(2

3 zyxa

b

)(2

1

3211 nnnik eS

2 n1+n2+n3 twice an even number

1i n1+n2+n3 odd0 n1+n2+n3 twice an odd number

Reciprocal lattice of the Diamond structure

reciprocal lattice :

bcc with two lattice points per cell


19/31

Diamond structure d1=0d2= a/4 (x+y+z)

Reciprocal lattice:

Body Centered Cube

of 4/a


20/31

Powder diffraction method


21/31

Laue method

Using white X-rays we fill the Ewald sphere

Ewald construction

Forward scattering

Elliptical shape

Backscattering

Hyperbolic shape


22/31

Ewald construction for the

rotating crystal method


23/31

Diffraction from non periodic structures: amorphous solids, liquids

)()( * kSkS ji

Structure factor:

N

i

rki

i

i

ekS 1)(

N

j

rki

jjekS

1

*)(

Radial distribution function

(isotropic distribution liquid):)(4 2

2 RgR

g2(R) Correlation

between 2 atoms

g2(R) and S are related by Fourier Transform


24/31

In 1984 quasicrystals are predicted by Penrose

and observed for quenched Al6Mn crystals (105 K/sec)

It is impossible to tile a 2D space with pentagons as well as

with decagons alone

Impossible symmetries: 5 fold axisr1 works for the decagon

but not for the pentagon

However, r2doesnt work

either since shifting it to thecenter of the cell one can

see that it does not reach

the border of the cell

S l i b P diff h bi i h i i l


25/31

Solution by Penrose: use two different rhombi with an irrational

ratio in the areas (1,618 golden mean)


26/31

Explanation of the

diffraction events

in quasicrystals

Lines containing bonds inone given direction are

evenly spaced

Electron diffraction

pattern of an icosahedral

Ho-Mg-Zn quasicrystal

Dan Schlechtmann,

Chemistry Nobel Prize

Winner 2011
http://en.wikipedia.org/wiki/Electron_crystallographyhttp://en.wikipedia.org/wiki/Electron_crystallographyhttp://en.wikipedia.org/wiki/Electron_crystallographyhttp://en.wikipedia.org/wiki/Electron_crystallography


27/31

3D quasicrystal construction rules are

more complicated but follow similar

reasoning

Ho-Mg-Zn icosahedral quasicrystal


28/31

An important area of application is the use of quasicrystals as materials for

surface coatings, which benefit from the hardness of quasicrystals. The

most prominent example is the use of quasicrystalline coatings in frying

pans - an application famous in the quasicrystal community as it has served

as a key example. Recently, quasicrystal-coated frying pans appeared onthe market, and are sold by the French company Sitram under the

trademark Cybernox.

Due to their particular physical and chemical properties, quasicrystalline

coatings are suited for this kind of application. They are also rather cheap

which makes them even more interesting for industrial applications.Other application concern hard tool, like blades coated with quasicrystals.

A third, and maybe more speculative, application concerns the use of quasicrystals as a

reversible storage medium for hydrogen. The most promising quasicrystal materials for

hydrogen storage are Zr-based quasicrystals.

Copied from the web page of Technion Institute
http://www.inductionsystems.com/cookware/cybernox.htm.http://www.inductionsystems.com/cookware/cybernox.htm.


29/31

5 fold symmetry axis are possible also for cluster

of atoms as long as they are not too big, i.e. < 3

nm diameter or


30/31

Unit cell of the reciprocal lattice:

3D Brillouin Zone

Special pointsfcc

bcc

hcp


31/31

Reciprocal Space and Quasiparticles (electrons, phonons and

magnons)

Electrons and Excitations have a wave nature which is

described by a wavevector k.

In a crystal k is limited to the first Brillouin zone.

Documents

Reciprocal Lattice (1)