– Retational energy of a diatomic molecule:
\[
E_{J}=\frac{\mathrm{H}^{2}}{2 I} J(J+1)
\]
where $I$ is the molecule’s mement of inertia.
– Vibrational energy of a diatemie melecules
\[
\varepsilon_{*}=H_{0}\left(r+\frac{1}{2}\right) \text {. }
\]
where of is the satural irequency of oscillatiens of the melecale.
$2 e 4$
– Meas energy of a suantum harmonic escillator at a lemperature $r$ :
– Debye formula for molar vibrational energy of a crystal:
\[
U=9 R \theta\left[\frac{1}{8}+\left(\frac{T}{\theta}\right)^{4} \int^{e r} \frac{x^{2} d x}{e^{x}-1}\right] .
\]
shere $\theta$ is the Debye temperaturs,
\[
\theta=\operatorname{Nomax}^{k} / \text {. }
\]
– Molar vibrational heat capacity of a cryatal for $T<\theta$ :
\[
c=\frac{12}{5} a^{4} R\left(\frac{T}{6}\right)^{3} .
\]
nero:
– Distribution of free electrons in metal in the vicinity of the absolute
\[
\ln =\frac{\sqrt{2} m^{3 / 2}}{\left.\pi^{3}\right)^{3}} \sqrt{Z} d E .
\]
where in is the cencentration of electrons whoop energy folls within the interval $E, E+d E$. The energy $E$ is counted of the bottom of the conduction basd.
– Fermil hevel at $T=0$ :
\[
E_{r}=\frac{n^{2}}{2 m}\left(B n^{n}\right)^{2 / 3},
\]
where $s$ is the coscentration of tre electrons in metul
6.167. Determine the anzular rotation velocity of an $S_{\text {, }}$ molecule promoted to the first excited rotational level if the distance between its auclei is $d=189 \mathrm{pm}$.
6.168. For an $\mathrm{HCl}$ molecule find the rotational quantum numbers of two neighbouring levels whose energies difler by $7.86 \mathrm{meV}$. The nuclei of the molecule are separated by the distance of $127.5 \mathrm{pm}$.
6.169. Find the angular momentum of an oxyzen molecule whose rotational energy is $E=2.16 \mathrm{meV}$ and the distance between the nuclei is $d=121 \mathrm{pm}$.
6.170. Show that the frequency intervals between the neighbouring spectral lines of a true rotational spectrum of a diatomic molecule are equal. Find the moment of inertia and the distance between the nuclei of a $\mathrm{CH}$ molecule if the intervals between the neighbouring lines of the true rotational spectrum of these molecules are equal to $\Delta \omega=5.47 \cdot 10^{11} \mathrm{~s}-1$.
6.171. For an HF molecule find the number of rotational levels located between the zeroth and first excited vibrational levels assuming rotational states to be independent of vibrational ones. The natural vibration frequency of this molecule is equal to $7.79 \cdot 10^{44} \mathrm{rad} / \mathrm{s}$, and the distance between the nuclei is $91.7 \mathrm{pm}$.
6.172. Evaluate how many lines there are in a true rotational spectrum of $\mathrm{CO}$ molecules whose natural vibration frequency is $=-4.09 \cdot 10^{41} \mathrm{~s}-1$ and moment of inertia $I=1.44 \cdot 10^{-7} \mathrm{~g} \cdot \mathrm{cm}^{2}$.
6.173. Find the number of rotational levels per unit energy interval, $d N / d E$, for a diatomic molecule as a function of rotational energy $E$. Calculate that magnitude for an iodine molecule in the state with rotational quantum number $J=10$. The distance between the nuclei of that molecule is equal to $267 \mathrm{pm}$.
6.174. Find the ratio of energies required to excite a diatomic molecule to the first vibrational and to the first rotational level. Calculate that ratio for the following molecules:

Here $\omega$ is the natural vibration frequency of a molecule, $d$ is the distance between nuclei.
