Physics 308: Statistical, Molecular, and SolidState Physics

Final Exam

Wednesday, April 23, 2008, 09:00–12:00

Student’s Name:

Constants

kB = 8.617× 10−5 eV/K

kBTroom = 0.025 eV

h = 1.055× 10−34 J s

hc = 1.24× 10−6 eV m = 1240 eV nm

mec2 = 5.11× 105 eV = 0.511 MeV

h2

2me

=(hc)2

8π2(mec2)= 0.0381 eV nm−2

1 u = 1.661× 10−27 kg = 931.4 MeV/c2

c = 299792458 m/s

a0 =4πε0h

2

mee2= 5.292× 10−2 nm

e = 1.602× 10−19 C

µB =eh

2me

= 5.7883× 10−5 eV T−1

1

Formulas

volume and surface area of a sphere:

V =4

3πR3 A = 4πR2

particle in a box:

ψ ∼ sinnπx

LEn =

h2n2

8mL2

quantum of light:

E = hω = hf c = λf

dipole moment:

p = (∆q)r0

binding energy in an ionic solid:

U = − α

4πε0a

Leonard-Jones potential:

U(r) = U0

[(a

r

)12

− 2

(a

r

)6]reduced mass:

1

µ=

1

m1

+1

m2

current density and resistivity:

j = nqv ρ =mv

nq2l

2

Ref

eren

ce T

able

s fo

r Ph

ysic

al S

ettin

g/C

HE

MIS

TR

Y9

8R

efer

ence

Tab

les

for

Phys

ical

Set

ting/

CH

EM

IST

RY

(223

) Fr87 -1

8-32

-18-

8-1+1

226.

025 Ra

88 -18-

32-1

8-8-

2+222

7.02

8 Ac89 -1

8-32

-18-

9-2+3

(261

) Rf10

4

138.

906 La

57 2-8-

18-1

8-9-

2+317

8.49 Hf

72 **18

-32-

10-2

+4

91.2

24 Zr40 2-

8-18

-10-

2

+4

47.8

8 Ti22 2-

8-10

-2

+2 +3 +4

50.9

415 V

23 2-8-

11-2

+2 +3 +4 +5

51.9

96 Cr24 2-

8-13

-1

+2 +3 +6

95.9

4 Mo

42 2-8-

18-1

3-1

+3 +6

183.

85 W74 -1

8-32

-12-

2

+6

54.9

380 Mn

25 2-8-

13-2

+2 +3 +4 +7

55.8

47 Fe26 2-

8-14

-2

+2 +358

.933

2 Co27 2-

8-15

-2

+2 +358

.69 Ni

28 2-8-

16-2

+2 +363

.546 Cu

29 2-8-

18-1

+1 +2

107.

868 Ag

47 2-8-

18-1

8-1

+1

65.3

9 Zn30 2-

8-18

-2

+269

.72 Ga

31 2-8-

18-3

+3

26.9

8154 Al

13 2-8-

3

+3

10.8

1

B5 2-

3

+312

.011

1 C6 2-

4

–4 +2 +4

15.9

994 O

8 2-6

–218

.998

403 F

9 2-7

–120

.179 Ne

10 2-8

0

4.00

260 He

2 2

0

28.0

855 Si

14 2-8-

4

–4 +2 +4

72.5

9 Ge32 2-

8-18

-4

–4 +2 +4

74.9

216 As

33 2-8-

18-5

–3 +3 +5

78.9

6 Se34 2-

8-18

-6

–2 +4 +6

127.

60 Te52 2-

8-18

-18-

6

–2 +4 +6

126.

905 l

53 2-8-

18-1

8-7

–1 +1 +5 +7

131.

29 Xe54 2-

8-18

-18-

8

0 +2 +4 +6

(209

) Po84 -1

8-32

-18-

6

+2 +4(2

10) At

85 -18-

32-1

8-7

79.9

04 Br35 2-

8-18

-7

–1 +1 +5

83.8

0 Kr36 2-

8-18

-8

0 +2

(222

) Rn86 -1

8-32

-18-

8

0

174.

