Alternate Derivation
Suppose that a single monochromatic wave (of any type) is incident on aligned planes of lattice points, with separation, at angle . Points A and C are on one plane, and B is on the plane below. Points ABCC' form a parallelogram.
There will be a path difference between the ray that gets reflected along AC' and the ray that gets transmitted, then reflected, along AB and BC respectively. This path difference is
The two separate waves will arrive at a point with the same phase, and hence undergo constructive interference, if and only if this path difference is equal to any integer value of the wavelength, i.e.
where the same definition of and apply as above.
Therefore,
from which it follows that
Putting everything together,
which simplifies to
which is Bragg's law.
Read more about this topic: Bragg's Law
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