Modern Physics

Paul A. Tipler, Ralph A. Llewellyn

Chapter 10

Solid State Physics - all with Video Answers

Educators

Chapter Questions

Problem 1

Find the value of $n$ in Equation $10-6$ that gives the measured dissociation energy of $741 \mathrm{~kJ} / \mathrm{mol}$ for $\mathrm{LiCl}$, which has the same structure as $\mathrm{NaCl}$ and for which $r_{0}=0.257 \mathrm{nm}$

Keshav Singh

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Problem 2

Calculate the distance $r_{0}$ between the $\mathrm{K}^{+}$ and $\mathrm{Cl}^{-}$ ions in $\mathrm{KCl}$, assuming that each ion occupies a cubic volume of side $r_{0}$. The molar mass of $\mathrm{KCl}$ is $74.55 \mathrm{~g} / \mathrm{mol}$ and its density is $1.984 \mathrm{~g} / \mathrm{cm}^{3}$.

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Problem 3

The distance between the $\mathrm{Li}^{+}$ and $\mathrm{Cl}^{-}$ ions in $\mathrm{LiCl}$ is $0.257 \mathrm{nm} .$ Use this and the molecular mass of $\mathrm{LiCl}(42.4 \mathrm{~g} / \mathrm{mol}$ ) to compute the density of $\mathrm{LiCl}$.

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Problem 4

The crystal structure of $\mathrm{KCl}$ is the same as that of $\mathrm{NaCl}$. ( $a$ ) Calculate the electrostatic potential energy of attraction of $\mathrm{KCl}$, assuming that $r_{0}$ is $0.314 \mathrm{nm}$. ( $b$ ) Assuming that $n=9$ in Equation $10-6,$ calculate the dissociation energy in $\mathrm{eV}$ per ion pair and in kcal/mol. ( $c$ ) The measured dissociation energy is $165.5 \mathrm{kcal} / \mathrm{mol}$. Use this to determine $n$ in Equation $10-6 .$

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Problem 5

The observed dissociation energy of solid LiBr is $788 \mathrm{~kJ} / \mathrm{mol}$. Compute the cohesive energy of $\mathrm{LiBr}$ and compare the result with the value in Table $10-1$. (Ionization energies for $\mathrm{Li}$ and $\mathrm{Br}$ are in Table $9-1 .)$
within 1 percent.

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Problem 6

The density of $\mathrm{NaCl}$ (an foc crystal) is $2.16 \mathrm{~g} / \mathrm{cm}^{3}$. Find the distance between ions that are nearest neighbors.

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Problem 7

The separation of nearest-neighbor ions in the KCl crystal (an fcc structure) is $0.315 \mathrm{nm}$. Use this information to determine the density of $\mathrm{KCL}$.

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Problem 8

Using the data for ionic and metallic crystals from Table $10-1,$ ( $a$ ) graph cohesive energy versus melting point and put the best straight line through the points. (b) Determine the cohesive energies of cobalt, silver, and sodium, whose melting temperatures are $1495^{\circ} \mathrm{C}, 962^{\circ} \mathrm{C},$ and $98^{\circ} \mathrm{C},$ respectively. (The measured values are cobalt $4.43 \mathrm{eV},$ silver $2.97 \mathrm{eV},$ and sodium $1.13 \mathrm{eV} .)$

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Problem 9

Figure $10-56$ shows a one-dimensional ionic lattice consisting of doubly charged positive ions and twice as many singly charged negative ions. Compute the Madelung constant for this "crystal" to

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Problem 10

Given a mean free path $\lambda=0.4 \mathrm{nm}$ and a mean speed $\langle v\rangle=1.17 \times 10^{5} \mathrm{~m} / \mathrm{s}$ for the current flow in copper at a temperature of $300 \mathrm{~K}$, calculate the classical value for the resistivity $\rho$ of copper. $(b)$ The classical model suggests that the mean free path is temperature independent and that $\langle v\rangle$ depends on temperature. From this model, what would $\rho$ be at $100 \mathrm{~K} ?$

