I can’t see why there remains a term 1/a in the exponent. Using the primitive lattice vectors for \delta\ \neq 0 and the following basis: (0,0,0), (1/2,0,1/2), (0, 1/2, 1/2), (1/2, 1/2, \delta) I constructed the following vectors:
- \vec{r_0}=\vec{0}
- \vec{r_1}=(\vec{a_1}+\vec{a_2})/2
- \vec{r_2}=(\vec{a_2}+\vec{a_3})/2
- \vec{r_3}=\vec{a_1}/2+\vec{a_2}/2+\vec{a_3}\delta
When using: \vec{G}=h\vec{b_1}+k\vec{b_2}+l\vec{b_3} and: a_i\cdot b_j = 2 \pi \delta_{ij} I get the following expression:
S(\vec{G})=\sum_j f_j e^{i(\vec{r_j} \cdot \vec{G})}=f\Big(1+e^{\pi i(h+l)}+e^{\pi i(k+l)}+e^{\pi i(h+k)} e^{2 \pi \delta l}\Big)
Which is missing the factor 1/a in the exponent.
