Reciprocal LatticeReciprocal Lattice

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Ewald Sphere ConstructionEwald Sphere Construction

MATERIALS SCIENCEMATERIALS SCIENCE&&

ENGINEERING ENGINEERING

Anandh Subramaniam & Kantesh Balani

Materials Science and Engineering (MSE)

Indian Institute of Technology, Kanpur- 208016

Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh

AN INTRODUCTORY E-BOOKAN INTRODUCTORY E-BOOK

Part of

http://home.iitk.ac.in/~anandh/E-book.htmhttp://home.iitk.ac.in/~anandh/E-book.htm

A Learner’s GuideA Learner’s GuideA Learner’s GuideA Learner’s Guide

The concepts of reciprocal lattice (and sometimes Ewald sphere construction) often ‘strike terror’ in the hearts of students.

However, these concepts are not too difficult if the fundamentals are understood in 1D and then extended to 3D.

A crystal resides in real space. The diffraction pattern resides in Reciprocal Space.

In a diffraction experiment (powder diffraction using X-rays, selected area diffraction in a TEM), a part of this reciprocal space is usually sampled.

The diffraction pattern from a crystal (in Fraunhofer diffraction geometry), consists of a periodic array of spots (sharp peaks of intensity). (Click here to know the conditions under which this is satisfied).

From the real lattice the reciprocal lattice can be geometrically constructed. The properties of the reciprocal lattice are ‘inverse’ of the real lattice → planes ‘far away’ in the real crystal are closer to the origin in the reciprocal lattice.

As a real crystal can be thought of as decoration of a lattice with motif; a reciprocal crystal can be visualized as a Reciprocal Lattice decorated with a motif* of Intensities. Reciprocal Crystal = Reciprocal Lattice + Intensities as Motif*

The reciprocal of the ‘reciprocal lattice’ is nothing but the real lattice!

Planes in real lattice become points in reciprocal lattice and vice-versa.c

Reciprocal Lattice and Reciprocal Crystals

I.e. the information needed is the geometry of the lattice.* Clearly, this is not the crystal motif- but a motif consisting of “Intensities”.

Why study reciprocal lattices?

Often the concepts related to reciprocal lattice strikes terror in the minds of students. As we shall see this is not too difficult if concepts are first understood in 1D.

In diffraction patterns (Fraunhofer geometry & under conditions listed here: (Click here to know the conditions under which this is

satisfied).) (e.g. SAD), planes are mapped as spots (ideally points). This as you will remember is the Bragg’s viewpoint of diffraction. Hence, we would like to have a construction which maps planes in a real crystal as points.

Apart from the use in ‘diffraction studies’ we will see that it makes sense to use reciprocal lattice when we are dealing with planes.

The crystal ‘resides’ in Real Space, while the diffraction pattern ‘lives’ in Reciprocal Space.

One motivation for constructing reciprocal lattices

As the index of the plane increases → the interplanar spacing decreases → and ‘planes start to crowd’ in the real lattice (refer

figure). Hence, it is a ‘nice idea’ to work in reciprocal space (i.e. work with the reciprocal lattice), especially when dealing with planes.

As index of the plane increases, the interplanar spacing decreases

We will construct reciprocal lattices in 1D and 2D before taking up a formal definition in 3D

We will construct reciprocal lattices in 1D and 2D before taking up a formal definition in 3D

Let us start with a one dimensional lattice and construct the reciprocal lattice

Reciprocal Lattice

Real Lattice

The periodic array of points with lattice parameter ‘a’ is transformed to a reciprocal lattice with periodicity of ‘1/a’.

The reciprocal lattice point at a distance of 1/a from the origin (O), represents the whole set of points (at a, 2a, 3a, 4a,….) in real space.

The reciprocal lattice point at ‘2/a’ comes from a set of points with fractional lattice spacing a/2 (i.e. with periodicity of a/2). The lattice with periodicity of ‘a’ is a subset of this lattice with periodicity of a/2. (Refer next slide).

O

How is this reciprocal lattice constructed?

The plane (2) has intercept at ½, plane (3) has intercept at 1/3 etc. As the index of the plane increases the interplanar spacing decreases

and the first in the setgets closer to the origin (there is overall crowding).

What do these planes with fractional indices mean? We have already noted the answer in the topic on Miller indices and XRD.