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Problem 11

Find ( $a$ ) the current density and ( $b$ ) the drift velocity if there is a current of $1 \mathrm{~mA}$ in a No. 14 copper wire. (The diameter of No. 14 wire, which is often used in household wiring, is 0.064 in $=0.163 \mathrm{~cm} .)$

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Problem 12

A measure of the density of the free-electron gas in a metal is the distance $r_{s}$, which is defined as the radius of the sphere whose volume equals the volume per conduction electron. ( $a$ ) Show that $r_{s}=\left(3 / 4 \pi n_{a}\right),$ where $n_{a}$ is the free-electron number density.
(b) Calculate $r_{s}$ for copper in nanometers.

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Problem 13

Calculate the number density of free electrons in $(a)$ Ag $\left(\rho=10.5 \mathrm{~g} / \mathrm{cm}^{3}\right)$ and
(b) $\mathrm{Au}\left(\rho=19.3 \mathrm{~g} / \mathrm{cm}^{3}\right)$, assuming one free electron per atom, and compare your results with the values listed in Table $10-3 .$

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Problem 14

Calculate the number density of free electrons for $(a) \mathrm{Mg}\left(\rho=1.74 \mathrm{~g} / \mathrm{cm}^{3}\right)$ and
(b) $\mathrm{Zn}\left(\rho=7.1 \mathrm{~g} / \mathrm{cm}^{3}\right),$ assuming two free electrons per atom, and compare your results with the values listed in Table $10-3 .$

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Problem 15

(a) Using $\lambda=0.37 \mathrm{nm}$ and $\langle v\rangle=1.08 \times 10^{5} \mathrm{~m} / \mathrm{s}$ at $T=300 \mathrm{~K}$, calculate $\sigma$ and $\rho$
for copper from Equations $10-13 .$ Using the same value of $\lambda,$ find $\sigma$ and $\rho$ at $(b) T=200 \mathrm{~K}$ and $(c) T=100 \mathrm{~K}$

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Problem 16

Find the average energy of the electrons at $T=0 \mathrm{~K}$ in $(a)$ copper $\left(E_{F}=7.06 \mathrm{eV}\right)$ and $(b) \operatorname{Li}\left(E_{F}=4.77 \mathrm{eV}\right)$

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Problem 17

Calculate the Fermi energy for magnesium in a long, very thin wire.

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Problem 18

Compute $(a)$ the Fermi energy and $(b)$ the Fermi temperature for silver and for iron and compare your results with the corresponding values in Table $10-3 .$ 10-19. Show that for $T=300 \mathrm{~K},$ about 0.1 percent of the free electrons in metallic silver have an energy greater than $E_{F}$.

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Problem 19

Show that for $T=300 \mathrm{~K},$ about 0.1 percent of the free electrons in metallic silver have an energy greater than $E_{F}$.

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Problem 20

What is the Fermi speed, that is, the speed of a conduction electron whose energy is equal to the Fermi energy $E_{F},$ for $(a) \mathrm{Na},(b) \mathrm{Au},$ and $(c) \mathrm{Sn} ?$ (See Table $\left.10-3 .\right)$

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Problem 21

The resistivities of $\mathrm{Na}, \mathrm{Au},$ and $\mathrm{Sn}$ at $T=273 \mathrm{~K}$ are $4.2 \mu \Omega \cdot \mathrm{cm}, 2.04 \mu \Omega \cdot \mathrm{cm}$
and $10.6 \mu \Omega \cdot \mathrm{cm},$ respectively. Use these values and the Fermi speeds calculated in Problem $10-20$ to find the mean free paths $\lambda$ for the conduction electrons in these elements.

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Problem 22

At what temperature is the heat capacity due to the electron gas in copper equal to 10 percent of that due to lattice vibrations?

