1) The document discusses crystal structures including simple cubic (sc), body centered cubic (bcc), and face centered cubic (fcc) unit cells. It notes the allowed reflections and nearest neighbor distances for each structure. 2) Methods are described for determining the zone axis of a crystal from diffraction patterns, including calculating ratios of d-spacings and identifying systematic absences. 3) An example is given of using powder diffraction data to index peaks and determine the lattice constant of cubic CdSe nanoparticles.

1. Allowed reflections: cubic
sc: any , ,
bcc: even
fcc: , , all even or all odd
h k
h k
h k
  



Note the dependence of atomic
concentration on nearest-neighbor
distance:
n-n
3 3
n-n
1 1
d a
a d

  
n-n
3 3 3
n-n n-n
3 2 0.87
2 3 3 8 0.65
d a a
a d d
 
   
n-n
3 3 3
n-n n-n
2 0.71
4 2 1.41
d a a
a d d
 
   
sc:
bcc:
fcc:

2. The Zone Axis
1 2 3
uvw u v w
  
r a a a
1 2 3
hk h k
  
g b b b
 
0
hk uvw hu kv w
    
g r
 
Direct lattice vector:
Reciprocal lattice vector:
For all RLVs within the ZOLZ:
If 1 0
uvw
 
g r
2 0
uvw
 
g r
 
1 2 0
uvw
  
g g r
and
then
This is the [uvw] zone axis.
Any reflection (hkl) in the [uvw] ZOLZ has: 0
hu kv w
  


3. 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 k
V
h k
 
a a a
g 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 j
V
      
 
 
a a a a a a a a a
i j
k
V


a a
b
j
k i
V
 
a
b b
j k
i
V


a a
b

4. 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 reflections
5) Select indices, based on angle
Assume we know the structure (fcc), but not the lattice constant or orientation:
1
2
0.208 nm
0.293 nm
d
d


2
2 2 2
2 1 1 1
2 2 2
1 2 2 2
2
1.98
1
d h k
d h k
   
  
 
 
 


4
2

6
3

2 2 2
2 2 2
8 2 2 0
4 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 1 110
uvw            B  110

 

6) Find zone axis
Reconsider choice

5. Revise and index
zone axis:      
1 1 0 0,0 0 1 1,1 0 0 1 110
uvw           
B  110
 
2 002
002
2
2 0.586 nm
a
d d
a d
 
 
Find lattice constant:
   
   
1
002 220 111
2
 
     
   
111 002 000 113
  
220 220

change:
     
   
000 220 000 220
  
index:

6. Powder pattern (I): CdSe
(1) :111
(2): 220
(3): 311
CdSe dots
2 2 2 2
2
2 2 2
1
8 2 2 0
2.66
3 1 1 1
d
d
 
 
  
   
 
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 d
d
2 2 2 2
3
2 2 2
1
11 3 1 1
3.66
3 1 1 1
d
d
 
    
 
 
 

7. Powder Pattern (II): Gd2Ti2O4

8. Structure factor and intensity
F f
  
1 e 0
i
F f 
   
destructive
interference
constructive
interference
Inserting planes can reduce the intensities of some peaks.
Waves scattered from inserted planes are 180° out-of-phase.