6.175. The natural vibration frequency of a hydrogen molecule is equal to $8.25 \cdot 10^{14} \mathrm{~s}^{-1}$, the distance between the nuclei is $74 \mathrm{pm}$. Find the ratio of the number of these melecules at the first exelted vibrational level $(v-1)$ to the number of molecules at the first exeited rotational level $(J-1)$ at a temperature $T=875 \mathrm{~K}$. It should be remembered that the degeneracy of rotational levels is equal to $2 j+1$.
6.176. Derive Eq. (6.4c), making use of the Boltumanh distribution. From Eq. (6.4c) obtain the expression for molar vibration heat capacity $C_{V}, \ldots$ of diatomic zas. Calculate $C_{V}, 10$ for $\mathrm{Cl}_{3}$ zas at the temperature $300 \mathrm{~K}$. The natural vibration frequency of these molecules is equal to $1.064 \cdot 10^{14} \mathrm{~s}-1$.
6.177. In the middle of the rotation-vibration band of emission spectrum of $\mathrm{HCl}$ molecule, where the \”zeroth\” line is forbidden by the selection rules, the interval between neighbouring lines is $\Delta \omega-$ $=0.79 \cdot 10^{13} \mathrm{~s}^{-1}$. Calculate the distance between the nuclei of an $\mathrm{HCl}$ molecule.
6.178. Calculate the wavelengths of the red and violet satellites, closest to the fixed line, in the vibration spectrum of Raman scattering by $\mathrm{F}$, molecules if the incident light waveleagth is equal to $\lambda_{0}=$ $=404.7 \mathrm{~nm}$ and the natural vibration frequency of the molecule is $\omega=2.15 \cdot 10^{14} \mathrm{~s}^{-1}$.
6.179. Find the natural vibration frequency and the quasielastic force coefficient of an $\mathrm{S}_{2}$ molecule if the wavelengths of the red and violet satellites, closest to the fixed line, in the vibration spectrum ‘of Raman scattering are equal to 346.6 and $330.0 \mathrm{~nm}$.
6.180. Find the ratio of intensities of the violet and red satellites, elosest to the fixed line, in the vibration spectrum of Raman scattering by $\mathrm{Cl}_{2}$ molecules at a temperature $\mathrm{T}=300 \mathrm{~K}$ if the natural
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vibration frequency of these molecules is $6=1.06 \cdot 10^{44} \mathrm{~s}-1$. By what factor will this ratio change if the temperature is doubled? 6.181. Consider the possible vibration modes in the following linear molecules:
(a) $\mathrm{CO}_{2}(\mathrm{O}-\mathrm{C}-\mathrm{O}) ;$ (b) $\mathrm{C}_{2} \mathrm{H}_{2}(\mathrm{H}-\mathrm{C}-\mathrm{C}-\mathrm{H})$.
6.182. Find the number of natural transverse vibrations of a string of length $l$ in the frequency interval from $\omega$ to $\omega+d \omega$ if the propagation velocity of vibrations is equal to $\mathrm{v}$. All vibrations are supposed to occur in one plase.
6.183. There is a square membrane of area $S$. Find the number of natural vibrations perpendicular to its plane in the frequency interval from $\omega$ to $\omega+d \omega$ if the propagation velocity of vibrations is equal to $v$.
6. 184. Find the number of natural transverse vibrations of a rightangled parallelepiped of volume $V$ in the frequency interval from $\omega$ to $\Theta+d \omega$ if the propagation velocity of vibrations is equal to $v$.
6.185. Assuming the propagation velocities of longitudinal and transverse vibrations to be the same and equal to $\mathrm{F}$, find the Debye temperature
(a) for a unidimensional crystal, i.e. a chain of identical atoms, incorporating $n_{0}$ atoms per unit length;
(b) for a two-dimensional crystal, i.e. a plane square grid consisting of identical atoms, containing $n_{0}$ atoms per unit area;
(e) for a simple cubie lattice consisting of identical atoms, containing $n$, atoms per unit volume.