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Problem 23

Use Equation $10-29$ with $\alpha=\pi^{2} / 4$ to calculate the average energy of an electron in copper at $T=300 \mathrm{~K}$. Compare your result with the average energy at $T=0$ and the classical result of $(3 / 2) k T .$

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Problem 24

Compute the maximum fractional contribution to the heat capacity of solid iron that can be made by the electrons.

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Problem 25

The magnetic polarization $P$ of any material is defined as $P=\left(\rho_{+}-\rho_{-}\right) / \rho$. Compute the high-temperature polarization of a paramagnetic solid at $T=200 \mathrm{~K}$ in a magnetic field of $2.0 \mathrm{~T}$

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Problem 26

Show that the magnetic susceptibility $\chi$ is a dimensionless quantity.

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Problem 27

The energy gap between the valence band and the conduction band in silicon is $1.14 \mathrm{eV}$ at room temperature. What is the wavelength of a photon that will excite an electron from the top of the valence band to the bottom of the conduction band? Do the same calculation for $(b)$ germanium, for which the energy gap is $0.72 \mathrm{eV},$ and $(c)$ for diamond, for which the energy gap is $7.0 \mathrm{eV}$

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Problem 28

The energy-band gap in germanium is $0.72 \mathrm{eV}$. What wavelength range of visible light will be transmitted by a germanium crystal? (Think about it carefully!)
( $b$ ) Now consider a crystal of an insulator whose energy-band gap is $3.6 \mathrm{eV}$. What wavelength range of visible light will this crystal transmit? ( $c$ ) Justify each of your answers to
$(a)$ and $(b)$

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Problem 29

A photon of wavelength $3.35 \mu \mathrm{m}$ has just enough energy to raise an electron from the valence band to the conduction band in a lead sulfide crystal. ( $a$ ) Find the energy gap between these bands in lead sulfide. $(b)$ Find the temperature $T$ for which $k T$ equals this
energy gap.

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Problem 30

Consider a small silicon crystal measuring $100 \mathrm{nm}$ on each side. ( $a$ ) Compute the total number $N$ of silicon atoms in the crystal. (The density of silicon is $\left.2.33 \mathrm{~g} / \mathrm{cm}^{3} .\right)$ ( $b$ ) If the conduction band in silicon is $13 \mathrm{eV}$ wide and recalling that there are $4 N$ states in this band, compute an approximate value for the energy spacing between adjacent conductionband states for the crystal.

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Problem 31

Arsenic has five valence electrons. If arsenic is used as a dopant in silicon, compute
(a) the ionization energy and ( $b$ ) the orbit radius of the fifth arsenic electron. The effective mass for electrons in silicon is $0.2 \mathrm{~m}_{e} .$ ( $c$ ) What is the ratio of the ionization energy of the fifth electron to the energy gap in silicon?

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Problem 32

Gallium has three valence electrons. If gallium is used to dope germanium, compute
(a) the ionization energy of the hole and ( $b$ ) the orbit radius of the hole. The effective mass of holes in germanium is $0.34 \mathrm{~m}_{e^{*}}$

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Problem 33

What type of semiconductor is obtained if silicon is doped with $(a)$ aluminum and
( $b$ ) phosphorus? (See Appendix $\mathrm{C}$ for the electron configurations of these elements.)

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Problem 34

The donor energy levels in an $n$ -type semiconductor are 0.01 eV below the conduction band. Find the temperature for which $k T=0.01 \mathrm{eV}$.

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Problem 35

A strip of tin is $10 \mathrm{~mm}$ wide and $0.2 \mathrm{~mm}$ thick. When a current of $20 \mathrm{~A}$ is established in the strip and a uniform magnetic field of $0.25 \mathrm{~T}$ is oriented perpendicular to the plane of the strip, a Hall voltage of $2.20 \mu \mathrm{V}$ is measured across the width of the strip. Compute ( $a$ ) the density of charge carriers in tin and $(b)$ the average number of charge carriers contributed by each tin atom. The density of tin is $5.75 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ and its molecular mass is 118.7

