Chapter 42

Germanium has a band gap of 0.67 eV. Doping with arsenic adds donor levels in the gap 0.01 eV below the bottom of the conduction band. At a temperature of 300 K, the probability is 4.4 $\times$ 10${}^{-}\${\$}^4$ that an electron state is occupied at the bottom of the conduction band. Where is the Fermi level relative to the conduction band in this case?

(a) Suppose a piece of very pure germanium is to be used as a light detector by observing, through the absorption of photons, the increase in conductivity resulting from generation of electron-hole pairs. If each pair requires 0.67 eV of energy, what is the maximum wavelength that can be detected? In what portion of the spectrum does it lie? (b) What are the answers to part a if the material is silicon, with an energy requirement of 1.12 eV per pair, corresponding to the gap between valence and conduction bands in that element?

A hypothetical diatomic molecule of oxygen $(mass=2.656\times {10}^{-26}kg)$ and hydrogen $(mass=1.67\times {10}^{-27}kg)$ emits a photon of wavelength 2.39 $\mu$m when it makes a transition from one vibrational state to the next lower state. If we model this molecule as two point masses at opposite ends of a massless spring, (a) what is the force constant of this spring, and (b) how many vibrations per second is the molecule making?

When a diatomic molecule undergoes a transition from the $l=2$ to the $l=1$ rotational state, a photon with wavelength 54.3 $\mu$m is emitted. What is the moment of inertia of the molecule for an axis through its center of mass and perpendicular to the line connecting the nuclei?

(a) The equilibrium separation of the two nuclei in an NaCl molecule is 0.24 nm. If the molecule is modeled as charges $+e$ and $-e$ separated by 0.24 nm, what is the electric dipole moment of the molecule (see Section 21.7)? (b) The measured electric dipole moment of an NaCl molecule is $3.0\times {10}^{-29}$ $C\cdot m$. If this dipole moment arises from point charges $+q$ and $-q$ separated by 0.24 nm, what is $q$? (c) A definition of the $fractional$ $ionic$ $character$ of the bond is $q/e$. If the sodium atom has charge $+e$ and the chlorine atom has charge $-e$, the fractional ionic character would be equal to 1. What is the actual fractional ionic character for the bond in NaCl? (d) Theequilibrium distance between nuclei in the hydrogen iodide (HI) molecule is 0.16 nm, and the measured electric dipole moment of the molecule is $1.5\times {10}^{-30}$ $C\cdot m$. What is the fractional ionic character for the bond in HI? How does your answer compare to that for NaCl calculated in part (c)? Discuss reasons for the difference in these results.

Our galaxy contains numerous $molecular$ $clouds$, regions many lightyears in extent in which the density is high enough and the temperature low enough for atoms to form into molecules. Most of the molecules are H${}_2$, but a small fraction of the molecules are carbon monoxide (CO). Such a molecular cloud in the constellation Orion is shown in Fig. P42.42. The upper image was made with an ordinary visiblelight telescope; the lower image shows the molecular cloud in Orion as imaged with a radio telescope tuned to a wavelength emitted by CO in a rotational transition. The different colors in the radio image indicate regions of the cloud that are moving either toward us (blue) or away from us (red) relative to the motion of the cloud as a whole, as determined by the Doppler shift of the radiation. (Since a molecular cloud has about 10,000 hydrogen molecules for each CO molecule, it might seem more reasonable to tune a radio telescope to emissions from H${}_2$ than to emissions from CO. Unfortunately, it turns out that the H${}_2$ molecules in molecular clouds do not radiate in either the radio or visible portions of the electromagnetic spectrum.) (a) Using the data in Example 42.2 (Section 42.2), calculate the energy and wavelength of the photon emitted by a CO molecule in an $l$ $=$ 1$\to$ $l$ $=$ 0 rotational transition. (b) As a rule, molecules in a gas at temperature $T$ will be found in a certain excited rotational energy level, provided the energy of that level is no higher than $kT$ (see Problem 42.39). Use this rule to explain why astronomers can detect radiation from CO in molecular clouds even though the typical temperature of a molecular cloud is a very low 20 K.

When an OH molecule undergoes a transition from the $n=0$ to the $n=1$ vibrational level, its internal vibrational energy increases by 0.463 eV. Calculate the frequency of vibration and the force constant for the interatomic force. (The mass of an oxygen atom is $2.66\times {10}^{-26}$ kg, and the mass of a hydrogen atom is $1.67\times {10}^{-27}kg.)$

The hydrogen iodide (HI) molecule has equilibrium separation 0.160 nm and vibrational frequency $6.93\times {10}^{13}$ Hz. The mass of a hydrogen atom is $1.67\times {10}^{-27}$ kg, and the mass of an iodine atom is 2.11 $\times$ 10${}^{-}\${\$}^2\${\$}^5$ kg. (a) Calculate the moment of inertia of HI about a perpendicular axis through its center of mass. (b) Calculate the wavelength of the photon emitted in each of the following vibrationrotation transitions: (i) $n=1$, $l=1\to n=0$, $l=0$; (ii) $n=1$, $l=2\to n=0$, $l=1$; (iii) $n=2$, $l=2\to n=1$, $l=3$.

Compute the Fermi energy of potassium by making the simple approximation that each atom contributes one free electron. The density of potassium is $851kg/m^3$, and the mass of a single potassium atom is $6.49\times {10}^{-26}$ kg.

To determine the equilibrium separation of the atoms in the HCl molecule, you measure the rotational spectrum of HCl. You find that the spectrum contains these wavelengths (among others): $60.4\mu m$, $69.0\mu m$, $80.4\mu m$, $96.4\mu m$, and $120.4\mu m$. (a) Use your measured wavelengths to find the moment of inertia of the HCl molecule about an axis through the center of mass and perpendicular to the line joining the two nuclei. (b) The value of $l$ changes by $\pm 1$ in rotational transitions. What value of $l$ for the upper level of the transition gives rise to each of these wavelengths? (c) Use your result of part (a) to calculate the equilibrium separation of the atoms in the HCl molecule. The mass of a chlorine atom is $5.81\times {10}^{-26}$ kg, and the mass of a hydrogen atom is $1.67\times {10}^{-27}$ kg. (d) What is the longest-wavelength line in the rotational spectrum of HCl?