6.185. \”Calculate the Debye temperature for iron in which the propagation velocities of longitudinal and transverse vibrations are equal to 5.85 and $3.23 \mathrm{~km} / \mathrm{s}$ respectively.
6.187. Evaluate the propagation velocity of acoustic vibrations in aluminium whose Debye temperature is $\theta=396 \mathrm{~K}$.
6.188. Derive the formula expressing molar heat capacity of a unidimensional crystal, a chain of identical atoms, as a function of temperature $T$ if the Debye temperatare of the chain is equal to $\theta$. Simplify the obtained expression for the case $T>\theta$.
6.183. In a chain of identical atoms the vibration frequency a depends on wave number $k$ as $6-0 \operatorname{mag}$ sin $(k a / 2)$, where $\omega$ mas is the maximum vibration frequency, $k=2 \pi / \lambda$ is the wave number corresponding to frequency $a, a$ is the distance between neighbouring atoms. Making use of this dispersion relation, find the dependence of the number of longitudinal vibrations per unit frequency interval on $\omega$, i.e. $d N /$ de, if the length of the chain is $l$. Having obtained $d N / d o$, find the total number $\boldsymbol{N}$ of possible longitudinal vibrations of the chain.
6.190. Calculate the rero-point energy per one gram of copper whose Debye temperature is $\Theta=330 \mathrm{~K}$.
6.191. Fig. 6.10 shows heat capacity of a crystal vs temperature in terms of the Debye theory. Here $C_{d}$ is classical heat capacity, $\theta$ is the Debye temperature. Using this plot, find:
(a) the Debye temperature for silver if at a temperature $T=65 \mathrm{~K}$ its molar heat capacity is equal to $15 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K})$ :
(b) the molar heat capacity of aluminium at $T=80 \mathrm{~K}$ if at $T=250 \mathrm{~K}$ it is equal to $22.4 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K})$;
(c) the maximum vibration frequency for copper whose heat capacity at $T=125 \mathrm{~K}$ differs from the classical value by $25 \%$.
ne. 6.10 .
6.192. Demonstrate that molar heat capacity of a crystal at a temperature $T \ll \Theta$, where $\theta$ is the Debye temperature, is defined by $\mathrm{Eq} .(6.4 \mathrm{f})$.
6.193. Can one consider the temperatures 20 and $30 \mathrm{~K}$ as low for a crystal whose heat capacities at these temperatures are equal to 0.225 and $0.760 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K})$ ?
6.194. Calculate the mean tero-point energy per one escillater of a crystal in terms of the Debye theory if the Debye temperature of the crystal is equal to $\theta$.
6.195. Draw the vibration energy of a crystal as a function of frequency (neglecting the zero-point vibrations). Consider two cases: $T=\Theta / 2$ and $T=\Theta / 4$, where $\theta$ is the Debye temperature.
6.196. Evaluate the maximum values of energy and momentum of a phonon (acoustie quantum) in copper whose Debye temperature is equal to $330 \mathrm{~K}$.
6.197. Employing Eq. $(6.4 \mathrm{~g})$, find at $T=0$ :
(a) the maximum kinetic energy of free electrons in a metal if their concentration is equal to $n$ :
(b) the mean kinetic energy of free electrons if their maximum kinetic energy $T_{\text {max }}$ is known.
6.198. What fraction (in per cent) of free electrons in a metal at $T=0$ has a kinetic energy exceeding half the maximum energy?
6.199. Find the number of tree electrons per one sodium atom at $T=0$ if the Fermi level is equal to $E_{\boldsymbol{r}}=3.07 \mathrm{eV}$ and the density of sodium is $0.97 \mathrm{~g} / \mathrm{cm}^{3}$.
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6.200. Up to what temperature has one to heat classical electronic gas to make the mean energy of its electrons equal to that of free electrons in copper at $T=6$ ? Only one free electron is supposed to correspond to each copper atom.
