Quinn J.J., Yi K.-S. Solid State Physics: Principles and Modern Applications
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Uncorrected Proof
BookID 160928 ChapID BM Proof# 1 - 29/07/09
518 A Operator Method for the Harmonic Oscillator Problem
Assume that |0 > is normalized (see (A.20)). Then, we can write 51
n|n = |C
n
|
2
0
a
n
a
†
n
0
!
. (A.23)
Using the relations given by (A.12) allows one to show that
52
a
n
a
†
n
|0 >= n!|0 >. (A.24)
So that
53
|n>=
1
√
n!
a
†
n
|0 > (A.25)
is the normalized eigenfunction for the nth excited state.
54
One can use Ψ
0
(x)=α
1/2
π
−1/4
e
−
1
2
α
2
x
2
and express a
†
n
in terms of p and 55
x to obtain 56
Ψ
n
(x)=
1
√
n!
−ı (−ı¯h∂/∂x)+mωx
√
2m¯hω
n
α
1/2
π
1/4
e
−
α
2
x
2
2
, (A.26)
This can be simplified a little to the form
57
Ψ
n
(x)=
(α/
√
π)
1/2
(−)
n
α
n
(2
n
n!)
1/2
∂
∂x
− α
2
x
n
e
−
α
2
x
2
2
. (A.27)
Summary
58
The Hamiltonian of the simple harmonic oscillator can be written 59
H =¯hω
a
†
a +
1
2
. (A.28)
and H|n>=¯hω(n +
1
2
)|n>. The excited eigenkets can be written 60
|n>=
1
√
n!
a
†
n
|0 >. (A.29)
The eigenfunctions (A.29) form a complete orthonormal set, i.e.,
61
n|m = δ
nm
, (A.30)
and
62
n
|nn| =1. (A.31)
The creation and annihilation operators satisfy the commutation relation
63
[a, a
†
]=1.
Problems 64
A.1. Prove that [
ˆ
A,
ˆ
B
ˆ
C]
−
=
ˆ
B[
ˆ
A,
ˆ
C]
−
+[
ˆ
A,
ˆ
B]
−
ˆ
C,where
ˆ
A,
ˆ
B,and
ˆ
C are
65
quantum mechanical operators. 66
A.2. Prove that [ˆa, (ˆa
†
)
n
]
−
= n(ˆa
†
)
n−1
. 67
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BookID 160928 ChapID BM Proof# 1 - 29/07/09
B 68
Neutron Scattering 69
A beam of neutrons interacts with a crystal through a potential 70
V (r)=
R
i
v(r − R
i
), (B.1)
where r is the position operator of the neutron, and R
i
is the position operator 71
of the i
th
atom in the crystal. It is common to write v(r −R
i
)intermsofits 72
Fourier transform v(r)=
k
v
k
e
ik·r
. Then, (B.1) can be rewritten 73
V (r)=
k,R
i
v
k
e
ik·(r−R
i
)
. (B.2)
The potential v(r) is very short-range, and v
k
is almost independent of k. 74
The k-independent coefficient v
k
is usually expressed as v =
2π¯h
2
a
M
n
,wherea 75
is defined as the scattering length and M
n
is the mass of the neutron. 76
The initial state of the system can be expressed as 77
Ψ
i
(R
1
, R
2
,...,r)=V
−1/2
e
i
p
¯h
·r
|n
1
,n
2
,...,n
N
. (B.3)
Here, V
−1/2
e
i
p
¯h
·r
is the initial state of a neutron of momentum p.Theket 78
|n
1
,n
2
,...,n
N
represents the initial state of the crystal, with n
i
phonons in 79
mode i. The final state, after the neutron is scattered, is 80
Ψ
f
(R
1
, R
2
,...,r)=V
−1/2
e
i
p
¯h
·r
|m
1
,m
2
, ··· ,m
N
. (B.4)
The transition rate for going from Ψ
i
to Ψ
f
can be calculated from Fermi’s 81
golden rule. 82
R
i→f
=
2π
¯h
|Ψ
f
