2.3.2.3 Out-Scattering and In-Scattering Terms

It is convenient to define a quantity $(\partial f(\vec{r},\vec{k},t)/\partial t)_{out}$ to express the electron number per unit volume with quasi-momenta in the infinitesimal volume $d\vec{k}$ around $\vec{k}$ and which have been scattered during the infinitesimal time interval $dt$:

$$\biggl(\frac{\partial f(\vec{r},\vec{k},t)}{\partial t}\biggr)_\mathrm{out}\frac{d\vec{k}}{(2\pi)^{3}}dt.$$ (2.50)

As the volume $d\vec{k}$ is infinitesimal, the scattering results in an electron being removed from this volume. Therefore (2.50) can also be considered as the number of electrons which are lost from the volume $d\vec{k}$ around $\vec{k}$ during the time interval $dt$ due to scattering.

The quantity $(\partial f(\vec{r},\vec{k},t)/\partial t)_\mathrm{out}$ can be found from the fact that the expression

$$dt\int (1-f(\vec{r},\vec{k}^{'},t))S(\vec{k},\vec{k}^{'},\vec{r},t)\,d\vec{k}^{'}$$

is the probability that any electron from the vicinity of point $\vec{k}$ has been scattered during the time interval $dt$ and thus the total number of the scattered electrons in $d\vec{k}$ around $\vec{k}$ is equal to

$$f(\vec{r},\vec{k},t)\frac{d\vec{k}}{(2\pi)^{3}}dt\int (1-f(\vec{r},\vec{k}^{'},t))S(\vec{k},\vec{k}^{'},\vec{r},t)\,d\vec{k}^{'}.$$ (2.51)

Comparison with (2.50) gives

$$\biggl(\frac{\partial f(\vec{r},\vec{k},t)}{\partial t}\biggr)_\m... ...nt (1-f(\vec{r},\vec{k}^{'},t)) S(\vec{k},\vec{k}^{'},\vec{r},t)\,d\vec{k}^{'},$$ (2.52)

where the minus sign shows that this quantity describes the loss of electrons.

Scattering processes can change the distribution function in the opposite way. In addition to the scattering out of the domain $d\vec{k}$ there also exist scattering processes leading to a gain of electrons in $d\vec{k}$. To describe these processes it is natural to introduce the quantity $(\partial f(\vec{r},\vec{k},t)/\partial t)_\mathrm{in}$ defined so that the expression

$$\biggl(\frac{\partial f(\vec{r},\vec{k},t)}{\partial t}\biggr)_\mathrm{in}\frac{d\vec{k}}{(2\pi)^{3}}dt$$ (2.53)

gives the number of electrons per unit volume which are scattered into the volume $d\vec{k}$ around $\vec{k}$ during the infinitesimal time interval $dt$. In order to find $(\partial f(\vec{r},\vec{k},t)/\partial t)_\mathrm{in}$ it is necessary to consider electrons in $d\vec{k}^{'}$ near $\vec{k}^{'}$ which are scattered into $d\vec{k}$ and sum over all possible $\vec{k}^{'}$. The total number of electrons in $d\vec{k}^{'}$ is equal to $f(\vec{r},\vec{k}^{'},t)\,d\vec{k}^{'}/(2\pi)^{3}$. From this number of electrons only $S(\vec{k}^{'},\vec{k},\vec{r},t)\,dtd\vec{k}$ would be scattered into $d\vec{k}$ around $\vec{k}$ during $dt$ if the corresponding states were not occupied. However only the fraction $1-f(\vec{r},\vec{k},t)$ of the states are available. Thus, the total number of electrons per unit volume scattered into $d\vec{k}$ around $\vec{k}$ from $d\vec{k}^{'}$ around $\vec{k}^{'}$ during $dt$ is equal to

$$f(\vec{r},\vec{k}^{'},t)\frac{d\vec{k}^{'}}{(2\pi)^{3}}S(\vec{k}^{'},\vec{k},\vec{r},t)\,d\vec{k}dt(1-f(\vec{r},\vec{k},t)).$$ (2.54)

Summing over all possible $\vec{k}^{'}$ and comparing with (2.53) gives:

$$\biggl(\frac{\partial f(\vec{r},\vec{k},t)}{\partial t}\biggr)_\m... ...)) \int S(\vec{k}^{'},\vec{k},\vec{r},t)f(\vec{r},\vec{k}^{'},t)\,d\vec{k}^{'}.$$ (2.55)

Now the collision integral in (2.46) can be expressed as a sum of two terms:

$$\biggl(\frac{\partial f(\vec{r},\vec{k},t)}{\partial t}\biggr)_\m... ...}+ \biggl(\frac{\partial f(\vec{r},\vec{k},t)}{\partial t}\biggr)_\mathrm{out}=$$

$$=(1-f(\vec{r},\vec{k},t))\int S(\vec{k}^{'},\vec{k},\vec{r},t)f(\vec{r},\vec{k}^{'},t)\,d\vec{k}^{'}-$$ (2.56)

$$-f(\vec{r},\vec{k},t)\int (1-f(\vec{r},\vec{k}^{'},t))S(\vec{k},\vec{k}^{'},\vec{r},t)\,d\vec{k}^{'}.$$

It should be noted that in the non-degenerate case when $f(\vec{r},\vec{k},t)\ll 1$ the scattering operator can be rewritten as:

$$\biggl(\frac{\partial f(\vec{r},\vec{k},t)}{\partial t}\biggr)_\m... ...},\vec{k}^{'},t)\,d\vec{k}^{'}- f(\vec{r},\vec{k},t)\lambda(\vec{r},\vec{k},t),$$ (2.57)

where the total scattering rate $\lambda(\vec{r},\vec{k},t)$ is defined as follows:

$$\lambda(\vec{r},\vec{k},t)=\int S(\vec{k},\vec{k}^{'},\vec{r},t)\,d\vec{k}^{'}.$$ (2.58)

The Boltzmann equation takes now the form:

$$\frac{\partial f(\vec{r},\vec{k},t)}{\partial t}+\vec{v}(\vec{k})... ...ar}\vec{F}(\vec{r})\cdot\frac{\partial f(\vec{r},\vec{k},t)}{\partial \vec{k}}=$$

$$=(1-f(\vec{r},\vec{k},t))\int S(\vec{k}^{'},\vec{k},\vec{r},t)f(\vec{r},\vec{k}^{'},t)\,d\vec{k}^{'}-$$ (2.59)

$$-f(\vec{r},\vec{k},t)\int (1-f(\vec{r},\vec{k}^{'},t))S(\vec{k},\vec{k}^{'},\vec{r},t)\,d\vec{k}^{'}.$$

S. Smirnov: