dear all
can anyone explain to me or give me a reference how they find that the expression of the Zn-O bond length in the wurtzite structure is given by [1] :
$ l=\sqrt{(\frac{a^2}{3}+(\frac{1}{2})-u)^2 c^2} $
where u is the positional parameter
$u=\frac{a^2}{3c^2}+1/4$
thanks in advance :)
[1] Wang XS, Wu ZC, Webb JF, Liu ZG (2003) Appl Phys A 77:561
There are two ways to calculate the bond-length.
1. The simple approach is, using the relation between interplanar spacing (dhkl) of a particular Bragg plane {hkl}, involving the lattice parameter a and c for a hexagonal system. There are two unknowns in the equation: a & c. So, you need at least two equations to solve them. So consider the two most intense peaks. Write the equation for each one. And solve it to calculate the lattice parameters.
You can also take a number of intense peaks, and calculate a & c for each pair. Then take their mean.
2. The more rigorous and accurate method is Rietveld refinement of the XRD data. You can obtain lattice parameters and many other things from the best fit.
In the Wurtzite structure type you have four atoms per unit cell:
Zn 1/3, 2/3, 0
Zn 2/3, 1/3, 1/2
O 1/3, 2/3, u
O 2/3, 1/3, u + 1/2
(You can interchange Zn and O it makes no difference)
In a general triclinic structure the length of any vector squared is
EQ. 1:
d^2 = (u, v, z) G (u, v, z)^T (The second vector is a column matrix)
G is the metric tensor, whose 3x3 elements are the scalar products of the lattice vectors. in a hexagonal system you have a=b, alpha=beta=90 gamma=120 and thus
EQ. 2
( a^2 -1/2 a^2 0 )
G = ( -1/2 a^2 a^2 0 )
( 0 0 c^2)
In the Wurtzite type structure you have two vectors Zn-O:
[ 0 0 u ] and
[ -1/3 1/3 1/2-u ]
Plug these two vectors into the EQ. 1 you need to calculate
EQ. 3
( a^2 -1/2 a^2 0 ) [ 0 ]
d^2 = [ 0 0 u] ( -1/2 a^2 a^2 0 ) [ 0 ]
( 0 0 c^2 ) [ u]
and
EQ 4.
( a^2 -1/2 a^2 0 ) [ -1/3 ]
d^2 = [ -1/3 1/3 1/2-u] ( -1/2 a^2 a^2 0 ) [ 1/3 ]
( 0 0 c^2 ) [ 1/2-u]
(standard matrix algebra)
These are symmetrically NOT equivalent and NEED NOT have the same length!
The paper implicitly makes the (far fetched) assumption that the two distances are equal. If you force these two vectors to have the same length you obtain the relationship between u and the a/c ratio in eq. 1 in the paper.
And eq. 2 in the paper is simply the reslut of EQ. 4 written down in full.
You will find these type of calculations explained in any basic crystallographic text book! For example Giacovazzo et al. "Fundamental crystallography"), D.E.Sands "Vectors and tensors in Crystallography"
For exact visualization of unit cell of ZnO, first you need to refine the XRD spectra using Rietveld Refinement then create the CIF file. This CIF file can easily open in VESTA, where you can easily measure the bond length in any direction.
Regards
- Badges
- Science method
- Similar topics
- Methodology
- Crystallography
- X-ray Diffraction