lattice directions individual directions: [uvw] symmetry-related...

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Chem 253, UC, Berkeley

Crystallographic Directions And Planes

Lattice DirectionsIndividual directions: [uvw]Symmetry-related directions: <uvw>


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Miller Indices:

1. Find the intercepts on the axes in terms of the latticeconstant a, b, c

2. Take the reciprocals of these numbers, reduce to thethree integers having the same ratio(hkl)

Set of symmetry-related planes: {hkl}


(100) (111)

(200) (110)

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Miller-Bravais indices

[uvtw], (hkil)i=-(h+k)t=-(u+v)

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In cubic system,

[hkl] direction perpendicular to (hkl) plane


2

222

2

1

a

lkh

dhkl

For cubic system

Lattice spacing

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Crystal Structure Analysis

X-ray diffraction

Electron Diffraction

Neutron Diffraction

Essence of diffraction: Bragg Diffraction

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LightInterference fringes

Constructive

Destructive


Bragg’s Law

For cubic system: But not all planes have the diffraction !!!

sin2

sinsin

hkl

hklhkl

d

dd

QTSQn

222 lkh

adhkl

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/hchE

35KeV ~ 0.1-1.4ACu K 1.54 A

Mo:

X-Ray Diffraction


X-Ray Diffraction

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(200)

Powder diffraction

X-Ray


Powder diffraction

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Laue Diffraction

Singlecrystal


Electron Diffraction

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Example: La2CuO2

2*theta d

7.2 12.1

14.4 6.1

22 4.0

nd sin2

Layered CupratesThin film, growth oriented along c axis

(hkl)

(001)

(002)

(003)c=12.2 A

(00l)

Cu K 1.54 A


CuO2

LaO

LaO

CuO212.2 A

c

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Example: Ca0.5Sr0.5CuO2

2*theta d

12.7 6.96

26 3.42

42.2 2.15

nd sin2

Layered CupratesThin film, growth oriented along c axis

(hkl)

(001)

(002)

(003) c=6.96 Å

(00l)


SrCuO2Ca

6.96

c

Å

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What we will see in XRD of simple cubic, BCC, FCC?

)(75.0);(5.0sin

sin2

2

FCCBCCB

A

222 lkh

adhkl


Reciprocal Lattice

n

'nd

R

cnbnanR 321Path Difference:

mnnRnRnR

)( ''

mnnR

2)(2 '

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Reciprocal Lattice

d

R

cnbnanR 321

mnnR

2)(2 '

mkkR 2)( '

Correspond to plane wave:12)( '

mikkRi ee

nk2

'k


Reciprocal Lattice

n

'nd

R

cnbnanR 321

1)( '

kkRie Laue Condition

Reciprocal lattice vector

For all R in the Bravais Lattice

'k

k

kkK'

1 RiKe

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Reciprocal Lattice

d

R

1)( '

kkRie Laue Condition

Reciprocal lattice vector


'k

k

kkK'

1 RiKe

K Reciprocal lattice vector


Reciprocal Lattice


A reciprocal lattice is defined with reference to a particular Bravias Lattice.

a

b

c Primitive vectors

)(2

cba

cba

)(2

cba

acb

)(2

cba

bac

1 RiKe

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Reciprocal Lattice


)(2

cba

cba

)(2

cba

acb

)(2

cba

bac

Verify:

0

0

2

ca

ba

aa

For any

cnbnanR

ckbkakK

321

321

)(2 332211 nknknkRK

1 RiKe


Reciprocal Lattice

Simple cubic

Reciprocal lattice is always one of 14 Bravais Lattice.

xaa

yab

zac

)(2

cba

cba

)(2

cba

acb

)(2

cba

bac

ya

2

za

2

xa

2

Simple cubic

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Primitive Cell of FCC

•Angle between a1, a2, a3: 60o

)(2

11

yxaa

)(2

12

yzaa

)(2

13

xzaa


Primitive Cell of BCC

)(2

12

zyxaa

•Primitive Translation Vectors:

•Rhombohedron primitive cell

0.53a

109o28’)(2

11

zyxaa

)(2

13

zyxaa

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Reciprocal Lattice

FCC

)(2

cba

cba

)(2

cba

acb

)(2

cba

bac

Reciprocal lattice is always one of 14 Bravais Lattice.

BCC

)(2

zya

a

)(2

zxa

b

)(2

yxa

c

)(2

14

xzya

)(2

14

yzxa

)(2

14

zyxa


Reciprocal Lattice

BCC FCC

Simple hexagonal Simple hexagonal

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)(2

cba

cba

)(2

cba

acb

)(2

cba

bac


)(2

cba

cba

)(2

cba

acb

)(2

cba

bac

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Wigner-Seitz cells of reciprocal lattice


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X (a 0 0)

L


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Theorem:

For any family of lattice planes separated by distance d, there are

reciprocal lattice vectors perpendicular to the planes, the shortest

being 2/d.


Orientation of plane is determined by a normal vector

The miller indices of a lattice plane are the coordination at the reciprocal lattice vector normal to the plane.

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Plane (hkl)

lckbhaK

A

B

C ax1

bx2

cx3

)1

,1

,1

()(321 xxx

hkl

cx

bx

ax 321

222

)()( 3121 cxaxbxaxACABK

cx

bx

ax 321

111

lckbhaK

)(2

cba

cba

)(2

cba

acb

)(2

cba

bac

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