One unit cell

Reciprocal Lattice

Each one of these points correspond to a set of ‘planes’ in real space

Note that in reciprocal space index has NO brackets

Real Lattice

To construct the reciprocal lattice we need not ‘go outside’ the unit cell in real space! (We already

know that all the information we need about a crystal is present within the unit cell– in conjunction with translational symmetry). Just to get a ‘feel’ for the planes we will be dealing with in the construction of 3D reciprocal

lattices, we ‘extend’ these points perpendicular to the 1D line and treat them as ‘planes’.

Note there is only one “Miller” index in 1D

Note: in 1D planes are points and have Miller indices of single digit (they have been extended into the second dimension (as lines) for better visibility and for the reason stated above).

Note that the indices in reciprocal space have no brackets

What do the various points (with indices 1, 2, 3, 4… etc.) represent in real space?Funda Check

0

‘1’ represents these set of planes in reciprocal space (interplanar spacing ‘a’)

‘2’ represents these set of planes in reciprocal space (interplanar spacing a/2)

‘4’ represents these set of planes in reciprocal space (interplanar spacing a/4)

‘3’ represents these set of planes in reciprocal space (interplanar spacing a/3)

Reciprocal Lattice

Note again: in 1D planes are points and have Miller indices of single digit (they have been extended into the second dimension (as lines) for better visibility and for the reason stated before).

‘1’ represents these set of planes in reciprocal space (interplanar spacing ‘a’)

aReal Lattice

Recipro

cal L

attice

Real Lattice

Real Lattice

Real Lattice

Real Lattice

Reciprocal Lattice

(01)

(10)(11)

(21)

10 20

11

221202

01 21

00

The reciprocal lattice has an origin!

1a

2a

1a1

1a

*11g *

21g*b2

*b1

Each one of these points correspond to a set of ‘planes’ in real space

2

1a

Reciprocal Lattice

Real Lattice

Note that vectors in reciprocal space are perpendicular to planes in real space (as constructed!)

But do not measure distances from the figure!

Overlay of real and reciprocal lattices

g vectors connect origin to reciprocal

lattice points

Now let us construct some 2D reciprocal lattices

Example-1

1a

2a

*b2

*b1

Note that vectors in reciprocal space are perpendicular to planes in real space (as constructed!)

The real lattice

The reciprocal lattice

Example-2Reciprocal Lattice

Real Lattice

But do not measure distances from the figure!

Reciprocal LatticeProperties are reciprocal to the crystal lattice

32*

1

1aa

Vb

13

*2

1aa

Vb

21

*3

1aa

Vb

B

O

P

M

A

C

B

O

P

M

A

C

O

P

M

A

C

O

P

M

A

C

O

P

M

A

C

*b3

2a

1a

3a

OPCellHeight of OAMBArea

OAMBArea

aaV

bb

1

)(

)(

121

*3

*3

001

*3

1

db

The reciprocal lattice is created by interplanar spacings

** as written usuall ii ab

B

BASIS VECTORS

21*

3 to is aandab

The basis vectors of a reciprocal lattice are defined using the basis vectors of the crystal as below

A reciprocal lattice vector is to the corresponding real lattice plane

*3

*2

*1

* blbkbhghkl

hklhklhkl d

gg1**

The length of a reciprocal lattice vector is the reciprocal of the spacing of the corresponding real lattice plane

Planes in the crystal become lattice points in the reciprocal lattice Note that this is an alternate geometrical construction of the real lattice.

Reciprocal lattice point represents the orientation and spacing of a set of planes.

Some properties of the reciprocal lattice and its relation to the real lattice

Reciprocal lattice* is the reciprocal of a primitive lattice and is purely geometrical does not deal with the intensities decorating the points.

Physics comes in from the following:

For non-primitive cells ( lattices with additional points) and for crystals having motifs ( crystal = lattice + motif) the Reciprocal lattice points have to be weighed in with the corresponding scattering power (|Fhkl|2) (Where F is the structure factor).

Some of the Reciprocal lattice points go missing (or may be scaled up or down in intensity). Making of Reciprocal Crystal: Reciprocal lattice decorated with a motif of scattering power (as intensities).

The Ewald sphere construction further can select those points which are actually observed in a diffraction experiment.

