Question

Problem 2: X-ray diffraction: (a) Determine the maximum wavelength for which constructive interference can be observed in the Bragg model for a simple cubic crystal with a lattice constant of 3.6 A. (b)What is the energy of the X-rays in electron volts? (c) If you were to perform neutron diffraction, what would the energy of the neutrons have to be in order to obtain the same de Broglie wavelength? (d) You could argue that if you take X-rays with twice the wavelength, you would still get a Bragg peak because there would be constructive interference between the X-rays that are reflected from every other plane. Why is this argument not valid? (e) You could describe the same crystal by using a unit cell that is a bigger cube of twice the side length, containing eight atoms instead of one. The lattice constant would then be 7.2 Å. Discuss how this different description would affect the X-ray diffraction from the crystal.

Answer 1

For Bragg's diffraction, constructive interference can be observed as 2d sin theta = n lambda (1) (a) from equation (1) lambda = 2d sin theta/n = 2 a sin theta/squareroot h^2 + k^2 + l^2 times 1/h for maximum lambda, sin theta = 1, h = 1 and hh^2 + k^2 + l^2 = 1 gives lambda_max = 29 = 2 times 3.6 A degree (a = 3.6 A degree gives) lambda_max = 7.2 A degree (b) energy of x-rays E = h c/lambda_max = 4.1357 times 10^-15 ev - su times 3 times 10^8 m/s/7.2 times 10^-10 m E = 1.7232 times 10^3 ev E = 1723.2 ev (c) for neutron if de-Broglie wavelength is lambda = lambda_max = 7.2 A degree then energy of the neutron is mass of neutron m = 1.6749 times 10^-27 kg then using lambda = h/p implies p = h/lambda = 6.626 times 10^-34 J-sec/7.2 times 10^-10 m p = 9.203 times 10^-25 kg m/sec so energy of the neutron is E = p^2/2m = (9.203 times 10^-25 kg m/sec)^2/2 times 1.6749 times 10^-27 kg