Bragg-scattering angle from scattered beam of energy

[Weyoun9]

Homework Statement

A beam of thermal neutrons emerges from a nuclear reactor and is incident on a crystal as shown in the figure below. The beam is Bragg scattered, as in the figure, from a crystal whose scattering planes are separated by 0.253 nm. From the continuous energy spectrum of the beam we wish to select neutrons of energy 0.0109 eV.

Find the Bragg-scattering angle that results in a scattered beam of this energy.

Homework Equations

Bragg Scattering
2dsin(θ)=nλ
λ=$hc/E$= $hc/sqrt(2mc^2 * k)$
hc= 1240 eV.nm
2mc²=511 keV

The Attempt at a Solution

This is from a textbook chapter mostly on Heisenberg Uncertainty Relationships, so I feel like that must factor in, but I'm unsure as to how. Initially I just tried to solve for λ so I could plug that into the Bragg equation.
I assumed the energy given for the neutrons is kinetic energy, so I attempted to solve for λ this way: λ=$hc/E$=$(1240eV.nm)/sqrt(2 * 511E3 eV * 0.0109 eV)$=11.75nm. However, when you plug this value for λ into 2dsin(θ)=nλ (assuming n is 1), and take the inverse sin of 11.75nm/(2 * .253nm) you get an error. I also tried solving λ=$hc/E$, assuming that .0109 eV is just the E, but that also resulted in an answer for which you cannot take the inverse sin. I realize I'm probably missing something simple, but I've been stuck on this problem for a while. I don't understand what I'm doing wrong or how Heisenberg Uncertainty principles factor into this problem, if at all. Any help would be greatly appreciated.

 

[mfb]

The formulas you have don't apply to slow particles like neutrons. You'll need $\lambda = \frac{h}{p}$ and (since the neutrons are slow) you can use $p=mv$. You can get the velocity from the energy, and then solve everything for the angle.

 

[Weyoun9]

Thank you so much. Most the equations on this homework were relativistic and I didn't think about that. All I had to do for this was use p=sqrt(2mK) and plug and chug. Answer turned out to be 32.8°.

 

[Summer95]

The formulas you have don't apply to slow particles like neutrons. You'll need $\lambda = \frac{h}{p}$ and (since the neutrons are slow) you can use $p=mv$. You can get the velocity from the energy, and then solve everything for the angle.

I am doing a similar problem. Is 2dsin(θ)=nλ what I must use to find the angle?

What I don't understand is, neutrons of various energies are incident on the crystal at the same angle (correct?) And by the nature of this formula they must be incident and reflected at the same angle. How can this separate the beam if it relies on that symmetry?

 

[mfb]

You never have to use specific formulas (unless the problem statement requires it), but usually some formulas make the problem easier than others.

What I don't understand is, neutrons of various energies are incident on the crystal at the same angle (correct?) And by the nature of this formula they must be incident and reflected at the same angle. How can this separate the beam if it relies on that symmetry?

The condition you posted is satisfied for specific neutron energies only.