|V |Ψ
i
|
2
δ (E
f
− E
i
) . (B.5)
Here, E
i
and E
f
are the initial and final energies of the entire system. Let us 83
write ε
i
= E
i
−
p
2
2M
n
and ε
f
= E
f
−
p
2
2M
n
. The total rate of scattering out of
84
Uncorrected Proof
BookID 160928 ChapID BM Proof# 1 - 29/07/09
520 B Neutron Scattering
initial state Ψ
i
is given by 85
R
outof i
=
2π
¯h
f
δ (ε
f
− ε
i
− ¯hω) |Ψ
f
|V |Ψ
i
|
2
, (B.6)
where ¯hω =
p
2
−p
2
2M
n
is the change in energy of the neutron. If we write p
= 86
p +¯hk,where¯hk is the momentum transfer, the matrix element becomes 87
i,k
m
1
,m
2
,...,m
N
v
k
e
−ik·R
i
n
1
,n
2
,...,n
N
. (B.7)
But we can take v
k
= v outside the sum since it is a constant. In addition, 88
we can write R
j
= R
0
j
+ u
j
and 89
u
j
=
qλ
¯h
2MNω
qλ
1/2
e
iq·R
0
j
ˆε
qλ
a
qλ
− a
†
−qλ
. (B.8)
The matrix element of e
iq·u
j
between harmonic oscillator states |n
1
,n
2
,...,n
N
90
and |m
1
,m
2
,...,m
N
is exactly what we evaluated earlier in studying the 91
M¨ossbauer effect. By using our earlier results and then summing over the 92
atoms in the crystal, one can obtain the transition rate. The cross-section is 93
related to the transition rate divided by the incident flux. 94
One can find the following result for the cross-section: 95
dσ
dΩdω
=
p
p
N
a
2
¯h
S(q, ω), (B.9)
where dΩ is solid angle, dω is energy transfer, N is the number of atoms in the
96
crystal, a is the scattering length, and S(q, ω) is called the dynamic structure 97
factor.Itisgivenby 98
S(q, ω)=N
−1
f
j
m
1
,...,m
N
e
iq·u
j
n
1
,...,n
N
2
δ (ε
f
− ε
i
− ¯hω) .
(B.10)
Again, there is an elastic scattering part of S(q, ω), corresponding to
99
no-phonon emission or absorption in the scattering process. For that case 100
S(q, ω)isgivenby 101
S
0
(q, ω)=e
−2W
δ(ω)N
K
δ
q,K
. (B.11)
Here, e
−2W
is the Debye–Waller factor. W is proportional to 102
n
1
,...,n
N
[q · u
0
]
2
n
1
,...,n
N
!
.
Uncorrected Proof
BookID 160928 ChapID BM Proof# 1 - 29/07/09
B Neutron Scattering 521
From (B.11) we see that there are Bragg peaks. In the harmonic approximation 103
the peaks are δ-functions [because of δ(ω)] due to energy conservation. The 104
peaks occur at momentum transfer p
− p = K, a reciprocal lattice vector. 105
In the early days of X-ray scattering there was some concern over whether 106
the motion of the atoms (both zero point and thermal motion) would broaden 107
the δ-function peaks and make X-ray diffraction unobservable. The result, 108
in the harmonic approximation, is that the δ-function peaks are still there, 109
but their amplitude is reduced by the Debye–Waller factor e
−2W
. 110
For the one-phonon contribution to the cross-section, we obtain 111
dσ
dΩdω
= N e
−2W
p
p
a
2
λ
(q · ˆε
qλ
)
2
2Mω
qλ
{(1 + n
qλ
) δ (ω + ω
qλ
)+n
qλ
δ (ω − ω
qλ
)}.
