Solving Bragg's Law: X-Ray & Electron Measurement

[steve233]

I'm preparing for a midterm exam and this is one of the practise questions I'm having a bit of trouble with.

Homework Statement

We have access to X-ray and electrons for the incident beam.
$\theta$ is the inner angle between the sample and the incident beam.

Spacing in sample = d =10^-10 m
$\lambda$ = 10^-10 m

a) What values of $\theta$ will we measure X-rays at a detector, that is, when will there be constructive interference (assume elastic collision)?

b) If we use electrons instead of X-rays, at which energy should they be to detect the electrons at the same angle as the X-rays (assume elastic collision)?

Homework Equations

Bragg's Law:
$2 d sin(\theta) = m \lambda$
E = hf
p = h / $\lambda$
Compton effect

The Attempt at a Solution

For part (a) what I did was basically use:

$sin(\theta) = (1 \lambda) / (2d)$
$sin(\theta) = (2 \lambda) / (2d)$

Since $sin(\theta) = (2 \lambda) / (2d)$ gives 90 degrees, my answer is that the only angle is 30 degrees, that is, m = 1.

For part (b) I'm not really sure how to approach this question. Is this just the classical model of an elastic collision?

Thanks.

 

[anathema]

Dear student,

For part (a), your approach is correct. The only angle at which there will be constructive interference is 30 degrees, as this satisfies the Bragg's Law equation. However, it is important to note that this is the ideal case and in reality, there may be some deviation from this angle due to imperfections in the sample or experimental setup.

For part (b), you are correct in thinking that this involves the classical model of an elastic collision. In this case, the energy of the electrons can be calculated using the equation E = p^2 / (2m), where p is the momentum of the electron and m is its mass. Since we know the wavelength of the electrons (10^-10 m) and the angle at which they should be detected (30 degrees), we can calculate the momentum using the equation p = h / \lambda * sin(\theta). Once we have the momentum, we can use it to calculate the energy of the electrons.

I hope this helps. Good luck on your exam!