Specific heat capacitance of phonons (Bosons)

3.4 Specific heat capacitance of phonons (Bosons)

One-dimensional lattice vibrations
We investigate a chain of identical atoms which are coupled by different kinds of springs:

Always two atoms form a unit (basis) addressed by an index $i$ .
The lattice distance shall be $a$ .
The excursion of both atoms is denominated by $u_{1}$ and $u_{2}$ :
for the potential energy we find:

$U^{h a r m} = \frac{K}{2} \sum_{n} {[u_{1} (n a) - u_{2} (n a)]}^{2} + \frac{G}{2} \sum_{n} {[u_{2} (n a) - u_{1} ((n + 1) a)]}^{2} .$

(3.19)

The equations of motion are:

$\begin{matrix} \begin{aligned} M ü_{1} (n a) & = - \frac{\partial U^{h a r m}}{\partial u_{1} (n a)} \\ = - K [u_{1} (n a) - u_{2} (n a)] - G [u_{1} (n a) - u_{2} ((n - 1) a)] \end{aligned} \end{matrix}$

(3.20)

$\begin{matrix} \begin{aligned} M ü_{2} (n a) & = - \frac{\partial U^{h a r m}}{\partial u_{2} (n a)} \\ = - K [u_{2} (n a) - u_{1} (n a)] - G [u_{2} (n a) - u_{1} ((n + 1) a)] \end{aligned} \end{matrix}$

(3.21)

Due to the translational invariance we search for lattice periodic functions. Just for simplicity we use periodic boundary conditions:

$\begin{matrix} \begin{aligned} u_{1} (n a) & = ϵ_{1} exp (i (k n a - ω t)) \\ u_{2} (n a) & = ϵ_{2} exp (i (k n a - ω t)) \end{aligned} \end{matrix}$

(3.22)

ϵ_{1}

and

ϵ_{2}

are parameters which describe the relative amplitudes and phases between both atoms.
For

k

the following relation holds:

$exp (i k N a) = 1, i.e. k = \frac{2 π}{a} \frac{n}{N}, n = - \frac{N}{2}, . . . ., \frac{N}{2} .$

(3.23)

Including this into Eq. (3.20) and Eq. (3.21) we get:

$\begin{matrix} \begin{aligned} [M ω^{2} - (K + G)] ϵ_{1} + (K + G e^{- i k a}) ϵ_{2} & = 0 \\ (K + G e^{- i k a}) ϵ_{1} + [M ω^{2} - (K + G)] ϵ_{2} & = 0 \end{aligned} \end{matrix}$

(3.24)

This system of linear equations only has solutions if the coefficient determinant vanishes:

${[M ω^{2} - (K + G)]}^{2} = {| K + G e^{- i k a} |}^{2} = K^{2} + G^{2} + 2 K G cos (k a) .$

(3.25)

We find

$ω^{2} (k) = \frac{K + G}{M} \pm \frac{1}{M} \sqrt{K^{2} + G^{2} + 2 K G cos (k a)}$

(3.26)

and

$\frac{ϵ_{2}}{ϵ_{1}} = \mp \frac{K + G e^{i k a}}{| K + G e^{i k a} |} .$

(3.27)

For the $N$ different $k$ -values we always find two $ω$ -values, i.e. we find $2 N$ modes; this corresponds to the $2 N$ degrees of freedom (2 atoms in $N$ elementary cells).
We can choose

$K > G .$

(3.28)

The both solutions are:
1)

$ω (0) = \sqrt{2 \frac{K + G}{M}}, and \frac{ϵ_{2}}{ϵ_{1}} < 0,$

(3.29)

i.e. both atoms oscillate in anti-phase. If the atoms are charged, thus a dipole moment would be introduced. Light can couple at this oscillations. Therefor the solution is called the optical mode.
2) We find

$ω (0) = 0, and \frac{ϵ_{2}}{ϵ_{1}} > 0,$

(3.30)

i.e. both atoms oscillate in phase. We will find density oscillations within the crystal. A sound wave will move through the crystal. Therefor his solution is called the acoustic mode.

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