967 Lu

71

+317

3.04 Yb

70

+2 +316

8.93

4 Tm69

+316

7.26 Er

68

+316

4.93

0 Ho67

+316

2.50 Dy

66

+315

8.92

5 Tb65

+315

7.25 Gd

64

+315

1.96 Eu

63

+2 +315

0.36 Sm 62

+2 +3(1

45) Pm 61

+314

4.24 Nd

60

+314

0.90

8 Pr59

+314

0.12 Ce

58

+3 +4

232.

038 Th

90

+423

1.03

6 Pa91

+4 +523

8.02

9 U92

+3 +4 +5 +6

237.

048 Np

93

+3 +4 +5 +6

(244

) Pu94

+3 +4 +5 +6

(243

) Am 95

+3 +4 +5 +6

(247

) Cm 96

+3(2

47) Bk

97

+3 +4(2

51) Cf

98

+3(2

52) Es

99

(257

) Fm10

0

(258

) Md

101

(259

) No10

2

(260

) Lr10

3

30.9

7376 P

15 2-8-

5

–3 +3 +5

32.0

6

S16 2-

8-6

–2 +4 +6

39.9

48 Ar18 2-

8-8

035

.453 Cl

17 2-8-

7

–1 +1 +3 +5 +7

118.

71 Sn50 2-

8-18

-18-

4

+2 +4

14.0

067 N

7 2-5

–3 –2 –1 +1 +2 +3 +4 +5

112.

41 Cd48 2-

8-18

-18-

2

+211

4.82 In

49 2-8-

18-1

8-3

+3

200.

59 Hg80 -1

8-32

-18-

2

+1 +220

4.38

3 Tl81 -1

8-32

-18-

3

+1 +320

7.2 Pb

82 -18-

32-1

8-4

+2 +420

8.98

0 Bi83 -1

8-32

-18-

5

+3 +5

121.

75 Sb51 2-

8-18

-18-

5

–3 +3 +5

106.

42 Pd46 2-

8-18

-18

+2 +4

195.

08 Pt78 -1

8-32

-17-

1

+2 +419

6.96

7 Au79 -1

8-32

-18-

1

+1 +3

101.

07 Ru44 2-

8-18

-15-

1

+3

190.

2 Os76 -1

8-32

-14-

2

+3 +419

2.22

Ir77 -1

8-32

-15-

2

+3 +4

102.

906 Rh

45 2-8-

18-1

6-1

+3(9

8)

Tc43 2-

8-18

-14-

1

+4 +6 +7

186.

207 Re

75 -18-

32-1

3-2

+4 +6 +7

92.9

064 Nb

41 2-8-

18-1

2-1

+3 +5

180.

948 Ta

73 -18-

32-1

1-2

+5

(262

) Db10

5

(263

) Sg10

6

(264

) Bh10

7

(265

) Hs10

8

(268

) Mt

109

(269

) Uun

(272

) Uuu

111

(277

) Uub

112

(285

) Uuq

114

88.9

059 Y

39 2-8-

18-9

-2

+3

44.9

559 Sc

21 2-8-

9-2

+3

137.

33 Ba56 2-

8-18

-18-

8-2+2

87.6

2 Sr38 2-

8-18

-8-2

+2

40.0

8 Ca20 2-

8-8-

2

+2

24.3

05 Mg

12 2-8-

2

+2

9.01

218 Be

4 2-2

+2

132.

