A question regarding particle diffraction - Might have posted in wrong place?

[BruceSpringste]

Homework Statement

"A beam of neutrons that emerges from a nuclear reactor contains neutrons with a variety of energies. To obtain neutrons with an energy of 0.050 eV, the beam is passed through a crystal whose atomic planes are 0.20 nm apart. At what angles relative to the original beam will the desired netruons be diffracted?"

Homework Equations

Braggs law
mλ=sin2θ


3. The Attempt at a Solution [/b
My first thought was to use braggs law and simply solve for θ and m=1,2,3... but θ>90degrees.
However the problem seems a lot more complex than that. My book doesn't have any answers to the question so I can't double check if I have done this right!

 

[Mute]

Homework Statement

"A beam of neutrons that emerges from a nuclear reactor contains neutrons with a variety of energies. To obtain neutrons with an energy of 0.050 eV, the beam is passed through a crystal whose atomic planes are 0.20 nm apart. At what angles relative to the original beam will the desired netruons be diffracted?"

Homework Equations

Braggs law
mλ=sin2θ3. The Attempt at a Solution [/b
My first thought was to use braggs law and simply solve for θ and m=1,2,3... but θ>90degrees.
However the problem seems a lot more complex than that. My book doesn't have any answers to the question so I can't double check if I have done this right!

The expression you wrote for Bragg's law is not correct. It should be

$$m\lambda = 2d \sin\theta .$$

Does using that expression give you reasonable answers? By the way, the original (incorrect) expression you wrote for Bragg's law did not contain the plane spacing d. Was that a typo, or did you not notice that you weren't using the plane spacing in your calculation? It's handy to keep track of what pieces of information you have but didn't use so that you can perform a 'sanity check' and assess whether or not you think you should have used that piece of information in your calculation.

 

[BruceSpringste]

I of course meant what you wrote. It was a typo!

And no it doesn't feel like I get a reasonable answer. It feels like something is missing.

Since the neutrons are behaving like a wave I thought using braggs law would solve it for me.
However the problem is my book doenst provide any answers to the questions. And i was wondering if I have been thinking correctly!

 

[Mute]

I of course meant what you wrote. It was a typo!

They're not always typos! Sometimes people just have the wrong equation, so we have to double-check that first.

And no it doesn't feel like I get a reasonable answer. It feels like something is missing.

Since the neutrons are behaving like a wave I thought using braggs law would solve it for me.
However the problem is my book doenst provide any answers to the questions. And i was wondering if I have been thinking correctly!

Bragg's law does seem like the appropriate approach to the problem. Have you double checked all of the numbers you were given and their units? For example, are you sure the neutron energy is not supposed to be in MeV? That would give a more reasonable wavelength (on the order of ~ 0.1 nm) for the neutrons.

 

[BruceSpringste]

Haha I get that! And I'm very grateful that you're taking your time to answer this question.

The unit is eV, I have double checked.
To obtain lambda I feel I have to use de broglies equation: λ=h/γmv
Since 0.050 eV is small compared with its rest energy mc^2 I figured Y=1.
However from this point I am stuck!

mv^2/2 = KE
v=√(2mKE)
m = mass of a neutron
E = ? Neutrons do not have an electrif fieldbecause they are neutral. But it can't be 0?
K = 0.05 eV

 

[Mute]

Haha I get that! And I'm very grateful that you're taking your time to answer this question.

The unit is eV, I have double checked.
To obtain lambda I feel I have to use de broglies equation: λ=h/γmv
Since 0.050 eV is small compared with its rest energy mc^2 I figured Y=1.
However from this point I am stuck!

mv^2/2 = KE
v=√(2mKE)
m = mass of a neutron
E = ? Neutrons do not have an electrif fieldbecause they are neutral. But it can't be 0?
K = 0.05 eV

Hm, before I wasn't getting reasonable answers either, but I did it again and I get something that looks ok, so I'll assume I'm more correct this time. So, let's start with your suggestions. I agree that since the kinetic energy of the neutron is much smaller than its rest mass energy, that $\gamma$ is probably close to 1, which means we can use the non-relativistic equation for the kinetic energy.

I also agree that you will need to use de Broglie's law, $\lambda = h/p$, where p = mv if we can use the non-relativistic equations.

Now, I'm not sure what you mean by E here, but electric field doesn't come into this problem at all. Instead, I would suggest you look at the equations you wrote down again. You want to find $\lambda$, yes? Do you have enough information to find it from the equations you wrote down for kinetic energy and de Broglie's law?

Also, I'll give you a hint that may save you some algebra: you can write the kinetic energy as

$$KE = \frac{p^2}{2m}.$$

(You can derive this from the usual form KE = mv^2/2 by plugging in v = p/m).

So, see if you can solve the problem now, and if you get stuck again show us your work and we'll try to point out where you went wrong (if you did) or give you some hints to get you unstuck.