Find ##2\theta## Values from Rotated Crystal and Intensity

[GL_Black_Hole]

Homework Statement

Consider the crystal in the attached image (https://ibb.co/ftMrBH) (a triangular lattice of white atoms with a triangular basis of grey atoms attached to them at angles of 0, 60 and 120. From a previous problem the fractional coordinates of the atoms in the basis are (0,0), (1/2,0), (0,1/2) and (-1/2,1/2). Assume that the nearest neighbour distance between the atoms is A, and that the wavelength of the incoming x-rays is 1.5 A. If the incoming x-ray is along the $x_{lab}$ direction what are the angles $2\theta$ at which scattering can be observed if the crystal is rotated through an angle $\phi$ between 0 and 90 degrees? If the ratio of the form factor of white atoms to the form factor of grey is atoms is 4, what are the relative intensities of the scattered radiation at each $2\theta$? Use the reciprocal lattice and the condition $q=G$.

Homework Equations

$q=G$, $|G|<2k =\frac{4\pi}{1.5}$ ,$b_1 = \frac{2\pi}{a}(x-\frac{1}{\sqrt{3}} y)$, $b_2 =\frac{2\pi}{a}\frac{2}{\sqrt{3}} y$

The Attempt at a Solution

I tried to start by finding the angles when $\phi =0$. If I can do this then the rotated case should be nothing more complicated then applying a rotation matrix and following the same steps because $k_0$ doesn't change but the components of the reciprocal lattice vector in the lab frame do. Decomposing $k_0 = \frac{2\pi}{1.5} x$, and $k' = \frac{2\pi}{1.5} (cos(2\theta)x +sin(2\theta)y)$ I can form $G =hb_1 +kb_2$ and apply $q =G$ to give me the system of equations:

$\frac{2\pi}{1.5} cos(2\theta) = h\frac{2\pi}{a} + \frac{2\pi}{1.5}$

$\frac{2\pi}{1.5} sin(2\theta) = \frac{2\pi}{a} \frac{1}{\sqrt{3}}(2k-h)$

But I can't seem to be able to use these equations to find a set of allowed $2\theta$ values. After this I'm not sure how to handle the intensities either.

 

[anathema]

Hello, thank you for your post. It seems like you are on the right track with your solution so far. Here are a few suggestions to help you continue:

1. In order to find the angles at which scattering can occur, you need to use the Bragg condition: $2d_{hkl} sin(\theta) = \lambda$, where $d_{hkl}$ is the spacing between the lattice planes, and $\lambda$ is the wavelength of the incoming x-rays. In this case, the spacing between lattice planes can be expressed as $d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + hk}}$, where $a$ is the lattice constant.

2. Using the Bragg condition, you can then solve for the allowed values of $\theta$, which will give you the angles at which scattering can occur. Keep in mind that you will need to consider both the white atoms and the grey atoms in your calculations.

3. Once you have the allowed angles, you can then use the form factors of the white and grey atoms to determine the relative intensities of the scattered radiation. The ratio of the form factors will give you the relative intensity of the scattered radiation for each $2\theta$ value.

I hope this helps guide you in the right direction. Good luck with your calculations!