* as considered here

Going from the reciprocal lattice to diffraction spots in an experiment

A Selected Area Diffraction (SAD) pattern in a TEM is similar to a section through the reciprocal lattice (or more precisely the reciprocal crystal, wherein each reciprocal lattice point has been decorated with a certain intensity).

The reciprocal crystal has all the information about the atomic positions and the atomic species (i.e. I have to look into both the positions of the points and the intensities decorating them).

Revision+

Real Crystal

Reciprocal Lattice

Reciprocal Crystal

To summarize:

Diffraction Pattern

Purely Geometrical Construction

Decoration of the lattice with Intensities

Ewald Sphere construction

Selection of some spots/intensities from the reciprocal crystal

Structure factor calculation

Real Lattice

Decoration of the lattice with motif

In crystals based on a particular lattice the intensities of particular reflections are modified they may even go missing

Crystal = Lattice + Motif

Diffraction Pattern

Position of the diffraction spots RECIPROCAL LATTICE

Intensity of the diffraction spots MOTIF’ OF INTENSITIES

Position of the diffraction spots Lattice

Is d

eter

min

ed b

y

Intensity of the diffraction spots Motif

There are two ways of constructing the Reciprocal Crystal:

1) Construct the lattice and decorate each lattice point with appropriate intensity*

2) Use the concept as that for the real crystal**

Making of a Reciprocal Crystal

The above two approaches are equivalent for simple crystals (SC, BCC, FCC lattices decorated with monoatomic motifs), but for ordered crystals the two approaches are

different (E.g. ordered CuZn, Ordered Ni3Al etc.) (as shown soon).

* Point #1 has been considered to be consistent with literature– though this might be an inappropriate.

** Point #2 makes reciprocal crystals equivalent in definition to real crystals

Real Crystal

Reciprocal Lattice

Reciprocal Crystal

Real Lattice

Take real lattice and construct reciprocal lattice

Use motif to compute structure factor and hence

intensities to decorate reciprocal lattice points

Decorate with motif

Examples of 3D Reciprocal Lattices weighed in with scattering power (|F|2)

Figures NOT to Scale

000

100

111

001

101

011

010

110

SC

Lattice = SC

Reciprocal Crystal = SC

Selection rule: All (hkl) allowedIn ‘simple’ cubic crystals there are No missing

reflections

SC lattice with Intensities as the motif at each ‘reciprocal’ lattice point

+

Single sphere motif

=

SC crystal

Figures NOT to Scale

000

200

222

002

101

022

020

110

BCC

BCC crystal

Reciprocal Crystal = FCC

220

011

202

100 missing reflection (F = 0)

22 4 fF

Weighing factor for each point “motif”

FCC lattice with Intensities as the motif

Selection rule BCC: (h+k+l) even allowedIn BCC 100, 111, 210, etc. go missing

Important note: The 100, 111, 210, etc. points in the

reciprocal lattice exist (as the corresponding real lattice planes exist), however the intensity decorating these points is zero.

x

x

x

x

x

x

x

x

x

x

x

x

Figures NOT to Scale

000200

222

002022

020

FCC

Lattice = FCC

Reciprocal Crystal = BCC

220

111

202

100 missing reflection (F = 0)110 missing reflection (F = 0)

22 16 fF

Weighing factor for each point “motif”

BCC lattice with Intensities as the motif

When a disordered structure becomes an ordered structure (at lower temperature), the symmetry of the structure is lowered and certain superlattice spots appear in the Reciprocal Lattice/crystal (and correspondingly in the appropriate diffraction patterns). Superlattice spots are weaker in intensity than the spots in the disordered structure.

An example of an order-disorder transformation is in the Cu-Zn system: the high temperature structure can be referred to the BCC lattice and the low temperature structure to the SC lattice (as shown next). Another examples are as below.

Order-disorder transformation and its effect on diffraction pattern

Click here to know more about Ordered StructuresClick here to know more about Ordered StructuresClick here to know more about Superlattices & SublatticesClick here to know more about Superlattices & Sublattices

Disordered Ordered

- NiAl, BCC B2 (CsCl type)

- Ni3Al, FCC L12 (AuCu3-I type)

Positional Order

G = H TS

High T disordered

Low T ordered

470ºC

Sublattice-1 (SL-1)

Sublattice-2 (SL-2)

BCC

SC

SL-1 occupied by Cu and SL-2 occupied by Zn. Origin of SL-2 at (½, ½, ½)

In a strict sense this is not a crystal !!

