The equations of motion of the atoms of mass M which locates in the periodic lattice is

${\displaystyle M{\frac {d^{2}}{dt^{2}}}u_{n}=-k_{0}(u_{n-1}+u_{n+1}-2u_{n})}$ ,

where ${\displaystyle u_{n}}$  is the displacement of the nth atom from their equilibrium positions.

Defining the displacement ${\displaystyle u_{\ell }}$  of the ${\displaystyle \ell }$ th atom by ${\displaystyle u_{\ell }=x_{\ell }-\ell a}$ , where ${\displaystyle x_{\ell }}$  is the coordinates of the ${\displaystyle \ell }$ th atom and ${\displaystyle a}$  is the lattice constant,

the displacement is given by ${\displaystyle u_{l}=Ae^{i(q\ell a-\omega t)}}$ 

Then using Fourier transform:

${\displaystyle Q_{q}={\frac {1}{\sqrt {N}}}\sum _{\ell }u_{\ell }e^{-iqa\ell }}$ 

and

${\displaystyle u_{\ell }={\frac {1}{\sqrt {N}}}\sum _{q}Q_{q}e^{iqa\ell }}$ .

Since ${\displaystyle u_{\ell }}$  is a Hermite operator,

${\displaystyle u_{\ell }={\frac {1}{2{\sqrt {N}}}}\sum _{q}(Q_{q}e^{iqa\ell }+Q_{q}^{\dagger }e^{-iqa\ell })}$ 

From the definition of the creation and annihilation operator ${\displaystyle a_{q}^{\dagger }={\frac {q}{\sqrt {2M\hbar \omega _{q}}}}(M\omega _{q}Q_{-q}-iP_{q}),\;a_{q}={\frac {q}{\sqrt {2M\hbar \omega _{q}}}}(M\omega _{q}Q_{-q}+iP_{q})}$ 

${\displaystyle Q_{q}}$  is written as

${\displaystyle Q_{q}={\sqrt {\frac {\hbar }{2M\omega _{q}}}}(a_{-q}^{\dagger }+a_{q})}$ 

Then ${\displaystyle u_{\ell }}$  expressed as

${\displaystyle u_{\ell }=\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}(a_{q}e^{iqa\ell }+a_{q}^{\dagger }e^{-iqa\ell })}$ 

Hence, using the continuum model, the displacement operator for the 3-dimensional case is

${\displaystyle u(r)=\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}e_{q}[a_{q}e^{iq\cdot r}+a_{q}^{\dagger }e^{-iq\cdot r}]}$ ,

where ${\displaystyle e_{q}}$  is the unit vector along the displacement direction.

The electron-longitudinal acoustic phonon interaction Hamiltonian is defined as ${\displaystyle H_{\text{el}}}$ 

${\displaystyle H_{\text{el}}=D_{\text{ac}}{\frac {\delta V}{V}}=D_{\text{ac}}\,\mathop {\rm {div}} \,u(r)}$ ,

where ${\displaystyle D_{\text{ac}}}$  is the deformation potential for electron scattering by acoustic phonons.^[1]

Inserting the displacement vector to the Hamiltonian results to

${\displaystyle H_{\text{el}}=D_{\text{ac}}\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}(ie_{q}\cdot q)[a_{q}e^{iq\cdot r}-a_{q}^{\dagger }e^{-iq\cdot r}]}$ 

The scattering probability for electrons from ${\displaystyle |k\rangle }$  to ${\displaystyle |k'\rangle }$  states is

${\displaystyle P(k,k')={\frac {2\pi }{\hbar }}\mid \langle k',q'|H_{\text{el}}|\ k,q\rangle \mid ^{2}\delta [\varepsilon (k')-\varepsilon (k)\mp \hbar \omega _{q}]}$ 

${\displaystyle ={\frac {2\pi }{\hbar }}\left|D_{\text{ac}}\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}(ie_{q}\cdot q){\sqrt {n_{q}+{\frac {1}{2}}\mp {\frac {1}{2}}}}\,{\frac {1}{L^{3}}}\int d^{3}r\,u_{k'}^{\ast }(r)u_{k}(r)e^{i(k-k'\pm q)\cdot r}\right|^{2}\delta [\varepsilon (k')-\varepsilon (k)\mp \hbar \omega _{q}]}$ 

Replace the integral over the whole space with a summation of unit cell integrations

${\displaystyle P(k,k')={\frac {2\pi }{\hbar }}\left(D_{\text{ac}}\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}|q|{\sqrt {n_{q}+{\frac {1}{2}}\mp {\frac {1}{2}}}}\,I(k,k')\delta _{k',k\pm q}\right)^{2}\delta [\varepsilon (k')-\varepsilon (k)\mp \hbar \omega _{q}],}$ 

where ${\displaystyle I(k,k')=\Omega \int _{\Omega }d^{3}r\,u_{k'}^{\ast }(r)u_{k}(r)}$ , ${\displaystyle \Omega }$  is the volume of a unit cell.

${\displaystyle P(k,k')={\begin{cases}{\frac {2\pi }{\hbar }}D_{\text{ac}}^{2}{\frac {\hbar }{2MN\omega _{q}}}|q|^{2}n_{q}&(k'=k+q;{\text{absorption}}),\\{\frac {2\pi }{\hbar }}D_{\text{ac}}^{2}{\frac {\hbar }{2MN\omega _{q}}}|q|^{2}(n_{q}+1)&(k'=k-q;{\text{emission}}).\end{cases}}}$ 

• Phonon scattering
• Umklapp scattering

• Hamaguchi, Chihiro (2017). Basic Semiconductor Physics. Graduate Texts in Physics (3 ed.). Springer. pp. 265–363. doi:10.1007/978-3-319-66860-4. ISBN 978-3-319-88329-8.
• Yu, Peter Y.; Cardona, Manuel (2005). Fundamentals of Semiconductors (3rd ed.). Springer.