# Dipole moment in a bulk crystal

When calculating the dipole moment in a crystal, there is always one term neglected. This neglection is never discussed and I do not understand it at all.

Assuming the Bloch functions $$\Psi_{n,\vec{k}}=\frac{1}{\sqrt{N V_\text{uc}}}e^{i \vec{k}\cdot\vec{r}}u_{n,\vec{k}}(\vec{r})$$, the dipole moment in a crystal is given via \begin{align} \int_{\rm I\!R^3} d^3r' \quad \Psi_{n',\vec{k}'}^*(\vec{r}')\,\vec{r}'\,\Psi_{n,\vec{k}}(\vec{r}') =& \frac{1}{N V_\text{uc}}\sum_{\vec{R}} \int_{\text{uc}} d^3r \quad e^{i (\vec{k}-\vec{k}')\cdot (\vec{R}+\vec{r})} \,(\vec{R}+\vec{r})\, u_{n',\vec{k}'}^*(\vec{r})u_{n,\vec{k}}(\vec{r}) \\ =& \frac{1}{N} \sum_{\vec{R}} e^{i (\vec{k}-\vec{k}')\cdot \vec{R}} \frac{1}{V_\text{uc}} \int_{\text{uc}} d^3r \quad e^{i (\vec{k}-\vec{k}')\cdot \vec{r}} \vec{r} u_{n',\vec{k}'}^*(\vec{r})u_{n,\vec{k}}(\vec{r}) \\ &+ \frac{1}{N} \sum_{\vec{R}} e^{i (\vec{k}-\vec{k}')\cdot \vec{R}} \vec{R} \frac{1}{V_\text{uc}} \int_{\text{uc}} d^3r \quad e^{i (\vec{k}-\vec{k}')\cdot \vec{r}} u_{n',\vec{k}'}^*(\vec{r})u_{n,\vec{k}}(\vec{r}) \end{align}

Here $$N$$ is the number of unit cells and $$\vec{R}$$ all lattice vectors.

The first term is easily solved via $$\frac{1}{N}\sum_{\vec{R}} e^{i (\vec{k}-\vec{k}')\cdot \vec{R}}=\delta_{\vec{k},\vec{k}'}$$, whereafter we have the same $$\vec{k}$$ in the two functions $$u$$, set $$e^{i (\vec{k}-\vec{k}')\cdot \vec{r}}=1$$ and solve the remaining integral over the unit cell, which give the typical dipole moments describing interband transitions $$n\to n'$$. This term is raughly on the size scale of the unit cell, as we have $$\vec{r}$$ within the integral over the unit cell.

My question is about the second term. Its size scale is on the order of the dimension of the crystal, as $$\vec{R}$$ is summed over all unit cells of the crystal. Lets try so solve this using a Taylor expansion around $$\vec{r}=0$$ on the $$e^{i (\vec{k}-\vec{k}')\cdot \vec{r}}=1+i (\vec{k}-\vec{k}')\cdot \vec{r}+...$$ within the integral over the unit cell. In first order this is equivalent to the application of the envelope function approximation. We get intraband transitions $$n\to n$$, which are typically neglected in my area of semiconductor physics (quantum dots). However, the really interesting part is the second order term of the Taylor expansion, which gives us the expression \begin{align} \frac{1}{N} \sum_{\vec{R}} e^{i (\vec{k}-\vec{k}')\cdot \vec{R}} i \vec{R} (\vec{k}-\vec{k}') \cdot \frac{1}{V_\text{uc}} \int_{\text{uc}} d^3r \quad \vec{r} u_{n',\vec{k}'}^*(\vec{r})u_{n,\vec{k}}(\vec{r}) \end{align} This term provides, as the first term, typical interband transitions $$n'\to n$$ in the microscopic integral. This term is also of the order of the unit cell (as $$\vec{k} \vec{R}\approx 1$$). Why is this term always neglected?