The intensity of x-ray scattering peaks from a crystal depends on the structure factor S of the crystal.
The structure factor of the Si crystal (face centered
cubic or fcc lattice), in particular, is discussed in many standard textbooks
on solid state physics, such as Kittel's Introduction to Solid State Physics
(Chapter 2 in 6th Edition). The basis of the fcc structure is
usually referred to the cubic unit cell with 4 atoms per unit cube.
These four atoms can be chosen to have the locations at (0,0,0);
(0,1/2,1/2); (1/2,0,1/2) and (1/2,1/2,0) [in units of the size of the cube:
a]. The structure factor Sfcc(hkl) for a wave vector
(h,k,l) in reciprocal space then vanishes if the intergers h, k and l contain
a mixture of even and odd number. In case of the Si crystal there
are now 8 atoms per unit cube since there are two penetrating fcc sublattices
displaced from each other by the distance (1/4,1/4,1/4). As
a result the structure factor of the Si crystal SSi(hkl) is
given by:
SSi(hkl)=Sfcc(hkl)[1+exp(ip/2)(h+k+l)].
This implies that SSi(hkl) will be zero if the sum (h+k+l)
is equal to 2 times an odd integer. When combining the above two
conditions together one obtains the result that SSi(hkl)
will be non-zero only if (1) (h,k,l) contains only even numbers and (2)
the sum (h+k+l) is equal to 4 times an integer. See,
for example, Kittel's Introduction to Solid State Physics (Chapter 2 in
6th Edition),Problem 5 at the end of Chapter 2. Based on this
result one expects that the diffraction spot corresponding to (2,2,2) in
the x-ray diffraction pattern of Si will have zero intensity since h+k+l=6.
It has been known since 1959 that the so-called forbidden (2,2,2) diffraction spot in diamond has non-zero intensity (see Ref. [3.19] or Kittel's Introduction to Solid State Physics, p.73 in 3rd Edition). It is now well established that the presence of this forbidden (2,2,2) diffraction spot can be explained by the existence of bond charges located approximately mid-way between the atoms in diamond or silicon. The question raised in this supplementary problem is:
What is the structure factor of the bond charges in the Si crystal if one assumes that they are located exactly in the mid-way between two Si atoms?
SOLUTION:
To solve this problem we first has to decide how many
bonds there are in the unit cube and then determine the location of the
bond charges associated with these bonds.
To answer the first question we note that each Si atoms
is surrounded by four nearest neigbors on the second sublattice forming
4 covalent bonds directed towards the corners of a tetrahedron. Thus
we can construct the bond charges by starting with the fcc lattice and
then for each Si atom on this lattice we place four bond charges at the
corners of a tetrahedron. In terms of the lattice constant a
the location of the 4 bond charges surrounding the Si atom at (1/4,1/4,1/4)
are:
(1/8,1/8,1/8), (3/8,3/8,1/8),(1/8,3/8,3/8) and (3/8,1/8,3/8).
Thus the bond charges form a "crystal" with a fcc lattice
with 4 "atoms" per unit cell. To calculate the structure factor Sbond-charge(hkl)
of the bond charge it will be convenient to translate the coordinate so
that the origin is located at the bond charge (1/8,1/8,1/8). In this
new coordinate system the location of the 4 bond charges in the unit cell
are:
(0,0,0), (1/4,1/4,0),(0,1/4,1/4) and (1/4,0,1/4).
With this choice of location for the bond charges it
is straight forward to show that:
Sbond-charge(hkl)=Sfcc(hkl)[1+exp(ip/2)(h+k)+exp(ip/2)(k+l)+exp(ip/2)(h+l)].
For h=k=l=2 we find:
Sbond-charge(222)=Sfcc(222)[1+3exp(ip/2)4]=4Sfcc(222)
and is non-zero.
Copyrighted by Peter Y. Yu, 1997. You are allowed to download this
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