Powder sample crystal is analyzed using Debye-Scherrer method....

[steroidjunkie]

Homework Statement

Powder sample of monoatomic cubic lattice crystal is analyzed using Debye-Scherrer method. Primitive vetors of direct lattice are: a1 = (a, 0, 0), a2 = (0, a, 0) i a3 = (0, 0, a). Wavelength of x-ray radiation is 1 Å.
a) Find primitive vectors of reciprocal lattice.
b) Find the first four shortest vectors of reciprocal latitce.
c) First diffraction ring is observed at an angle $17,9^\circ$ with regard to incident radiation angle. Find lattice constant a.
d) Find angles for the next three difraction rings.

Homework Equations

a) $b_1 = \frac{2 \pi \cdot \vec{a_2} \times \vec{a_3}}{\vec{a_1} \cdot \vec{a_2} \times \vec{a_3}} = \frac{2 \pi}{a} \hat{x}$
$b_2 = \frac{2 \pi \cdot \vec{a_3} \times \vec{a_1}}{\vec{a_1} \cdot \vec{a_2} \times \vec{a_3}} = \frac{2 \pi}{a} \hat{y}$
$b_3 = \frac{2 \pi \cdot \vec{a_1} \times \vec{a_2}}{\vec{a_1} \cdot \vec{a_2} \times \vec{a_3}} = \frac{2 \pi}{a} \hat{z}$

b), c), d) ?

The Attempt at a Solution

[/B]
I have no idea how to proceed. I've found the $\vec{k} = \frac{2 \pi}{d}$ on the internet, where $d=\lambda$, but I'm not sure if this equation can give me the shortest vector or how I'd find the other three shortest vectors.

 

[anathema]

Also, I'm not sure how to use the given angle to find the lattice constant. Can someone help me out?a) The reciprocal lattice vectors can be found using the formula: $\vec{b_i} = \frac{2 \pi \vec{a_j} \times \vec{a_k}}{\vec{a_i} \cdot (\vec{a_j} \times \vec{a_k})}$ where i,j,k are cyclic permutations of 1,2,3. In this case, we have: $\vec{b_1} = \frac{2 \pi \vec{a_2} \times \vec{a_3}}{\vec{a_1} \cdot (\vec{a_2} \times \vec{a_3})} = \frac{2 \pi}{a} \hat{x}$, $\vec{b_2} = \frac{2 \pi \vec{a_3} \times \vec{a_1}}{\vec{a_2} \cdot (\vec{a_3} \times \vec{a_1})} = \frac{2 \pi}{a} \hat{y}$, and $\vec{b_3} = \frac{2 \pi \vec{a_1} \times \vec{a_2}}{\vec{a_3} \cdot (\vec{a_1} \times \vec{a_2})} = \frac{2 \pi}{a} \hat{z}$.

b) The first four shortest vectors of the reciprocal lattice can be found by considering the lengths of the reciprocal lattice vectors. The shortest vector will have a length of $\frac{2 \pi}{a}$, and the next shortest vectors will have lengths of $\sqrt{2} \frac{2 \pi}{a}$, $\sqrt{3} \frac{2 \pi}{a}$, and $2 \frac{2 \pi}{a}$, respectively.

c) The first diffraction ring is observed at an angle of $17.9^\circ$ with regard to the incident radiation angle. This angle can be used to find the lattice constant a using the formula: $\sin(\theta) = \frac{\lambda}{2a}$. Rearranging