Probabilistic occupation of each BCC lattice site: 50% by Cu, 50% by Zn

Diagrams not to scale

Disordered Ordered

- NiAl, BCC B2 (CsCl type)

BCC SC

FCC

Ordered

Reciprocal crystal

Reciprocal Crystal = ‘FCC’FCC lattice with Intensities as the motif

‘Diffraction pattern’ from the ordered structure (3D)

This is like the NaCl structure in Reciprocal Space!

Click here to see structure factor calculation for NiAl (to see why some spots have weak intensity) → Slide 27

Click here to see XRD powder pattern of NiAl → Slide 5

For the ordered structure:

Notes: For the disordered structure (BCC) the reciprocal crystal is FCC. For the ordered structure the reciprocal crystal is still FCC but with

a two intensity motif: ‘Strong’ reflection at (0,0,0) and superlattice (weak) reflection at (½,0,0) .

So we cannot ‘blindly’ say that if lattice is SC then reciprocal lattice is also SC.

BCCFCC

SC

Ordered

Disordered Ordered

- Ni3Al, FCC L12 (AuCu3-I type)

Reciprocal crystal

Reciprocal Crystal = BCCBCC lattice with Intensities as the motif

Diffraction pattern from the ordered structure (3D)

Click here to see structure factor calculation for Ni3Al (to see why some spots have weak intensity) → Slide 29

Click here to see XRD powder pattern of Ni3Al → Slide 6

There are two ways of constructing the Reciprocal Crystal:

1) Construct the lattice and decorate each lattice point with appropriate intensity

2) Use the concept as that for the real crystal (lattice + Motif)

1) SC + two kinds of Intensities decorating the lattice

2) (FCC) + (Motif = 1FR + 1SLR)

1) SC + two kinds of Intensities decorating the lattice

2) (BCC) + (Motif = 1FR + 3SLR)

FR Fundamental Reflection SLR Superlattice Reflection

Motif

Motif

SAD patterns from a BCC phase (a = 10.7 Å) in as-cast Mg4Zn94Y2 alloy showing important zones

[111] [011]

[112]

The spots are ~periodically arranged

Example of superlattice spots in a TEM diffraction pattern

Superlattice spots

Example of superlattice peaks in XRD pattern

NiAl pattern from 0-160 (2)

Superlattice reflections (weak)

The Ewald Sphere

* Paul Peter Ewald (German physicist and crystallographer; 1888-1985)

Reciprocal lattice/crystal is a map of the crystal in reciprocal space → but it does not tell us which spots/reflections would be observed in an actual experiment.

The Ewald sphere construction selects those points which are actually observed in a diffraction experiment

organisiert von:Max-Planck-Institut für MetallforschungInstitut für Theoretische und Angewandte Physik,Institut für Metallkunde,Institut für Nichtmetallische Anorganische Materialiender Universität Stuttgart Programm

13:30 Joachim Spatz (Max-Planck-Institut für Metallforschung) Begrüßung

13:45 Heribert Knorr (Ministerium für Wissenschaft, Forschung und Kunst Baden-Württemberg Begrüßung

14:00 Stefan Hell (Max-Planck-Institut für Biophysikalische Chemie)Nano-Auflösung mit fokussiertem Licht

14:30 Antoni Tomsia (Lawrence Berkeley National Laboratory)Using Ice to Mimic Nacre: From Structural Materials to Artificial Bone

15:00 Pause Kaffee und Getränke

15:30 Frank Gießelmann(Universität Stuttgart) Von ferroelektrischen Fluiden zu geordneten Dispersionen von Nanoröhren: Aktuelle Themen der Flüssigkristallforschung

16:00 Verleihung des Günter-Petzow-Preises 2008

16:15 Udo Welzel (Max-Planck-Institut für Metallforschung) Materialien unter Spannung: Ursachen, Messung und Auswirkungen- Freund und Feind

ab 17:00 Sommerfest des Max-Planck-Instituts für Metallforschung

7. Paul-Peter-Ewald-Kolloquium

Freitag, 17. Juli 2008

Circular of a Colloquium held at Max-Planck-Institut für Metallforschung (in honour of Prof.Ewald)

The Ewald Sphere The reciprocal lattice points are the values of momentum transfer for which the

Bragg’s equation is satisfied. For diffraction to occur the scattering vector must be equal to a reciprocal lattice

vector. Geometrically if the origin of reciprocal space is placed at the tip of ki then

diffraction will occur only for those reciprocal lattice points that lie on the surface of the Ewald sphere.

