I managed to get an answer close to the correct one, but there is something I am not geting. It's a pretty easy problem, but I have a concern. Here is the question:

A crystal has a set of planes spaced 0.30 nm apart. A beam of neutrons strikes the crystal at a normal incidence and the first maximum of the diffraction pattern occurs at φ = 42° . What are the de Broglie wavenlength and kinetic energy of the neutrons?


I approached the question with the plan to find the wavelength first, and then compute the kinetic energy from that.

For the first part- finding the de Broglie wavelength:

d = 0.30 nm

φ = 42°

Now, my text has the following relation:

nλ = D sinφ

and then this one:

d = D sinα

where 2α = φ ==> α = φ/2

So I wanted D, and I did this:

D = d/sin(φ/2)
'
which gave me an answer of: 0.837 nm for D.


I then plugged this in to nλ = D sinφ (with n = 1) and got:

0.560 nm.


My concern is that the text got an answer of 0.523 nm, and they used a slightly different method. However, I see no reason why my method shouldn't work. Hopefully, someone can explain where my reasoning was flawed.


Anyway, for the second part, I did this:

λ = h/√(2mE)

* h is Planck's constant
* mass of neutron is 1.67 x 10^-27 kg

solving for E

E = h^2/(2mλ^2)

E = 2.6 x 10^-3 eV

which is very close to the text answer of 3.0 x 10 ^-3 eV.

I am assuming the difference is due to my mistake on the first part, so my understanding of why my method in part one is flawed is pretty important for getting this basic problem down.

Thanks for the help!