Electron Diffraction Intensity and Reciprocal Lattic

[GL_Black_Hole]

Homework Statement

Low-energy electron diffraction (LEED) experiments are carried on to study a deposition of argon (Ar) and
xenon (Xe) on the surface of a graphite single crystal. In the regime of vapor pressure considered, 75% of Ar
and 25% of Xe are adsorbed on the (hexagonal) crystalline surface of graphite. The Ar-Ar distance and Ar-Xe distances are both $a$. The Ar atoms and the Xe atoms organize themselves in the two-dimensional lattice structure shown below. We take the relative form factor of Xe, over that of Ar to be R.
(img at https://ibb.co/cb76ac)
a)
Identify a space lattice and a primitive basis suitable to describe the arrangements of argon and xenon
atoms in the figure above.
b) Considering the Xe atoms alone, what is the space lattice formed by the Xe atoms?
c)Express the lattice translation vectors $a_1$ and $a_2$ in terms of the cartesian unit vectors $x$ and $y$, and the nearest neighbour distance $a$.
d)
What are the fractional coordinates of the Ar atoms and Xe atoms in the basis.
e)
Determine the basis structure factor for a diffracted beam with the scattering vector $q$ in the
xy plane of the lattice
f) What are the conditions on the $G=q$ indices h and k for observable diffraction peaks?
g) Determine the relative intensity ratio for the (h,k,0) peaks.
h) What would be the relative intensity if the lattice were made entirely of Ar?
i) What would be the relative intensity if all of the Ar was replaced with vacancies?

Homework Equations

$S(q) = \sum_{j=1}^{s} f_j exp(-i2\pi({x_j}h+{y_j}k)$

The Attempt at a Solution

a) The space lattice is triangular and the primitive basis is also triangular with one Xe atom(white) and three Ar atoms(grey).
b)The Xe atoms also form a triangular space lattice.
c) The lattice translation vectors are therefore $a_1 =a(1,0)$ and $a_2 = a(\frac{1}{2},\frac{\sqrt{3}}{2})$.
d) Choosing the Xe atom to be at (0,0), the other fractional coordinates are $(\frac{1}{2},0), (0,\frac{1}{2})$, and $(\frac{-1}{2},\frac{1}{2})$.
e) With these coordinates the structure factor is:
$S(q) = f_{Xe} +f_{Ar} [exp(-i\pi(h)) +exp(-i\pi(k)) +exp(-i\pi(k-h))]$
f) The structure factor cannot vanish so the conditions have to come directly from $G=q$ but I have no clue how they would.
g) Not even sure what is meant by relative intensity( i.e. relative to what baseline?)
h) Now it is possible for the structure factor to vanish, so some previously valid (h,k,0) should have zero intensity. [/B]

 

[_coyote_]

i) With vacancies, the structure factor should reduce to something like $S(q)= f_{Xe} +f_{Ar}[exp(-i\pi(h)) +exp(-i\pi(k))]$.