1. The problem statement, all variables and given/known data A mono-energetic beam are of neutrons is directed perpendiculary at the surface of a metal crystal whose atom are a constant distance 0.1E-10m apart. The scattered beam of neutrons is found to have maximum intensity at an angle of 25degree to the initial direction. What is the energy (in eV) of the neutrons? 2. Relevant equations 2dsinθ = nλ E = hc/λ 3. The attempt at a solution 2(0.1E-10)*sin(25) = (1)λ λ = 8.45E-12 m E = hc/λ = (6.63E-34)(3E+8)/ 8.45E-12 m = 2.35E-14 J=146890eV but ans = 45.7eV Can anyone tell me where goes wrong? Is that the right equation uses to find the energy of the neutrons?
The θ used in your calculation is wrong. Check how it is defined in the 2dsinθ = nλ relation. E = hc/λ is correct for a photon, but this is a particle with mass and the c has to be replaced with the neutrons velocity. Use the de Broglie-relation λ = h/p and you should be able to get an expression for E. I got the answer E = 43.7eV using my pocket calculator just now which is in the ball park at least.
Φ = 180 - 2θ θ = (180-Φ)/2 = (180-25)/2 = 77.5 2dsinθ = nλ λ = 2dsinθ = 2(0.1E-10)sin(77.5) = 1.95E-11m λ = h/p λ = h/√(2mE) E = (h/λ)^2 / 2m E = (6.63E-34 / 1.95E-11)^2 / 2(1.67E-27) = 3.46E-19 J E = 3.46E-19 J/ eV = 2.16eV ?
Your problem text said that the angle between incoming and outgoing ray was 25°, but if you've deduced the relation 2dsinθ = nλ you should know that θ is defined as the angle against a perpendicular line against the surface, compare with the law of reflection.