905 Cs

55 2-8-

18-1

8-8-

1+1

85.4

678 Rb

37 2-8-

18-8

-1

+1

39.0

983 K

19 2-8-

8-1

+1

22.9

8977 Na

11 2-8-

1

+1

6.94

1 Li3 2-

1

+1

1.00

794 H

1 1

+1 –1

*The

sys

tem

atic

nam

es a

nd s

ymbo

ls fo

r el

emen

ts o

f ato

mic

num

bers

abo

ve 1

09

will

be

used

unt

il th

e ap

prov

al o

f triv

ial n

ames

by

IUPA

C.

**D

enot

es th

e pr

esen

ce o

f (2-

8-)

for

elem

ents

72

and

abov

e

Period 1 2

11

2G

roup

1716

1514

131818

Gro

up

109

87

65

43

1112

Gro

up3 4 5 6 7

Per

iodi

c Ta

ble

of t

he E

lem

ents

C12

.011

–4 +2 +4

6 2-4

Ato

mic

Mas

s

Sym

bol

Ato

mic

Num

ber

Ele

ctro

n C

onfig

urat

ion

Sel

ecte

d O

xida

tion

Sta

tes

Rel

ativ

e at

omic

mas

ses

are

base

don

12 C

= 1

2.00

0

Not

e:M

ass

num

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in p

aren

thes

esar

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ass

num

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of t

he m

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stab

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on is

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3

Multiple Choice Questions (30 points)

Answer by circling one of (a), (b), (c), etc. Please be clear about which oneyou have selected.

1. Free electrons with wavefunction ψ ∼ eik·r are characterized by a quan-tum number k = (kx, ky, kz) and have energy E(k) = h2|k|2/2m.Which of the following correctly describes the ground state of N elec-trons in a volume V with Fermi energy EF = (h2/2m)(3π2N/V )2/3 ?

(a) all states with (k2x + k2

y + k2z)

1/2 < (3π2N/V )1/3 are filled

(b) all states with (k2x + k2

y + k2z)

1/2 > (3π2N/V )1/3 are filled

(c) all states with −EF/2 < E(k) < EF/2 are filled

2. Blackbody radiation consists of photons populated according to

(a)1

eαeβE − 1(α 6= 0)

(b)1

eαeβE − 1(α = 0)

(c)1

eαeβE + 1

(d) e−βE

3. A Bose-Einstein condensate of 4He atoms with N0 atoms in the groundstate is populated according to

(a)1

eαeβE − 1(α ∼ 1/N0)

(b)1

eαeβE − 1(α = 0)

(c)1

eαeβE + 1

(d) e−βE

4

4. A system with two energy levels ε1 = 0 and ε2 = ∆ is in thermalequilibrium. Which picture correctly illustrates the total energy as afunction of T?

(a) (b) (c)

0 T

0

∆

0

∆

∆/2∆/2 ∆/2

0 T 0 T

0

∆

5. A system with many energy levels (some degenerate) has a lowest levelε1 = 0 and additional levels spaced above it (in increments of ∆ or 2∆,as indicated in the diagrams). Which has average energy ∼ 2∆e−∆/kBT

when the system is very cold?

! 2!

6. The quantum numbers of an electron in an atomic orbital describethe radial character of the wavefunction (n), the angular momentum(l,ml), and the spin (mS). What is the total number of states withn = 3?

(a) 32

(b) 30

(c) 24

(d) 18

5

7. LiF has a dipole moment of 2.09 × 10−29 C m. The atomic separationis 0.1539 nm. What is the percentage ionic character of the bond?

(a) 10%

(b) 15%

(c) 85%

(d) 90%

(e) 115%

8. Van der Waals forces arise from the dipole-dipole interaction betweenthe fluctuations of the electron cloud (shaded grey) around the nu-cleus (black circle). Which diagram shows the strongest dipole-dipoleattraction?

9. The total heat capacity of a material is the sum C = Cel + Cph ofits electron and phonon contributions. Which of the following graphsdescribes a metal?

C/T

T 2

C/T

T 2

C/T

T 2

6

10. (2 points) Which diagram is consistent with the energy levels described?Write the corresponding letter in the spaces provided.