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Problem 35

A strip of tin is $10 \mathrm{~mm}$ wide and $0.2 \mathrm{~mm}$ thick. When a current of $20 \mathrm{~A}$ is established in the strip and a uniform magnetic field of $0.25 \mathrm{~T}$ is oriented perpendicular to the plane of the strip, a Hall voltage of $2.20 \mu \mathrm{V}$ is measured across the width of the strip. Compute ( $a$ ) the density of charge carriers in tin and $(b)$ the average number of charge carriers contributed by each tin atom. The density of tin is $5.75 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ and its molecular mass is 118.7

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Problem 36

For a temperature of $300 \mathrm{~K}$, use Equation $10-49$ to find the bias voltage $V_{b}$ for which the exponential term has the value $(a) 10$ and $(b) 0.1$

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Problem 37

For what value of bias voltage $V_{b}$ does the exponential in Equation $10-49$ have the value $(a) 5$ and $(b) 0.5$ for $T=200 \mathrm{~K} ?$

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Problem 38

Compute the fractional change in the current through a $p n$ junction diode when the forward bias is changed from $+0.1 \mathrm{~V}$ to $+0.2 \mathrm{~V}$.

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Problem 39

For $T=300 \mathrm{~K}$, use Equation $10-49$ to find the bias voltage $V_{b}$ for which the exponential term had the value $(a) 10$ and $(b)$ 0.1.

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Problem 40

When light of wavelength no larger than $484 \mathrm{nm}$ illuminates a CdS solar cell, the cell produces electric current. Determine the energy gap in CdS.

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Problem 41

Three naturally occurring isotopes of lead are ${ }^{206} \mathrm{~Pb},{ }^{207} \mathrm{~Pb},$ and ${ }^{208} \mathrm{~Pb}$. Using the value of $\alpha$ from Table $10-7$ and the isotopic masses from Appendix $A$, compute the critical temperatures of these isotopes.

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Problem 42

Compute $(a)$ the superconducting energy gap for indium and $(b)$ the wavelength of a photon that could just break up a Cooper pair in indium at $T=0 \mathrm{~K}$.

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Problem 43

Use Equation $10-56$ to calculate the superconducting energy gap for tin and compare your result with the measured value of $6 \times 10^{-4} \mathrm{eV} .$ ( $b$ ) Use the measured value to calculate the wavelength of a photon having sufficient energy to break up a Cooper pair in tin at $T=0 \mathrm{~K}$.

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Problem 44

Use the BCS curve in Figure $10-53$ to estimate the energy gaps in (a) tin,
(b) niobium, (c) aluminum, and ( $d$ ) zinc, all at $T=0.5 T_{c}$.

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Problem 45

Expressing the temperature $T$ as a fraction of the critical temperature $T_{c}$, according to BCS theory at what temperature is $(a) B_{c}(T)=0.1 B_{c}(0),(b) B_{c}(T)=0.5 B_{c}(0),(c) B_{c}(T)=$
$0.9 B_{c}(0) ?$

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Problem 46

Approximating atoms in an fcc crystal as hard spheres of radius $r$ with $a$ being the length of each side of the unit cube, what fraction of the volume of the cube (and hence the crystal) is occupied by atoms?

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Problem 47

Estimate the fraction of free electrons in copper that are in excited states above the Fermi energy at $(a)$ room temperature of $300 \mathrm{~K}$ and $(b) 1000 \mathrm{~K}$.

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Problem 48

A one-dimensional model of an ionic crystal consists of a line of alternating positive and negative ions with distance $r_{0}$ between adjacent ions. ( $a$ ) Show that the potential energy of attraction of one ion in the line is
$$
V=-\frac{2 k e^{2}}{r_{0}}\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots\right)
$$

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Problem 49

Estimate the Fermi energy of zinc from its electronic molar heat capacity of $\left(3.74 \times 10^{-4} \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}\right) T$

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Problem 50

The density of the electron states in a metal can be written $g(E)=A E^{\frac{1}{2}}$ where $A$ is a constant and $E$ is measured from the bottom of the conduction band.
(a) Show that the total number of states is $(2 / 3) A\left(E_{F}\right)^{3 / 2} .$ ( $b$ ) About what fraction of the conduction electrons is within $k T$ of the Fermi energy? ( $c$ ) Evaluate this fraction for copper at $T=300 \mathrm{~K}$.

