Allowed reflections: cubicsc: any , ,bcc: evenfcc: , , all even or all odd

h kh kh k

Note the dependence of atomic concentration on nearest-neighbor distance:

n-n3 3

n-n1 1

d a

a d

n-n

3 3 3n-n n-n

3 2 0.87

2 3 3 8 0.65

d a a

a d d

n-n

3 3 3n-n n-n

2 0.71

4 2 1.41

d a a

a d d

sc:

bcc:

fcc:

The Zone Axis1 2 3uvw u v w r a a a

1 2 3hk h k g b b b

0hk uvw hu kv w g r

Direct lattice vector:

Reciprocal lattice vector:

For all RLVs within the ZOLZ:

If 1 0uvw g r

2 0uvw g r

1 2 0uvw g g r

and

then

This is the [uvw] zone axis.

Any reflection (hkl) in the [uvw] ZOLZ has: 0hu kv w

Finding the zone axis

1 2||uvw g g

If g1 and g2 reside in the [uvw] ZOLZ, then:

Notice that:

1 2 3

1 2 1 1 1

2 2 2

1 h kV

h k

a a ag g

So:

1 2 1 2 1 2 1 2 1 2 1 2|| , ,uvw k k h h h k k h

The zone axis of a ZOLZ in any crystal system satisfies the condition:

i j j k i j k j jV a a a a a a a a a

i jk V

a a

b

jk i V

ab b

j ki V

a a

b

Determining Orientation

Procedure:1) measure d1 and d2 2) Determine (d2/d1)2

3) Look for integer ratio,consistent with fcc

4) Determine types of reflections5) Select indices, based on angle

Assume we know the structure (fcc), but not the lattice constant or orientation:

1

2

0.208 nm0.293 nm

dd

2 2 2 22 1 1 1

2 2 21 2 2 2

21.981

d h kd h k

42

63

2 2 2

2 2 2

8 2 2 04 2 0 0

(1): {220}, (2): {200}

1 2 0 (1) : 220, (2) : 002 g g

[ ] 1 1 0 0,1 0 1 1,1 0 0 1 110uvw B 110

6) Find zone axis

Reconsider choice

Revise and index

zone axis: 1 1 0 0,0 0 1 1, 1 0 0 1 110uvw

B 110

2 002

002

22 0.586 nm

ad d

a d

Find lattice constant:

1 002 220 1112

111 002 000 113

220 220change:

000 220 000 220

index:

Powder pattern (I): CdSe

(1) :111(2): 220(3): 311

CdSe dots

2 2 2 22

2 2 21

8 2 2 02.663 1 1 1

dd

Test fcc:

ring 2/d (1/nm) d (nm)

1 5.677 0.352

2 9.251 0.216

3 10.858 0.1842

2 dd

2 2 2 23

2 2 21

11 3 1 13.663 1 1 1

dd

Powder Pattern (II): Gd2Ti2O4

Structure factor and intensity

F f 1 e 0iF f

destructiveinterference

constructiveinterference

Inserting planes can reduce the intensities of some peaks.

Waves scattered from inserted planes are 180° out-of-phase.