E E E

( ) electronic donor levels in a doped semiconductor

( ) rotational energy levels in a diatomic molecule

( ) vibrational energy levels in a diatomic molecule

11. A diatomic gas is held at room temperature. Which of the followingexcitation modes is NOT significantly occupied?

(a) rotational

(b) vibrational

(c) electronic

(d) vibrational and electronic

(e) rotational, vibrational, and electronic

12. The number of different Bravais lattices is

(a) 3

(b) 14

(c) 167

(c) unlimited

7

13. The band structure diagram on the left shows the dispersion (energyversus k vector) for electrons in a crystal. The dotted line is the re-sult for noninteracting electrons; the solid line includes the effect ofthe electrons’ Coloumb attraction to the positive ions. Which pointcorresponds to the electronic wavefunction shown on the right?

x

!(x)

ionic potential

Brillouin zone edge

14. (2 points) Match the band structure with the electronic behaviour.Write the appropriate letter in the spaces provided.

EF

EFEF

( ) metal ( ) semiconductor ( ) insulator

8

15. Which diagram best represents the band structure of a light-emittingdiode (LED)?

16. Which plot of current versus bias voltage is that of a diode rectifier?

0

V0

I

0

V0

I

0

V0

I

0

V0

I

17. An electron orbiting a proton in the H atom is analogous to an electronorbiting its donor impurity in a doped semiconductor. The main dif-ference is that inside the semiconductor the relative dielectric constantis enhanced: κ = ε/ε0 1. How do the sizes of these orbits compare?

(a) the orbital radius in the semiconductor is much larger than in theH atom

(b) the orbital radius in the semiconductor is much smaller than inthe H atom

(c) the two orbits are roughly equal in size

9

18. The superconducting current is carried by so-called Cooper pairs. Cooperpairs are what?

(a) two strongly bound phonons of opposite polarization

(b) two weakly bound phonons of equal polarization

(c) two strongly bound electrons of equal spin and momenta

(d) two weakly bound electrons of opposite spin and momenta

19. (2 points) Meissner effect: A metallic cylinder is held at temperatureT in a magnetic field H. Match the three illustrations with the threepoints in the phase diagram.

Hc2Hc1

Tc

H

T

normal

superconducting

normal

superconducting

20. Is the material in the previous question type-I or type-II?

(a) type-I

(b) type-II

10

21. A thin slab of superconductor is threaded by a single flux quantum,as shown. Where the magnetic field line penetrates, the system isnonsuperconducting (i.e., in the normal state). The superconductorresponds by setting up a vortex of circulating supercurrent. Whichdiagram—representing a view from above the slab—gives a correct ac-count of those currents.

!0

22. Do the two vortices attract or repel one another?

(a) attract

(b) repel

(c) neither; they don’t interact

23. Which of the following outer-shell electronic configurations correspondsto an atom with no magnetic moment?

3d

3d

3d

4s

4s

4s

11

24. Identify the magnetic susceptibility of a paramagnet.

0T0

! ! !

0T0

0

T0

25. Circulation of current produces a magnetic moment. Accordingly, thereis an orbital magnetic moment µ associated with the motion of anelectron in an atomic orbital. How is that moment directed?

p+

µ

e!p+

e!p+

e!

µ

12

26. (2 points) Hysteresis and ferromagnetic domains: The arrows representthe magnetic moments of the ferromagnetic domains of an iron disc.The graph below shows the total magnetization as a function of theapplied field H. Positive M and H are directed to the right. Matchthe domain configurations with the points on the graph.

M

H

13

Long Answer Questions (45 points)

27. (3 points) An electron in a parabolic potential V (x) = 12mω2x2 has

evenly-spaced energy levels En = hω(n+ 12) with n = 0, 1, 2, 3, . . . The

ground state wavefunction ψ0(x) is drawn below on the left. In theremaining two panels, sketch the first and second excited-state wave-functions.

