Solid State: Diffraction Conditions

[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 $$d$$?

 

[_coyote_]

The relationship d_{hkl}=\tfrac{1}{h}\hat{n}\cdot\vec{a}_1 is due to the fact that the reciprocal lattice vector \vec{G} is normal to the lattice plane given by the indices (hkl). This means that the lattice plane spacing d_{hkl} is equal to the magnitude of the projection of \vec{a}_1 onto \vec{G}, which is equal to \tfrac{1}{h}\hat{n}\cdot\vec{a}_1, where \hat{n} is a unit vector normal to \vec{G}.The relationship d_{hkl}=\tfrac{d}{n} means that for any given lattice plane (hkl), the spacing between successive planes is equal to the spacing between the same plane in successive unit cells, divided by the number of unit cells in that plane. That is, the distance between two adjacent planes of (hkl) index is equal to n times the distance between the same plane in successive unit cells. The physical meaning here is that the distance between two adjacent planes of (hkl) index is equal to the average distance between the same plane in successive unit cells.