Solve X-ray Reflection Homework: Find Crystal Plane Spacing

[roam]

Homework Statement

X-rays of wavelength 0.09 nm are incident on a crystal. A reflected beam is observed at 36 degrees from the incident beam. What is the spacing of the planes in the crystal?

Homework Equations

Bragg's law:

[itex]2 d \ sin \theta = m \lambda[/itex]

The Attempt at a Solution

I thought this is a simple application of Bragg's law but I get

[itex]2 d \ sin \ 36 = 0.09 \times 10^{-9} \implies d=7.65 \times 10^{-11}[/itex]

But the answer should be 4.73 x 10^-11 m. What's wrong with my calculation?

 

[Simon Bridge]

Draw a picture of the situation.

You have been given the angle between the incident and reflected beams. That is not the angle indicated by the theta in Bragg's law though is it?
http://en.wikipedia.org/wiki/Bragg's_law

 

[roam]

Draw a picture of the situation.

You have been given the angle between the incident and reflected beams. That is not the angle indicated by the theta in Bragg's law though is it?
http://en.wikipedia.org/wiki/Bragg's_law

I see, thank you. It makes perfect sense now. Thanks!

 

[Simon Bridge]

Well done :) No worries.

 

[asteriatic]

As a scientist, it is important to carefully review your calculations and make sure all variables and units are accounted for. In this case, it appears that you have made a small error in your calculation. The correct equation to use for Bragg's law is 2d sinθ = mλ, where d is the spacing of the planes, θ is the angle of incidence, m is the order of the reflection, and λ is the wavelength of the incident X-rays. In your attempt, you have mistakenly used the value of λ as the coefficient for sinθ, instead of the wavelength itself. This has resulted in a slightly larger value for d.

The correct calculation would be:

2d sin36 = (1)(0.09 x 10^-9)

d = 0.045 x 10^-9 m = 4.5 x 10^-11 m

This is much closer to the given answer of 4.73 x 10^-11 m. It is also important to make sure that all units are consistent, so be sure to convert all values to meters before plugging them into the equation.

In summary, it is important to double-check your calculations and equations to ensure accuracy in your scientific work.