386

Analytical Methods for Materials

Lesson 15Reciprocal Lattices and Their Roles in Diffraction Studies

Suggested ReadingChs. 2 and 6 in Tilley, Crystals and Crystal Structures, Wiley (2006)

Ch. 6 – M. DeGraef and M.E. McHenry, Structure of Materials, Cambridge (2007). Appendix A from Cullity & Stock

387

Corresponding to any crystal lattice in ‘real’ space is a reciprocal lattice

388

Our interest stems from the fact that many properties are reciprocal to those of the crystal lattice.

Do you remember what I said on Day 1 of the class:

crystals diffraction is inverse to lattice spacings!

389Things can look very different in reciprocal space than in real space.

Fun in reciprocal space, © The New Yorker Collection, 1991. John O’Brien, from http://www.cartoonbank.com. All rights reserved.

390Things can look very different in reciprocal space than in real space.

391

Reciprocal Lattice

• Provides a vector representation of crystal directions and spacing between diffracting planes.

• Real Space: (lattice parameters define lattice)– a, b, c, , ,

• Reciprocal Space: (just another type of lattice)– a*, b*, c*, *, *, *

0,0,2

0,2,0

0,2,2

1,1,1

2,0,0

2,0,2

2,2,0

2,2,2

a* b*

c*

0,0,0

Real Space Reciprocal Space

FCC Crystal

393

Recall some simple vector operations

• Dot product (scalar product):

• Cross product (vector product):

• Other useful relations (and there are more…)

is the direction planeC BA

A

B

C

2 2 2h k l 2 2 2h k l

cosB BA A

sinA AB C B

( )B B

BA C

C C CA

A A AB B

394

0

*

*

01

area of baseunit cell volume

1height1

a bc

c a b

cd

Reciprocal Vectors

b

a

c

O

B

AC

c*

d001 in real

space

x

yz

unit cell volume c c cV a b b a a b

395

Thus, we can define entire reciprocal lattices

* plane in real space; it is the reciprocal lattice vector for * plane in real space; it is the reciprocal lattice vector for

plane in real space; it is the reciprocal lattice vect r f* o

a b ab a b

ab

cc

c

or c

• For an arbitrary lattice [a b c and ( 90°)] in real space.

ba

c

O

c*

*

*

*

c cc

c cc

b ba

a b Va a

bb a V

a b a bc

a bc V

a*

b*

396

Example of the relationship between the real lattice and the reciprocal lattice.

From Leng, p. 52

397

Example of the relationship between the real lattice and the reciprocal lattice.

From Leng, p. 53

398

* * *

100

* * *

010

* * *

001

1 cos cos coscossin sin

1 cos cos coscossin sin

1 cos cos coscossin sin

a ad

b bd

c cd

_____ _________ __ ___ __________ _______Useful Properties of the reciprocal lattice

399

Relationship between real and reciprocal

In orthogonal real lattices(a= b = c and = = (= 90°))

Not necessarily so in non-orthogonal lattices

We can draw an analogy between reciprocal and real lattices:

* ; * ; *a a b b c c

* * * *hkl

uvwr ua vb wc

r ha kb lc

We use this to build a real lattice from unit cells

We can use this principle to build a reciprocal lattice

400

Construction of a 2D reciprocal lattice

(a) draw the plane lattice and mark the unit cell

Adapted from R. Tilley, Crystals and Crystal Structures, John Wiley & Sons, Hoboken, NJ, 2006

10

01b

a

401

Construction of a 2D reciprocal lattice

(b) draw lines perpendicular to the two sides of the unit cell to give the axialdirections of the reciprocal lattice basis vectors.

10

01

*

*a

*b

00

Adapted from R. Tilley, Crystals and Crystal

Structures, John Wiley & Sons, Hoboken, NJ, 2006

402

Construction of a 2D reciprocal lattice

(c) determine the perpendicular distances from the origin of the direct lattice to theend faces of the unit cell, d10 and d01, and take the inverse of these distances, 1/d10and 1/d01, as the reciprocal lattice axial lengths, a* and b.

Sets axes and interaxial

angle

What you’re doing first is finding the spacing between the planes making up the unit cell sides.

10

01

*

*a

*b

00

Adapted from R. Tilley, Crystals and Crystal

Structures, John Wiley & Sons, Hoboken, NJ, 2006

403

Construction of a 2D reciprocal lattice

(d) mark the lattice points at the appropriate reciprocal distances, and complete thelattice.

Draw reciprocal

lattice using axes

a* = 1/d10b* = 1/d01

Take reciprocals to get reciprocal lattice parameters.

Adapted from R. Tilley, Crystals and Crystal

Structures, John Wiley & Sons, Hoboken, NJ, 2006

*10

01

00

01

1 *ad

01* 1/b d

404

Construction of a 2D reciprocal lattice

The vector joining the origin of the reciprocal lattice to a lattice point hk isperpendicular to the (hk) planes in the real lattice and of length 1/dhk.

Real lattice

Every point on a reciprocal lattice represents a set of planes in the

real space crystal!

