# Solid State: Diffraction Conditions

1. Sep 11, 2009

### Itserpol

In my Solid State course we've reached the topic of crystal diffraction and reciprocal lattices. I haven't had any problems so far, but I've hit a little snag in understanding how the diffraction condition $$2\vec{k} \cdot \vec{G}=G^2$$ is equivalent to the Bragg law $$2d\sin{\theta}=n\lambda$$.

In the textbook (Introduction to Solid State Physics - Kittel 8th Ed.), the math is as follows:

1. The reciprocal lattice vector $$\vec{G}=h\vec{b}_1+k\vec{b}_2+l\vec{b}_3$$ is normal to the lattice plane given by the indices $$(hkl)$$.
2. The spacing between these planes is $$d_{hkl}=\tfrac{1}{h}\hat{n}\cdot\vec{a}_1=\tfrac{1}{Gh}\vec{G}\cdot\vec{a}_1=\tfrac{2\pi}{G}$$ where $$\vec{a}_1$$ is a primitive axis for the direct lattice.
3. The diffraction condition $$2\vec{k} \cdot \vec{G}=G^2$$ can then be written as $$2k\sin{\theta}=2(\tfrac{2\pi}{\lambda})\sin{\theta}=\tfrac{2\pi}{d_{hkl}} \Rightarrow 2d_{hkl}\sin{\theta}=\lambda$$. Here, $$\theta$$ is the angle between $$\vec{k}$$ and the plane $$(hkl)$$, not between $$\vec{k}$$ and $$\vec{G}$$.
4. Somehow $$d_{hkl}=\tfrac{d}{n}$$, so this becomes $$2d\sin{\theta}=n\lambda$$ which is just the Bragg law.

Now, most of this I understand. The only things I don't get are why the lattice plane spacing $$d_{hkl}$$ is equal to $$\tfrac{1}{h}\hat{n}\cdot\vec{a}_1$$ and what the relationship $$d_{hkl}=\tfrac{d}{n}$$ means. What's the difference in physical meaning between $$d_{hkl}$$ and $$d$$?