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Problem 51

High-purity germanium (HPGe) crystals are used as detectors for $x$ rays and gamma rays. On interacting with the crystal, incoming photons produce electron-hole pairs, exciting many electrons across the $0.72 \mathrm{eV}$ energy gap into the conduction band. The decay of the radioisotope ${ }^{60}$ Co results in the emission of two gamma rays with energies $1.17 \mathrm{MeV}$ and $1.33 \mathrm{MeV}$ (see Chapter 11 ). ( $a$ ) Compute the numbers of electrons $N_{1}$ and $N_{2}$ excited across the energy gap by each of the two gamma rays. ( $b$ ) The numbers $N_{1}$ and $N_{2}$ are subject to statistical fluctuations of $\pm \sqrt{N_{1}}$ and $\pm \sqrt{N_{2}}$. Compute the fractional uncertainties in $N_{1}$ and $N_{2}$. ( $c$ ) Compute the corresponding fractional uncertainties in the energies of the two gamma rays. This is a measure of the energy resolution of the HPGe crystal.

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Problem 52

A doped $n$ -type silicon sample with $10^{16}$ electrons per cubic centimeter in the conduction band has a resistivity of $5 \times 10^{-3} \Omega \cdot \mathrm{m}$ at $300 \mathrm{~K}$. Find the mean free path of the electrons, using $0.2 m_{e}$ for the effective mass of the electron. Compare your result with the mean free path of electrons in copper at $300 \mathrm{~K}$.

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Problem 53

Crystallographers and materials scientists use the density of a metallic sample to infer its likely crystal structure. The density of copper $(\mathrm{Cu})$ is $8.96 \mathrm{~g} / \mathrm{cm}^{3}$ and its atomic radius is $0.128 \mathrm{nm}$. Is the copper crystal more likely to be face-centered cubic or bodycentered cubic? (See Figure $10-57 .)$

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Problem 54

A "good" silicon diode has a current-voltage characteristic given by
$$
I=I_{0}\left(e^{e V_{b} / k T}-1\right)
$$
Let $k T=0.025 \mathrm{eV}$ (room temperature) and the saturation current $I_{0}=1 \mathrm{nA.}$ ( $a$ ) Show that for small reverse-bias voltages, the resistance is $25 \mathrm{M} \Omega$. (Hint: Do a Taylor expansion of the exponential function, or use your calculator and enter small values for $V_{b} .$ ) ( $b$ ) Find the dc resistance for a reverse bias of $0.5 \mathrm{~V}$. ( $\mathrm{c}$ ) Find the dc resistance for a $0.5 \mathrm{~V}$ forward bias. What is the current in this case? $(d)$ Calculate the ac resistance $d V / d I$ for a $0.5 \mathrm{~V}$ forward bias.

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Problem 55

The relative binding of the extra electron in the arsenic atom that replaces an atom in silicon or germanium can be understood from a calculation of the first Bohr orbit of this electron in these materials. Four of arsenic's outer electrons form covalent bonds, so the fifth electron "sees" a singularly charged center of attraction. This model is a modified hydrogen atom. In the Bohr model of the hydrogen atom, the electron moves in free space at a radius $a_{0}$ given by
$$
a_{0}=\frac{\epsilon_{0} h^{2}}{\pi m_{e} e^{2}}
$$

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Problem 56

InSb is a semiconductor. The energy gap $E_{g}$ between its valence and conduction bands is $0.23 \mathrm{eV},$ and its dielectric constant $\kappa=18 .$ In the InSb crystal the electron's effective mass $m^{*}=0.015 m_{e} .$ (a) Compute the ionization energy for an electron donor in InSb. (b) What is the radius of the ground-state orbit? (c) At approximately what donor concentration will the orbits of adjacent donor atoms begin to overlap?

