Questions regarding 2D reciprocal lattices translation vectors

Question

If I have a reciprocal net of a rectangular real space lattice with lattice parameters $a = 0.4$ nm and $b = 0.5 $ nm along $x$ and $y$. I am trying to find which are true (I have never had a background in surface science so this is all quite new to me)

1. the net is rectangular

2. has $a^* < b^*$

3. has a reciprocal lattice point with index $(20)$ with magnitude $10π$

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Considering the space lattices are rectangular, I presumed that the net of the lattices would consequently be rectangular.

Is this a correct assumption to make? What characterises the shape of the net?

For the second statement, I know that the primitive translation vectors, $a^*$ and $b^*$, are related to the primitive translation vectors of the real space lattices, $a$ and $b$ by:

$$a^* = 2\pi \frac{b \times n}{|a \times b|}\hspace{5mm} b^*= 2\pi\frac{n \times a}{|a \times b|}$$ where $n$ is a unit vector normal to the surface.

Considering this last statement I believe the value for $n$ will be $n=1$ and that therefore the values are: $a^*= 5\pi$ and $b=4\pi$. Making the second statement incorrect.

Is this take on the value of n correct? I am afraid that somehow these should be matrices but I don't see how?

Could someone give me a good source of information for learning regarding the last statement. I am currently using Modern Techniques of surface Science and Surface Science - An Introduction but I don't seem to find anything to clear all these doubts.

Answer 1

The lattice vector with index (20) is a vector $ \vec{G} =2\cdot \vec{a*} + 0 \cdot \vec{b*} $ so its magnitude is indeed 10$\pi $ assuming that your calculations for the unit vectors of the reciprocal lattice are correct. In my opinion you did the right thing. It's just the standard definition of the reciprocal lattice vectors used in solid state physics. The most famous reference in the field is Kittel's book "Introduction to Solid State Physics". Look it up on the internet. It is not the best book to explain things but is a great reference. Any book on Solid state physics will include the definition of the reciprocal lattice. It is not a matter specific to surface physics. The formula is usualy given for a 3D lattice. In order to adapt them for 2D you just assume a third lattice vector of magnitude 1, perpendicular to the plane defined by the other two vectors. This is your unit normal vector. Then the formulas for the 2D case follow right away. You can find the formulas for both 2D and 3D case in the wiki article about reciprocal lattice.