V(x)!0(x)

!1(x) !2(x)

x x x

28. (5 points) Draw the molecular diagrams for (a) H2S, (b) H2Te, (c) H3P,(d) H3Sb, and (e) CH4. As usual, use a line to denote a covalent bondand a dot to indicate any unpaired electrons.

14

29. (7 points) (a) A zig-zag chain of positive and negative ions has bond-length a and bending angle θ.

!

a

Show that its Madelung constant is

α =∞∑n=1

[1√

4n(n− 1) cos2 θ + 1− 1

2n cos θ

].

(Remember that (i) a right-angled triangle of hypotenuse a has an edgeof length a cos θ adjacent to a vertex of angle θ and an edge of lengtha sin θ across from the vertex; and (ii) sin2 θ + cos2 θ = 1.)

15

!3

!2

!1

0

0

0

"(#)

#

$/2$/4!$/2 !$/4

1

(b) Over the range −π/2 < θ < π/2, the Madelung constant has itsmaximum value at θ = 0. What is the significance of that?

(c) What does it mean that α is negative when the bending exceedsabout 65?

16

30. (6 points) (a) What is the Fermi speed uF for gold (Au)? Rememberthat uF is the speed of a conduction electron whose energy is equal tothe Fermi energy EF . For Au, EF = 5.55 eV.

(b) The conduction electron density of Au is 5.9× 1028 m−3. Its resis-tivity at room temperature is 2.04µΩ cm. Compute the mean free pathof the conduction electrons.

17

31. (6 points) The four panels below show the energy level diagrams for p-and n-type doped semiconductors in various junction configurations.

(a) The top-left panel represents two disjoint semiconductor samples.Indicate the forbidden region that constitutes the semiconductor gap.

(b) The top-right panel shows a pn junction in equilibrium. Draw anarrow to show the direction of the diffusive forces felt by the donorelectrons. Draw another arrow for the direction of the electrostaticforces felt by the acceptor holes.

(c) The bottom two panels show pn junctions in a circuit. Indicatewhich one is forward biased and which is reverse biased. In the forward-biased case, draw arrows to denote the drift velocities of the electronsand holes.

p n p n

p np n

+ –+–

18

32. (5 points) A doped n-type silicon sample with 1016 electrons per cubiccentimeter in the conduction band has a resistivity of 5× 10−3 Ω m at300 K. Find the mean free path of the electrons. Use 0.2me for theeffective mass of the electron.

19

33. (7 points) The rotational energy of a diatomic molecule is quantizedaccording to Erot(l) = h2l(l + 1)/2I, where l = 0, 1, 2, 3, . . . is theangular momentum quantum number and I is the molecule’s momentof intertia.

r1

r0

r2

m2m1

(a) The moment of inertia is I = m1r21 + m2r

22 = µr2

0, where µ isthe reduced mass. Carbon monoxide (CO) has a bond length r0 =0.1128 nm. Compute its moment of inertia.

20

(b) The rotational modes have degeneracy g(l) = 2l+ 1. If the systemis in thermal equilibrium at room temperature, what is the ratio n5/n1

of the occupation of the l = 5 and l = 1 modes?

21

(c) Draw all the transitions between rotational states l = 0, 1, . . . , 5that result in the emission of a photon. Compute the longest possiblephoton wavelength.

l = 0

l = 1

l = 2

l = 3

l = 4

l = 5

22

34. (3 points) Brass is an alloy of copper (Cu) and zinc (Zn). For low Znconcentrations, brass maintains the face-centred-cubic crystal structureof Cu; the Zn atoms simply substitute for Cu atoms in the lattice.When the temperature is lowered from 300 K to 4 K, the resistivity ofpure copper drops by a much greater factor than that of brass. Why?

35. (3 points) Explain why the mean free path of electrons in a pure metaldecreases as the temperature increases from zero.

23