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Suppose I have a crystal of which I like to determine the distance $d$ between the lattice planes. We can use a Bragg diffraction experiment to do so by making a plane wave of X-ray radiation with wavelength $\lambda$ incident on our crystal. It is easily derived that

$$ n\lambda = 2d\sin(\theta) $$

should hold, where $\theta$ is the angle between the crystal plane and incident radiation. If we know $\lambda$ and $\theta$, we can find $d$.

Since I do not know what the inside of the crystal looks like, I do not know how the crystal plane is oriented, thus I do not know what $\theta$ is.

Question How does an experimenter know $\theta$?

Scheme of a Bragg diffraction experiment

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If you have a single crystal, you can probably tell the orientation of the crystal plane(s) and then move both the X-ray source and the detector symmetrically so $\theta_\text{left}$ from the source and $\theta_\text{right}$ to the detector are equal.

For educational purposes there are devices that rotate the crystal and the detector simultaneously, assuring $\theta_\text{left} = \theta_\text{right}$: enter image description here

Your equation holds for the angle of constructive interference, so when you detect a peak in intensity, you have found $\theta = \theta_\text{left} = \theta_\text{right}$.

Another method is known as powder diffraction where you put a fine powder of the crystal in the (monochromatic) X-ray beam. Now the crystal planes are oriented at random angles, so there will be crystals creating the angle $\theta$ for constructive interference that can then be detected by a photographic film placed around the sample.

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  • $\begingroup$ "Your equation holds for the angle of constructive interference, so when you detect a peak in intensity, you have found $\theta$", I think this answers it! I just rotate my sample until I observe constructive interference. I then use the positions of the detector, source and sample to find $\theta$. $\endgroup$ – Meneer-Beer Nov 22 '17 at 20:33

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