Here, for illustration, we consider a 2D section thought the Ewald Sphere (the ‘Ewald Circle’)

See Cullity’s book: A15-4

hklhkl Sindn 2

2

12 hkl

hklhkl

d

dSin

Draw a circle with diameter 2/ Construct a triangle with the diameter

as the hypotenuse and 1/dhkl as a side (any triangle inscribed in a circle with the diameter as the hypotenuse is a right angle triangle: APO = 90): AOP

The angle opposite the 1/d side is hkl (from the rewritten Bragg’s equation)

Bragg’s equation revisited

RewriteThis is Bragg’s equation

in reciprocal spaceThis is Bragg’s equation

in reciprocal space

hklhklhkl d

gg1**

Now if we overlay ‘real space’ information on the Ewald Sphere. (i.e. we are going to ‘mix-up’ real and reciprocal space information).

Assume the incident ray along AC and the diffracted ray along CP. Then automatically the crystal will have to be considered to be located at C with an orientation such that the dhkl planes bisect the angle OCP (OCP = 2).

OP becomes the reciprocal space vector ghkl (often reciprocal space vectors are written without the ‘*’).

2

12 hkl

hklhkl

d

dSin

Radiation related information is present in the Ewald Sphere

Crystal related information is present in the reciprocal crystal

The Ewald sphere construction generates the diffraction pattern

The Ewald Sphere construction

Which leads to spheres for various hkl reflections

Chooses part of the reciprocal crystal which is observed in an experiment

01

10

02

00 20

2

(41)

Ki

KD

DK

Reciprocal Space

DK = K =g = Diffraction Vector

Ewald Sphere

The Ewald Sphere touches the reciprocal lattice (for point 41)

Bragg’s equation is satisfied for 41

When the Ewald Sphere (shown as circle in 2D below) touches the reciprocal lattice point that reflection is observed in an experiment (41 reflection in the figure below).

41

(Cu K) = 1.54 Å, 1/ = 0.65 Å−1 (2/ = 1.3 Å−1), aAl = 4.05 Å, d111 = 2.34 Å, 1/d111 = 0.43 Å−1

Ewald sphere X-rays

Row of reciprocal lattice pointsRows of reciprocal lattice points

Diffraction from Al using Cu K radiation

The 111 reflection is observed at a smaller angle 111 as compared to the 222 reflection

(Cu K) = 1.54 Å, 1/ = 0.65 Å−1 (2/ = 1.3 Å−1), aAl = 4.05 Å, d111 = 2.34 Å, 1/d111 = 0.43 Å−1

Ewald sphere X-raysNow consider Ewald sphere construction for two different

crystals of the same phase in a polycrystal/powder (considered next).

Click to compare themClick to compare them

Diffraction cones and the Diffractometer geometryPOWDER METHOD

In the powder method is fixed but is variable (the sample consists of crystallites in various orientations).

A cone of ‘diffraction beams’ are produced from each set of planes (e.g. (111), (120) etc.)(As to how these cones arise is shown in an upcoming slide).

The diffractometer moving in an arc can intersect these cones and give rise to peaks in a ‘powder diffraction pattern’.

Click here for more details regarding the powder methodClick here for more details regarding the powder method

Different cones for different reflections

Diffractometer moves in a semi-circle to capture the intensity of the diffracted beams

‘3D’ view of the ‘diffraction cones’

Cone of diffracted rays

THE POWDER METHOD

In a power sample the point P can lie on a sphere centered around O due all possible orientations of the crystals

The distance PO = 1/dhkl

Understanding the formation of the cones

The 440 reflection is not observed(as the Ewald sphere does not intersect the reciprocal lattice point sphere)

Circular Section through the spheres made by the hkl reflections

Ewald sphere construction for AlAllowed reflections are those for h, k and l unmixed

The 331 reflection is not observed

Ewald sphere construction for CuAllowed reflections are those for h, k and l unmixed