6.201. Calculate the interval (in eV units) between neighbouring levels of free electrons in a metal at $T=0$ near the Fermi level, if the concentration of free electrons is $n=2.0 \cdot 10^{n 1} \mathrm{~cm}^{-1}$ and the volume of the metal is $\boldsymbol{V}=\mathbf{1 . 0} \mathrm{cm}^{\prime}$.
6.202. Making use of $\mathrm{Eq}$. $(6.4 \mathrm{~g})$, find at $T=0$ :
(a) the velocity distribution of free electrons;
(b) the ratio of the mean velocity of free electrons to their maximum velocity.
6.203. On the basis of Eq. $(6.4 \mathrm{~g})$ find the number of free electrons in a metal at $T=0$ as a function of de Broglie waveleagths.
6.204. Calculate the electronie gas pressure in metallie sodium, at $T=0$, in which the concentration of free electrons is $n=$ $=2.5 \cdot 10^{\mathrm{h}} \mathrm{cm}^{-4}$. Use the equation for the pressure of ideal gas.
6.205. The increase in temperature of a cathode in electronic tube by $\Delta T=1.0 \mathrm{~K}$ from the value $T=2000 \mathrm{~K}$ results in the increase of saturation current by $\eta=1.4 \%$. Find the work function of electron for the material of the cathode.
6.206. Find the refractive index of metallic sodium for electrons with kiaetic energy $T=135 \mathrm{eV}$. Only one free electron is assumed to correspond to each sodium atem.
6.207. Find the minimum energy of electron-hole pair formation in an impurity-tree semiconductor whose electric conductance increases $\eta=5.0$ times, when the temperature increases from $T_{1}=$ $=300 \mathrm{~K}$ to $T_{1}=400 \mathrm{~K}$.
6.208. At very low temperatures the photeelectric threshold short wavelength in an impurity-free germanium is equal to $\lambda_{\mathrm{an}}=\mathbf{1 . 7} \mathrm{\mu m}$. Find the temperature coefficient of resistance of this germanium sample at room temperature.
6.209. Fig. 6.11 illustrates logarithmic electric conductance as a function of reciprocal temperature ( $T$ in $\mathrm{kK}$ units) for some
ค.e. 6.11.
n-type semiconductor. Using this plot, find the width of the forbidden band of the semiconductor and the activation energy of donor levels.
6.210. The resistivity of an impurity-free semiconductor at room temperature is $\rho=50 \Omega \cdot \mathrm{cm}$. It becomes equal to $\rho_{1}=40 \Omega \cdot \mathrm{cm}$ when the semiconductor is illuminated with light, and $t=8 \mathrm{~ms}$ after switehing off the light source the resistivity becomes equal to $\rho_{2}=45 \Omega \cdot \mathrm{cm}$. Find the mean lifetime of conduction electrons and holes.
6.211. In Hall effect measurements a plate of width $h=10 \mathrm{~mm}$ and length $l=50 \mathrm{~mm}$ made of $p$-type semiconductor was placed in a magnetic field with induction $B=5.0 \mathrm{kG}$. A potential difference $V=10 \mathrm{~V}$ was applied across the edges of the plate. In this case the Hall field is $V_{n}=50 \mathrm{mV}$ and resistivity $\rho=2.5 \Omega \cdot \mathrm{cm}$. Find the concentration of holes and hole mobility.
6.212. In Hall effect measurements in a magnetic field with induction $B=5.0 \mathrm{kG}$ the transverse electric field strength in an impurity-free germanium turned out to be $\eta=10$ times less than the longitudinal electric field strength. Find the difference in the mobilities of conduction electrons and holes in the given semiconductor.
6.213. The Hall eflect turned out to be not observable in a semiconductor whose conduction electron mebility was $\eta=2.0$ times that of the hole mobility. Find the ratio of hole and conduction electron concentrations in that semiconductor.