91Ó°ÊÓ

The Debye temperature is given as

Here, $h$is the Planck's constant, $c_s$is the speed of the sound in the liquid, $N$is the Avogadro number, V is the volume, and $k$is the Boltzmann's constant.

The density of the liquid ${}^4\mathrm{He}$is,$\mathrm{ÒÏ}=\frac{m}{V}$

Here, $m$is the mass of the liquid ${}^4\mathrm{He}$.

Solving the equation for $V$, $V=\frac{m}{\mathrm{ÒÏ}}$

Substituting value $4\mathrm{g}$for$m$and $0.145\mathrm{g}/{\mathrm{cm}}^3$for $\mathrm{ÒÏ}$.

role="math" localid="1647513805305" $$\begin{gathered} V=\frac{4g}{0.145g/c{m}^3} \\ V=27.6c{m}^3\left(\frac{1m^3}{{10}^{6}c{m}^3}\right) \\ V=2.76×{10}^{-5}{\mathrm{m}}^3 \end{gathered}$$

Where,

so, we get the $T_D$

$T_{\mathrm{D}}=\frac{\left(6.626×{10}^{-34}\mathrm{J}·\mathrm{s}\right)(238\mathrm{m}/\mathrm{s})}{2\left(1.38×{10}^{-23}\mathrm{J}/\mathrm{K}\right)}{\left(\frac{6\left(6.02×{10}^{23}\right)}{\mathrm{π}\left(2.76×{10}^{-5}{\mathrm{m}}^3\right)}\right)}^{1/3}$

$T_D=19.8K$

So, the energies of the allowed modes is given as

$U=\underset{n_s}{∑}\underset{n_y}{∑}\underset{n_r}{∑}\mathrm{Î\mu }{\stackrel{¯}{n}}_{\mathrm{PI}}\left(\mathrm{Î\mu }\right)$

Here, ${\stackrel{¯}{n}}_{\mathrm{P}}\left(\mathrm{Î\mu }\right)$is the average Planck's distribution. The number of polarization state tor the lıquid is only $1$for the triplet $\left(n_x,n_y,n_z\right)$.

As, the heat capacity in the low temperature limit for the liquid is equal to $\frac{1}{3}$times of the heat capacity at the lower temperature for the solid as in the formula.

$C_V=\frac{1}{3}\left(\frac{12{\mathrm{Ï€}}^4}{5}\right){\left(\frac{T}{T_{\mathrm{D}}}\right)}^{3}Nk$

$\frac{C_V}{Nk}=\frac{4{\mathrm{Ï€}}^4}{5}{\left(\frac{T}{T_{\mathrm{D}}}\right)}^3$

$\frac{C_V}{Nk}={\left(\frac{T}{{\left(\frac{5}{4{\mathrm{Ï€}}^4}\right)}^{1/3}19.8\mathrm{K}}\right)}^3$

$={\left(\frac{T}{4.64\mathrm{K}}\right)}^3$.

$\text{Hence,The value of the photon contribution to the heat capacity of}{}^4\mathrm{He}\text{is}\frac{C_V}{Nk}={\left(\frac{T}{4.64\mathrm{K}}\right)}^3\text{.}$

The measured value of $\frac{C_V}{Nk}$for the heat capacity of ${}^4\mathrm{He}$is ${\left(\frac{T}{4.67\mathrm{K}}\right)}^3$. So, the value found in the above is approximately similar with the measured value of the heat capacity for liquid${}^4\mathrm{He}$.