Inquiry regarding Ashcroft and Mermin's page 365

[MathematicalPhysicist]

I'll write the passage from the text which doesn't seem to follow.

Suppose, furthermore, that the crystal surface is a lattice plane perpendicular to the reciprocal lattice vector $b_3$.
Choose a set of primitive vectors including $b_3$ for the reciprocal lattice, and primitive vectors $a_i$ for the direct lattice, satisfying:
(18.24) $a_i\cdot b_j = 2\pi \delta_{ij}$.

If the electron beam penetrates so little that only scattering from the surface plane is significant, then the condition for constructive interference is that the change $q$ is the wave vector of the scattered electron satisfy:

(18.25) $q\cdot d = 2\pi\times integer, \ \ \ \ \ q=k'-k$
for all vectors $d$ joining lattice points in the plane of the surface (cf. Eq. (6.5)).
Since such $d$ are perpendicular to $b_3$, they can be written as:
(18.26) $d = n_1a_1+n_2a_2$.

Writing $q$ in the general form:
(18.27) $q= \sum_{i=1}^3 q_i b_i ,$

we find that conditions (18.25) and (18.26) require:
(18.28) $q_1=2\pi \times integer$
$q_2 = 2\pi \times integer$
$q_3 = arbitrary$

For q_1 and q_2 I get that they should be integers without the multiple of $2\pi$.

Here's my reasoning:
If I plug (18.27) back into the scalar product in (18.25) and also (18.26) and use the identity of (18.24) connecting $a_i$ and $b_j$, I get:
$$2\pi \times integer = q_1n_1a_1\cdot b_1+q_2n_2a_2\cdot b_2 = 2\pi(q_1n_1+q_2n_2)$$

Divide by $2\pi$ and get: $integer = q_1n_1+q_2n_2$.
So $q_i \in \mathbb{Q}$.

How did they get relation (18.28)?

Thanks.

 

[Cthugha]

You need to pay attention to the fact that 18.25 is valid for

all vectors d joining lattice points in the plane of the surface.

This means that it must also be valid for vectors with $n_1=0$ or $n_2=0$.

 

[MathematicalPhysicist]

You need to pay attention to the fact that 18.25 is valid for
This means that it must also be valid for vectors with $n_1=0$ or $n_2=0$.

Ok, this implies that $q_i\in \mathbb{Z}$, but they write that $q_i \in 2\pi \mathbb{Z}$; why is that?
For $i\in \{ 1,2\}$ obviously.

 

[hutchphd]

How did they get relation (18.28)?

I believe these are just Laue indices by this definition and should indeed be integers without the 2pi.