(B.12)
There are still unbroadened δ-function peaks at ε
f
±¯hω
qλ
= ε
i
, corresponding 112
to the emission or absorption of a phonon. The peaks occur at a scattering 113
angle determined from p
−p = q + K where K is a reciprocal lattice vector. 114
The amplitude again contains the Debye–Waller factor e
−2W
. Inelastic neu- 115
tron scattering allows a experimentalist to determine the phonon frequencies 116
ω
qλ
as a function of q and of λ. 117
The broadening of the δ-function peaks occurs only when anharmonic 118
terms are included in the calculation. Anharmonic forces lead to phonon– 119
phonon scattering and to finite phonon lifetimes. 120
Uncorrected Proof
BookID 160928 ChapID BM Proof# 1 - 29/07/09
121
References 122
N.W. Ashcroft, N.D. Mermin, Solid State Physics (Thomson Learning, 1976)
AQ: The
references are not
cited in the text.
Please check and
confirm.
123
O. Madelung, Introduction to Solid-State Theory (Springer, Berlin, 1978) 124
C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1996) 125
G. Grosso, G.P. Parravicini, Solid State Physics (Academic, London, 2000) 126
H. Ibach, H. L¨uth, Solid-State Physics An Introduction to Principles of Material 127
Science (Springer, Heidelberg, 1995) 128
F.S. Levin, An Introduction to Quantum Theory (Cambridge University Press, 129
Cambridge, 2002) 130
Chapter 1 Crystal Structures 131
R.W.G. Wyckoff, Crystal Structures, vol. 1–5 (Wiley, New York, 1963–1968) 132
G. Burns, Solid State Physics (Academic, New York, 1985) 133
Chapter 2 Lattice Vibrations 134
L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953) 135
R.A. Smith, Wave Mechanics of Crystalline Solids, 2nd edn. (Chapman & Hall, 136
London, 1969) 137
J.M. Ziman, Electrons and Phonons (Oxford University Press, Oxford, 1972) 138
P.F. Choquard, The Anharmonic Crystal (Benjamin, New York, 1967) 139
G.K. Wertheim, M¨ossbauer Effect: Principles and Applications (Academic, New 140
York, 1964) 141
Chapter 3 Free Electron Theory of Metals 142
P. Drude, Ann. Phys. (Leipzig) 1, 566 (1900) 143
K. Huang, Statistical Mechanics, 2nd edn. (Wiley, New York, 1987) 144
C. Kittel, Elementary Statistical Physics (Wiley, New York, 1958) 145
R. Kubo, Statistical Mechanics, 7th edn. (North-Holland, Amsterdam, 1988) 146
R.K. Pathria, Statistical Mechanics, 7th edn. (Pergamon, Oxford, 1972) 147
A.H. Wilson The Theory of Metals, 2nd edn. (Cambridge University Press, Cam- 148
bridge, 1965) 149
Uncorrected Proof
BookID 160928 ChapID BM Proof# 1 - 29/07/09
524 References
Chapter 4 Elements of Band Theory 150
J. Callaway, Energy Band Theory (Academic, New York, 1964) 151
J.M. Ziman, Electrons and Phonons (Oxford University Press, Oxford, 1972) 152
W.A. Harrision, Electronic Structure and the Properties of Solids (Freeman, San 153
Francisco, 1980) 154
J. Callaway, Quantum Theory of the Solid State, 2nd edn. (Academic, New York, 155
1991) 156
Chapter 5 Use of Elementary Group Theory in Calculating Band 157
Structure 158
V. Heine Group Theory in Quantum Mechanics – An Introduction to Its Present 159
Usage (Pergamon, Oxford, 1977) 160
S.L. Altmann, Band Theory of Solids: An Introduction from the Point of View of 161
