What Are the First Five Bragg Scattering Angles for Cu X-ray Analysis?

[fatherdaly]

Homework Statement

A thin polycrystalline film of Cu, with lattice parameter of 0.361 nm, is irradiated with X-rays with wavelength of 0.154 nm. Calculate the first 5 bragg scattering angles for which X-rays may be detected.

Homework Equations

2dsin(\theta) = n\lambda

Bragg condition for constructive interference.

n is an integer.

The Attempt at a Solution

That seems the obvious equation to be using here. I rearrange it for theta

\theta = arcsin(n\lambda/2d)

Like that. Then I put in the numbers for n = 1, 2, 3 etc

For n=1 I get, 12.3 ish degrees

For n=2, 25.3 degrees

n=3, 39.8 degrees

n=4, 58.6

n=5, well you can't arcsin something that's >1.

Therein lies the problem. Also I have access to the answers and they're not the same as mine. I can provide them if anyone wants to know.

Please help. Not even google is able to provide answers.

 

[kloptok]

You have to consider that there can be several scattering angles for n=1 (I would say definitely 5 angles) for different Miller indices. There are restrictions on which Miller indices are allowed depending on the crystal structure. The d in the Bragg equation is in general NOT the lattice constant a, but another number (a function of lattice constant and Miller indices).

The X-rays can scatter from different planes in the crystal for the same integer n, giving rise to different scattering angles. Therein lies the answer to your problems.

You can probably find more info about Bragg diffraction in your Solid State Physics book, or you can look up 'Bragg diffraction' on Wikipedia.

 

[fatherdaly]

Thanks a lot for a fast reply. Yeah I've read up a bit on the allowed hkl values for copper which is FCC structure. I guess I'll play with the numbers until I get something similar to the answer.

 

[kloptok]

You can find the restrictions on the Miller indices by calculating the structure factor S. The intensity of the reflected beams is proportional to |S|². As it turns out, for an fcc structure the structure factor will be zero for some Miller indices leading to restrictions on which Miller indices are allowed. As it turns out, all Miller indices must be even or all odd for a reflection to occur. For all cases when the Miller indices aren't all odd or all even the intensity of the reflected wave will be zero.