Why Δk is in that direction? 
Hello, can anybody explain to me why Δk is in that direction?
Can you explain it using the diffraction/scattering amplitude ?
Thank you!
 A: Here is how I like to see it. I don't like using Ewald's sphere to discuss it directly. Before that I like to use simple diffraction mathematics.
Considered a lattice structure with lattice translation vector $\vec{R}_{m,n}$ and basis vectors $\vec{R}_1$ and $\vec{R}_2$. So that I can write
$$\vec{R}_{m,n}=m\vec{R}_1+n\vec{R}_2\quad\forall m,n\in\mathbb{Z}$$
Now imagine an electron beam directed towards it and the electrons diffracted back from every lattice site on the structure. The assumption made is the diffracted electrons from every lattice site travels parallel to each other and hence hold a wave vector $\vec{k}$ each, since there propagation directions are parallel.
The diffracted wave from the $(m,n)^{\text{th}}$ lattice site can easily be said to be 
$$y(\vec{r})=A\exp{\left(-i\vec{k}\cdot\left(\vec{r}-\vec{R}_{m,n}\right)\right)}$$
Since there are diffracted waves from each lattice site which interferes with each other, hence the total diffracted wave $y_T(\vec{r})$ can be written as
$$y_T(\vec{r})=\sum_{m,n}A\exp{\left(-i\vec{k}\cdot\left(\vec{r}-\vec{R}_{m,n}\right)\right)}=A\sum_{m,n}e^{-i\vec{k}\cdot\vec{r}}\exp{\left(-i\vec{k}\cdot\vec{R}_{m,n}\right)}$$
$$=Ae^{-i\vec{k}\cdot\vec{r}}\sum_{m,n}\exp{\left(-i\vec{k}\cdot\vec{R}_{m,n}\right)}$$
This expression $y_T(\vec{r})$ is max when $$\vec{k}\cdot\vec{R}_{m,n}=2\pi N\quad\forall m,n,N\in\mathbb{Z}$$
This is your condition for constructive interference (the bright points you see on your LEED screen).
Now to solve the above two equations, two basis vectors $\vec{k_1}$ and $\vec{k_2}$ are selected for the vector $\vec{k}$. Thus $\vec{k}$ can be written as 
$$\vec{k}=\lambda_1\vec{k_1}+\lambda_2\vec{k_2}\quad\forall \lambda_1,\lambda_2\in\mathbb{R}$$
Expanding the inner product/dot product we get
$$m\lambda_1\vec{k_1}\cdot\vec{R}_1+n\lambda_1\vec{k_1}\cdot\vec{R_2}+m\lambda_2\vec{k_2}\cdot\vec{R}_1+n\lambda_2\vec{k_2}\cdot\vec{R}_2=2\pi N$$
To solve this, we select the vectors $\vec{k_1}$ and $\vec{k_2}$ to be perpendicular to $\vec{R}_2$ and $\vec{R}_1$ respectively. Thus the conditions
$$\vec{k_1}\cdot\vec{R}_2=0\quad\quad\vec{k_2}\cdot\vec{R}_1=0 \quad\text{hold}$$
The above equation on the other hand becomes
$$m\lambda_1\vec{k_1}\cdot\vec{R}_1+n\lambda_2\vec{k_2}\cdot\vec{R}_2=2\pi N$$
To solve this both the dot products/inner products are set to $2\pi$ and the $\lambda$ values are restricted to integers as shown below.
$$\vec{k_1}\cdot\vec{R}_1=2\pi\quad\quad\vec{k_2}\cdot\vec{R}_2=2\pi$$
$$\lambda_1=p,\lambda_2=q\quad\forall p,q\in\mathbb{Z}$$
Thus the intensity maxima is resticted to a very few points on your diffraction/LEED screen and the $\vec{k}$ vector for your $(p,q)^{\text{th}}$ diffraction spot is given by
$$\vec{k}_{p,q}=p\vec{k_1}+q\vec{k_2}$$.
This discretization of your $\vec{k}$ space is equivalent to your reciprocal lattice space. In fact reciprocal space is defined under the conditions
$$\vec{k_i}\cdot\vec{R_j}=2\pi\delta_{i,j}$$
Herein $\delta_{i,j}$ is the Kronecker Delta function.
Now the $\Delta k$ that you are talking about can be in any direction, not just the one you mentioned, but you need to remember that $\Delta k$ give you $\vec{k}_{p,q}$ for some $(p,q)$ and not $\vec{k_1}$ or $\vec{k_2}$ precisely. So, the $\Delta k$ that you are talking about is I think subject to whichever $\Delta k$ you want to use.