Symmetry (Clarendon, Oxford, 1994) 162
H. Jones, The Theory of Brillouin Zones and Electronic States in Crystals (North- 163
Holland, Amsterdam, 1975) 164
L.M. Falicov, group theory and Its applications (University of Chicago Press, 165
Chicago, 1966) 166
M. Tinkham, Group Theory and Quantum Mechanics (McGraw-Hill, New York, 167
1964) 168
P.Y. Yu, M. Cardona, Fundamentals of Semiconductors (Springer, Berlin, 1996) 169
Chapter 6 More Band Theory and the Semiclassical Approximation 170
J.C. Slater, Symmetry and Energy Band in Crystals (Dover, New York, 1972) 171
D. Long, Energy Bands in Semiconductors (Wiley, New York, 1968) 172
J. Callaway, Energy Bands Theory (Academic, New York, 1964) 173
Chapter 7 Semiconductors 174
J.C. Phillips, Bonds and Bands in Semiconductors (Academic, New York, 1973) 175
A. Anselm, Introduction to Semiconductor Theory (Prentice-Hall, Englewood Cliffs, 176
NJ, 1981) 177
P.S. Kireev, Semiconductor Physics (Mir, Moscow, 1978) 178
P.Y. Yu, M. Cardona, Fundamentals of Semiconductors (Springer, Berlin, 1996) 179
K.F. Brennan, THE PHYSICS OF SEMICONDUCTORS with applications to 180
optoelectronic devices (Cambridge University Press, Cambridge, 1999)
181
M. Balkanski, R.F. Wallis, Semiconductor Physics and Applications (Oxford Uni- 182
versity Press, Oxford, 2000) 183
K. Seeger, Semiconductor Physics – An Introduction, 8th edn. (Springer, Berlin, 184
2002) 185
S.M. Sze, Semiconductor Devices: Physics and Technology, 2nd edn. (Wiley, New 186
York, 2001) 187
G. Bastard, wave mechanics applied to semiconductor heterostructures (Halsted, 188
New York, 1986) 189
T. Ando, A. Fowler, F. Stern, Rev. Mod. Phys. 54, 437 (1982) 190
T. Chakraborty, P. Pietil¨ainen, The Quantum Hall Effects, Integral and Fractional, 191
2nd edn. (Springer, Berlin, 1995) 192
R.E. Prange, S.M. Girvin (eds.), The Quantum Hall Effect, 2nd ed., (Springer, New 193
York, 1990) 194
Uncorrected Proof
BookID 160928 ChapID BM Proof# 1 - 29/07/09
References 525
K.vonKlitzing,G.Dorda,M.Pepper,Phys. Rev. Lett. 45, 494 (1980) 195
D.C. Tsui, H.L. St¨ormer, A.C. Gossard, Phys. Rev. Lett. 48, 1559 (1982) 196
R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983) 197
Chapter 8 Dielectric Properties of Solids 198
J.M. Ziman, Principles of the Theory of Solids (Cambridge University Press, 199
Cambridge, 1972) 200
G. Grosso, G.P. Parravicini, Solid State Physics (Academic, London, 2000) 201
J. Callaway, Quantum Theory of the Solid State, 2nd edn. (Academic, New York, 202
1991) 203
P.Y. Yu, M. Cardona, Fundamentals of Semiconductors (Springer, Berlin, 1996) 204
A.A. Abrikosov, Fundamentals of the Theory of Metals (North-Holland, Amsterdam, 205
1988) 206
M. Balkanski, Optical Properties of Semiconductors (North-Holland, Amsterdam, 207
1994) 208
Chapter 9 Magnetism in Solids 209
D.C. Mattis, The Theory of Magnetism vols. I and II (Springer, Berlin, 1981) 210
R. Kubo, T. Nagamiya (eds.), Solid Sate Physics (McGraw-Hill, New York, 1969) 211
W. Jones, N.H. March, Theoretical Solid State Physics (Wiley-Interscience, New 212
York, 1973) 213
Chapter 10 Magnetic Ordering and Spin Waves 214
