Chemistry of solids: allowed wavevectors

Question

I'm a complete beginner in the chemistry of solids, and many of the things I came across don't make sense. I have two subquestions, which I think it would be most logical to post as a single question.

• The longest wavevector of a commensurate matter wave is $k_{max} = \pi / d$, where d – lattice-plane spacing, so that amplitude minima and maxima of the wave coincide with adjacent lines / planes of atoms. I don't understand this explanation – why is, for example, $k = 2\pi / d$ inconsistent?

• Bloch's equations dictate that the value of the electron wavevector isn't uniquely defined, and the vector is uncertain to modulo G, where G – a general Reciprocal Lattice (RL) vector. This means that wavevectors greater than $\pi / d$ are permitted and can be found in Brillouin's zones of order greater than 1 in the RL. This seems to be in contradiction to the point above – what am I missing?

Answer 1

In a crystal, electron waves are reflected off the periodic lattice and this (Bragg) reflection is the cause of the energy gaps also called a band structure. If there is a weak periodic potential in a 1D crystal then the energy gaps occur at wavevector $k=\pm \pi/d$ where d is the lattice constant.

If $\bar G$ represents the reciprocal lattice vector then the general Bragg condition for diffraction of a wave with wavevector $\vec K$ can be written as $(\vec k + \vec G)^2=k^2 $

In one dimension this condition becomes

$$k=\pm \frac{1}{2}G = \pm n\pi/d$$

and $G=2\pi n/d$ is a reciprocal lattice vector and $n=1,2,3..$ an integer. This is where your factor of 2 appears.

The reflection at $k=\pm \pi/d$ occurs because one wave is reflected of an atom and interferes constructively with phase difference $2\pi$ with one reflected from the nearest neighbour atom. The Brillouin zone is the $\bar k$ space between $-\pi/d$ and $+\pi/d$.

(The reciprocal lattice vectors have the form $\vec G =h\vec A+k\vec B + l\vec C$ for integers h, k, l. The vectors are defined as

$$\vec A= 2\pi \frac{b~ \times~ c}{a\cdot b~ \times~ c}~~ \vec B= 2\pi \frac{c~ \times~ a}{a\cdot b~ \times~ c}~~ \vec C= 2\pi \frac{a~ \times~ b}{a\cdot b~ \times~ c}$$ for primitive lattice vectors a, b, c)