Adapted from R. Tilley, Crystals and Crystal

Structures, John Wiley & Sons, Hoboken, NJ, 2006

11

00

111

d10

0100 01

1110

Reciprocal lattice

01

1 *ad

01* 1/b db

a

405

Construction of a 3D reciprocal lattice

Figure 2.10 The construction of a reciprocallattice: (a) the a-c section of the unit cell in amonoclinic (mP) direct lattice; (b) reciprocallattice aces lie perpendicular to the end facesof the direct cell; (c) reciprocal lattice pointsare spaced a* = 1/d100 and c* = 1/d001; (d)the lattice plane is completed by extendingthe lattice; (e) the reciprocal lattice iscompleted by adding layers above and belowthe first plane.

Adapted from R. Tilley, Crystals and Crystal Structures, John Wiley & Sons, Hoboken, NJ, 2006

a

c a*c*

100d001d

(b)

000100

200300 400

a*

c*

001

002

101201

301 401102202

302 401

102

101

100

101001

101201

301 401

*

(d)

a*c*

100* 1a d001* 1c d *

(c)

000

010

020

030

100 200

001002

011

110 210

012010* 1b d

(e)

a

c

(a)

Just like 2-D

406

REAL LATTICE

4 Å

4 Å

(210)

(110)

(010)

a

b

c

RECIPROCAL LATTICE

210110[110] [210]

200 400

020

040

a*

b*

c*

010

[010]

uvwr ua vb wc * * * *

hklr ha kb lc

220020 120

Cubic Reciprocal Lattice

• Every point on a reciprocal lattice represents a set of planes in the real space crystal!

• Reciprocal lattice vectors are 90° away from realspace planes!

0.25 Å-1

90°

407

NORMAL LATTICE

(210)

(110)

(100)a

bc

RECIPROCAL LATTICE

Hexagonal Reciprocal Lattice

(010)

200

100

000

010

020

120

220

210

110

110

*r120

*r

*b

*a4 Å

0.25 Å-1

• Every point on a reciprocal lattice represents a set of planes in the real space crystal!

Real Space Reciprocal Space

What does the BCC reciprocal lattice look like?

0,0,2

0,1,1

0,2,0

1,0,1

1,1,0

1,1,2

2,0,0

2,0,1

2,2,0

2,2,2

2,1,11,2,1

b*

0,2,2

c*

0,0,0

• When a diffraction event occurs, the diffracted waves/pattern will “match” the reciprocal lattice.

• REASON:

EM radiation scatters in inverse proportion to the spacing between diffraction centers (i.e., planes in crystals).

409

Importance of Reciprocal Space

410

Why use reciprocal space

• Bragg’s Law:2 sin

2 sin1

/ 2sin 2

hkl

hkl hkl

hklhkl

hkl

nn d

dd

d

d

d

n

d

We can combine and as follows:

This allows us to write Bragg's Law as:

ophypot

positeenuse

(Defines conditions where a crystal is oriented for coherent scattering)

411

Aθhkl

(0,0,0)

1/ 2sin 2

hklhkl

hkl Cd B

d AC

hypotenuse

oppositeEwald Sphere Construction

1

2 Origin of reciprocal lattice and center of limiting sphere

B

C2θhkl

See this web site for an explanation and examplehttp://www.msm.cam.ac.uk/doitpoms/tlplib/xray-diffraction/ewald.php

Reflection or Ewald sphere

*1hkl

hkl

dd

Limiting sphere

CRITICALOnly lattice points lying within the limiting sphere can diffract. Points lying on the Ewald sphere will satisfy the Bragg condition.

2

412

*

sin

Which can be re-written as:12 sin

Since is a reciprocal lattice point:1

1 and 2 s

2

i

/

n

hklhkl

hkl hkl

OB OBCO

OBB

OB d gd

d dOB

1

os

• Size of sphere corresponds to wavelength of radiation used (see next page).

• Rotation of the crystal will cause points to lie on the sphere.

• When points lie on the sphere, Bragg’s law is satisfied!

(0,0,0)

*010d

*100d

*010d

*110d

RECIPROCAL SPACE

*020d *

120d

*030d *

130d

*200d

*210d

*220d

*230d

Crystalline solid

*130d

Ewald’s sphere

A

B

OC

[Bragg’s Law]

413

Adapted from C. Hammond, The Basics of Crystallography and Diffraction, 3rd Edition (Oxford University Press, Oxford, UK, 2009) p. 201.

Reflecting sphere for smallest wavelength

Reflecting sphere for largest wavelength

If you change λ, you change the radius of the Ewald’s sphere.

This is how the Laue technique works.

Diffractometers generally use fixed λ and variable θ.

100200 000

001

002

101

102202

201

201 10 1 00 1

002102202

201reflected

beam

200 reflectedbeam

201reflected

beam

min1/

max1/

414

Adapted from A.D. Krawitz, Introduction to Diffraction in Materials Science and Engineering, Wiley, New York, 2001

Diffractometers generally use fixed λ and variable θ.

0000°

200θ1°

400θ2°

Increasing diffraction angle

000

200

400

415

Synopsis1. The reciprocal lattice allows us to compute the spacing

between successive lattice planes in a crystal lattice.

2. The reciprocal lattice vector d*hkl with components (hkl) is perpendicular to the plane with Miller indices (hkl).

3. The length of the reciprocal lattice vector is equal to the inverse of the spacing between the corresponding planes.

4. Diffraction of X-rays (and electrons) is described by the Bragg equation, which relates the radiation wavelength (λ) to the diffraction angle (θ) and the spacing between crystal planes (dhkl).