C. Herring, Direct exchange between well separated atoms in Magnetism,vol.2B,G. 215
Rado, H. Suhl (eds.) (Academic, New York, 1965) 216
R.M. White, Quantum Theory of Magnetism (Springer, Berlin, 1995) 217
D.C. Mattis, The Theory of Magnetism vols. I and II (Springer, Berlin, 1981) 218
C. Kittel, Quantum Theory of Solids, 2nd edn. (Springer, Berlin, 1995) 219
Chapter 11 Many Body Interactions – Introduction 220
D. Pines, Elementary Excitations in Solids, 2nd edn. (Benjamin, New York, 1963) 221
K.S. Singwi, M.P. Tosi, Correlations in Electron Liquids in Solid State Physics
36, 222
177 (1981), ed. by H. Ehrenreich, F. Seitz, D. Turnbull (Academic, New York, 223
1981) 224
D. Pines, P. Noizi`eres, The Theory of Quantum Liquids (Perseus Books, Cambridge, 225
1999) 226
P.L. Taylor, O. Heinonen, A Quantum Approach to Condensed Matter Physics 227
(Cambridge University Press, Cambridge, 2002) 228
J. Hubbard, Proc. Roy. Soc. A 276, 238 (1963); A 277, 237 (1964); A 281, 401 229
(1964) 230
A.W. Overhauser, Phys. Rev. Lett. 4, 462 (1960); Phys. Rev. 128, 1437 (1962) 231
Chapter 12 Many Body Interactions – Green’s Function Method 232
A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski, Method of Quantum Field Theory 233
in Statistical Physics (Prentice-Hall, Englewood Cliffs, NJ, 1963) 234
G. Mahan, Many-Particle Physics (Plenum, New York, 2000) 235
Uncorrected Proof
BookID 160928 ChapID BM Proof# 1 - 29/07/09
526 References
P.L. Taylor, O. Heinonen, A Quantum Approach to Condensed Matter Physics 236
(Cambridge University Press, Cambridge, 2002) 237
J. Hubbard, Proc. Roy. Soc. A 276, 238 (1963); A 277, 237 (1964); A 281, 401 238
(1964) 239
Chapter 13 Semiclassical Theory of Electrons 240
I.M.Lifshitz,A.M.Kosevich,Soviet Physics JETP 2, 636 (1956) 241
D. Shoenberg, Phil.Roy.Soc.(London), Ser A 255, 85 (1962) 242
J. Condon, Phys. Rev. 145, 526 (1966) 243
J.J. Quinn, Nature 317, 389 (1985) 244
M.H. Cohen, M.J. Harrison, W.A. Harrison, Phys. Rev. 117, 937 (1960) 245
M.P. Greene, H.J. Lee, J.J. Quinn, S. Rodriguez, Phys. Rev. 177, 1019 (1969) 246
Chapter 14 Electrodynamics of Metals 247
For a review of magnetoplasma surface waves see, for example, J.J. Quinn, K.W. 248
Chiu, Magnetoplasma Surface Waves in Metals and Semiconductors, Polaritons, 249
ed. by E. Burstein, F. DeMartini (Pergamon, New York, 1971), p. 259 250
P.M. Platzman, P.J. Wolff, Waves and Interactions in Solid State Plasmas, Solid 251
State Physics-Supplement 13, ed. by H. Ehrenreich, F. Seitz, D. Turnbull 252
(Academic, New York, 1973) 253
E.D. Palik, B.G. Wright, Free-Carrier Magnetooptic Effects, Semiconductors and 254
Semimetals 3, ed. by R.K. Willardson, A.C. Beer (Academic, New York, 1967), 255
p. 421 256
G. Dresselhaus, A.F. Kip, C. Kittel, Phys. Rev. 98, 368 (1955) 257
M.A. Lampert, S. Tosima, J.J. Quinn, Phys. Rev. 152, 661 (1966) 258
I. Bernstein, Phys. Rev. 109, 10 (1958) 259
Chapter 15 Superconductivity 260
F. London, Superfluids vols. 1 and 2, (Wiley, New York, 1954)
261
D. Shoenberg, Superconductivity, (Cambridge University Press, Cambridge, 1962)
AQ: Please check
if the insertion of
publisher name
and location is
correct.
