# Transformations 1

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```					Transformations
There’s still the 2-fold in the center of the cell. Move the origin along
the diagonal with the same 3-layer stacking to give:

I-centered
Monoclinic
Again since there is no symmetry in the a-c plane new axes can be
chosen to give a C centered cell.

aC  a I  cI
bC  bI
cC   a I

 1 0 1
 0 1 0
T         
  1 0 0
        
Th2Fe17 has been reported to be monoclinic with a = 9.68 Å, b = 8.56 Å, c
= 6.46 Å,  = 99.3 °, space group C2/m. The Th atoms are located in
position 4i: ±(x , 0 , z) + (0, 0, 0 ; ½, ½, 0), with x = 0.167, z = 0.333.

A different unit cell may be chosen for this structure. The transformations
are:

anew  1 a  1 b
2     2                         12
1
2  0
bnew  1 a  1 b
2     2
T  1  1
2   2  0
cnew  c                              0
    0  1


Calculate the new unit cell. What crystal system is suggested? What
are the new coordinates of the Th atoms?
We can use the metric tensor for the transformation…

 93.702    0     10.106                                3
G 0
         73.274     0  
V  G  528.24 A
 10.106
           0     41.732 


Gnew  TGT     T

41.74 5.11 5.05                                     3
Gnew   5.11 41.74 5.05 
                 
V  G  264.12 A
 5.05 5.05 41.74
                 
Now the new metric tensor suggests a new crystal system…

41.74 5.11 5.05 
Gnew   5.11 41.74 5.05 
                 
 5.05 5.05 41.74
                 
If 5.11 == 5.05 = 5.07 (average).


a  b  c  41.74  6.461A
      cos1 5.07 41.74  83.02
These values suggest a rhombohedral system.
Coordinates of atoms are represented by vectors:

r  xa  yb  zc
To get rnew from the original coordinates the following relation is used:

rnew  T    r1 T

 xnew                  x
 y   T 1
 new               T 
 y
 z new 
                      z
 
Transformation of Th Coordinates

0.167    0 0.333      →   0.167     0.167   0.667 

0.833    0 0.667      →    0.167     0.167   0.333 

0.667    0.5 0.333    →   0.167     0.167   0.667 

0.333    0.5 0.667    →    0.167     0.167   0.333 
If T is a transformation matrix that when multiplied by old unit cell
basis vectors gives a new set of unit basis vectors:

Gnew  TGT T

 a new       a     *                  xnew              x
b   T
 new 
b 
 
 y   T 1
 new           T 
 y
 cnew 
             c 
                         z new 
                  z
 

 hnew       h                       U new             U 
k   T      k 
 new         
 V   T 1
 new           T  
V 
 lnew 
            l 
                        Wnew              W 
                   
* a, b, and c are vectors not unit cell constants!

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