How is Bragg's Law Derived Using the Diffraction Condition?

[James Marquez]

Hello. I am reading "Introduction to Solid State Physics" by Kittel and there is a derivation in the textbook that I am understanding. This should be a fairly simple question but I am unable to see it.

1. Homework Statement
In Chapter 2, it derives the Bragg law using the diffraction condition and equation for spacing between parallel lattice planes. The Bragg law is given by:
$$2d\sin{\theta}=n\lambda$$

Homework Equations

The Diffraction Condition:
$$2\textbf{k}\bullet\textbf{G}=G^2$$

Spacing Between Parallel Lattice Planes:
$$d=\frac{2\pi}{G}$$

Wave Vector:
$$\textbf{k}=\frac{2\pi}{\lambda}\hat{\textbf{k}}$$

The Attempt at a Solution

I just don't see how we get the Bragg law using these two equations? I rearranged it so that
$$G=\frac{2\pi}{d}$$
But given by the diffraction condition, I see that:
$$G=2\frac{2\pi}{\lambda}\cos{\theta}$$
$$\frac{1}{d}=\frac{2\cos{\theta}}{\lambda}$$

I don't understand how the textbook arrives at that conclusion. Thank you.

 

[Charles Link]

I think the answer is that $G=(\frac{2 \pi}{d}) m$, where $m$ is an integer, defines the reciprocal lattice vectors for a cubic lattice. $\\$ Editing... That will not give them all because the reciprocal lattice vectors are found from the lattice vectors for the primitive cell of the reciprocal lattice by having integer numbers of primitive basis vectors in each direction... If you look at the previous page or two in Kittel, he defines a reciprocal lattice vector $G=h \vec{A}+k\vec{B}+l \vec{C}$ , $(h,k,l=$ integers$)$.