- #1
GL_Black_Hole
- 21
- 0
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.