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Problem 57

The mean free path of an electron in a metal depends on both the lattice oscillations of the metal ions and those of any impurity ions according to $1 / \lambda=1 / \lambda_{m}+1 / \lambda_{i} .$ The resistivity of pure copper is increased by about $1.2 \times 10^{-8} \Omega \cdot \mathrm{m}$ by the addition of 1 percent (by number of atoms) of a certain impurity dispersed evenly throughout the metal. ( $a$ ) Estimate $\lambda_{i}$ from this information. (b) The impurity atoms are "seen" by the electrons to have an effective diameter $d$. Estimate the scattering cross section $d^{2}$ from Equation $10-12,$ where $d=2 r$.

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Problem 58

When arsenic is used to dope silicon, the fifth arsenic electron and the As $^{+}$ ion act like a hydrogen atom system, except that the potential function $V(r)$ and the electron mass must be modified as described in Section $10-7$ to account for the crystal lattice. With these modifications, ( $a$ ) solve the Schrödinger equation, using the solution in Chapter 7 as a guide. ( $b$ ) Obtain Equation $10-43,$ and $(c)$ sketch a properly scaled energy-level diagram for the fifth electron for $n=1$ through 5 .

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Problem 59

The quantity $\kappa$ is the force constant for a "spring" consisting of a line of alternating positive and negative ions. If these ions are displaced slightly from their equilibrium separation $r_{0}$, they will vibrate with a frequency
$$
f=\frac{1}{2 \pi} \sqrt{\frac{K}{m}}
$$
(a) Use the values of $\alpha, n,$ and $r_{0}$ for $\mathrm{NaCl}$ and the reduced mass for the $\mathrm{NaCl}$ molecule to calculate this frequency. (b) Calculate the wavelength of electromagnetic radiation corresponding to this frequency, and compare your result with the characteristic strong infrared absorption bands in the region of about $\lambda=61 \mu \mathrm{m}$ that are observed for $\mathrm{NaCl}$.

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Problem 60

Consider a model for a metal in which the lattice of positive ions forms a container for a classical electron gas with $n$ electrons per unit volume. In equilibrium, the average electron velocity is zero, but the application of an electric field produces an acceleration of the electrons. If we use a relaxation time $\tau$ to account for the electron-lattice collisions, then we have the equation
$$
m \frac{d v}{d t}+\frac{m}{\tau} v=-e E
$$
(a) Solve the equation for the drift velocity in the direction of the applied electric field.
(b) Verify that Ohm's law is valid, and find the resistivity as a function of $n, e, m,$ and the relaxation time $\tau$.

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Problem 61

Imagine a cubic crystal like $\mathrm{NaCl}$, with a negative charge at the center of a Cartesian coordinate system with scale units equal to the interatomic distance. ( $a$ ) Show that an ion at a position $r$ units along the $x$ axis, $s$ units along the $y$ axis, and $t$ units along the $z$ axis has a charge of $e(-1)^{r} \cdot(-1)^{s} \cdot(-1)^{t}=e(-1)^{r+s+t},$ where $e$ is the electron charge.
(b) Using Equation $10-2$ as a guide, calculate the Madelung constant for a cube 2 units on a side. Do the same for cubes of sides $4,6,8,10,12,16,$ and 20 units. (You will probably want to use a computer spreadsheet to write a program to do the calculations for the larger cubes.) Are your answers approaching the value $\alpha=1.7476 ?$

RZ

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Problem 62

Show that for a paramagnetic solid with electron energies given by Equation $10-33,$ the magnetization per unit volume $M$ is given by
$$
M=\mu \rho \tanh (\mu B / k T)
$$
(b) For $\mu B \ll k T$ show that the susceptibility is given by Equation $10-35 .$

Keshav Singh

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