262
J.R. Schrieffer, Superconductivity, (W.A. Benjamin, New York, 1964) 263
J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108, 1175 (1957) 264
A.A. Abrikosov, Fundamentals of the Theory of Metals (North-Holland, Amsterdam, 265
1988) 266
M. Tinkham, Introduction to Superconductivity 2nd ed. (McGraw-Hill, New York, 267
1996) 268
G. Rickayzen, Theory of Superconductivity (Wiley-Interscience, New York, 1965) 269
R.D. Parks, ed., Superconductivity vols. I and II (Dekker, New York, 1969) 270
F. London, H. London, Proc. Roy. Soc. (London) A 149, 71 (1935) 271
Chapter 16 The Fractional Quantum Hall Effect 272
R.E. Prange, S.M. Girvin (eds.), The Quantum Hall Effect, 2nd edn. (Springer, 273
Berlin, 1990) 274
T. Chakraborty, P. Pietil¨ainen, The Quantum Hall Effects Fractional and Integral, 275
2nd edn. (Springer, Berlin, 1995) 276
K.vonKlitzing,G.Dorda,M.Pepper,Phys. Rev. Lett. 45, 494 (1980) 277
R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983) 278
D.C.Tsui,H.L.Stormer,A.C.Gossard,Phys.Rev.Lett.48, 1559 (1982) 279
Uncorrected Proof
BookID 160928 ChapID BM Proof# 1 - 29/07/09
References 527
A.Shapere,F.Wilczek(eds.),Geometric Phases in Physics (World Scientific, 280
Singapore, 1989) 281
F. Wilczek, Fractional Statistics and Anyon Superconductivity (World Scientific, 282
Singapore, 1990) 283
Z.F. Ezawa, Quantum Hall Effects: Field Theoretical Approach and Related Topics 284
(World Scientific, Singapore, 2000) 285
O. Heinonen, Composite Fermions: A Unified View of the Quantum Hall Regime 286
(Singapore, World Scientific, 1998) 287
P. Sitko, S.N. Yi, K.-S. Yi, J.J. Quinn, Phys. Rev. Lett. 76, 3396 (1996) 288
Uncorrected Proof
BookID 160928 ChapID BM Proof# 1 - 29/07/09
Index
acceptor, 185
acoustic attenuation, 417
acoustic wave, 441
electromagnetic generation of, 443
adiabatic approximation, 366
adiabatic demagnetization, 265
Aharanov–Bohm phase, 494
amorphous semiconductor, 206
Anderson localization, 205
Anderson model, 207
anharmonic effect, 69
anisotropy constant, 281
anisotropy energy, 280
antiferromagnet, 283
ground state energy, 299
antiferromagnetism, 282
anyon, 492
anyon parameter, 492
atomic polarizability, 217
atomic scattering factor, 21
attenuation coefficient, 381
Azbel–Kaner effect, 430
BCS theory, 462
ground state, 467
Bernstein mode, 435
binding energy, 27
Bloch electron
in a dc magnetic field, 391
semiclassical approximation for, 168
Bloch’s theorem, 112
Bogoliubov–Valatin transformation, 469
Bohr magneton, 250
effective number of, 257
Boltzmann’s equation, 83
linearized, 95
Bose–Einstein distribution, 56
Bragg reflection, 17
Bragg’s law, 17
Bravais lattice
three-dimensional, 10
two-dimensional, 9
Brillouin function, 256
bulk mode
for an infinite homogeneous medium,
230
longitudinal mode, 230
of coupled plasmon–LO phonon, 231
transverse mode, 230, 232
carrier concentration, 182
extrinsic case, 187
intrinsic case, 184
Cauchy’s theorem, 344
charge density, 334
external, 217
polarization, 217
chemical potential, 86
actual overall, 347
local, 347
Chern–Simons term, 492
Chern–Simons transformation, 493
effective magnetic field, 494
Clausius Mossotti relation, 221