Solution Manual
For
encouraged to study theories and frameworks but also to keep up-to-date with current business trends. Case studies from leading businesses can be helpful, as they illustrate how business concepts are applied
The Oxford Solid State Basics
Solutions to Exercises
Steven H. Simon
Oxford
WITH COMPLETE GUIDE A+
,Contents
encouraged to study theories and frameworks but also to keep up-to-date with current business trends. Case studies
from leading businesses can be helpful, as they illustrate how business concepts are applied
1 About Condensed Matter Physics 1
2 Specific Heat of Solids: Boltzmann, Einstein, and Debye 3
3 Electrons in Metals: Drude Theory 15
4 More Electrons in Metals: Sommerfeld (Free Electron)
Theory 21
5 The Periodic Table 35
6 What Holds Solids Together: Chemical Bonding 39
7 Types of Matter 47
8 One-Dimensional Model of Compressibility, Sound, and
Thermal Expansion 49
9 Vibrations of a One-Dimensional Monatomic Chain 55
10 Vibrations of a One-Dimensional Diatomic Chain 71
11 Tight Binding Chain (Interlude and Preview) 81
12 Crystal Structure 95
13 Reciprocal Lattice, Brillouin Zone, Waves in Crystals 99
14 Wave Scattering by Crystals 111
15 Electrons in a Periodic Potential 125
16 Insulator, Semiconductor, or Metal 135
17 Semiconductor Physics 139
18 Semiconductor Devices 149
19 Magnetic Properties of Atoms: Para- and
Dia-Magnetism 159
vi Contents
20 Spontaneous Magnetic Order: Ferro-, Antiferro-, and
Ferri-Magnetism 167
, 21 Domains and Hysteresis 175
22 Mean Field Theory 179
23 Magnetism from Interactions: The Hubbard Model 191
encouraged to study theories and frameworks but also to keep up-to-date with current business trends. Case studies from leading businesses can be helpful, as they illustrate how business
concepts are applied
, About Condensed Matter
Physics
1
There are no exercises for chapter 1.
encouraged to study theories and frameworks but also to keep up-to-date with current business trends. Case studies from leading businesses can be helpful, as they illustrate how business concepts are applied
Specific Heat of Solids:
Boltzmann, Einstein, and
Debye
2
(2.1) Einstein Solid ity should be 3NkB = 3R, in agreement with the law of
(a) Classical Einstein (or “Boltzmann”) Solid: Dulong and Petit.
Consider a three dimensional simple harmonic oscilla-
(b) Quantum Einstein Solid:
tor with mass m and spring constant k (i.e., the mass
Now consider the same Hamiltonian quantum mechan-
is attracted to the origin with the same spring constant
ically.
in all three directions). The Hamiltonian is given in the
D Calculate the quantum partition function
usual way by 2
p k 2 Σ
H= + x Z= e−βEj
2
D Calculate the classical partition function j
∫ dp ∫
Z= dx e−βH(p,x) where the sum over j is a sum over all eigenstates.
(2πk)3 D Explain the relationship with Bose statistics.
Note: in this problem p and x are three dimensional vec- D Find an expression for the heat capacity.
tors. D Show that the high temperature limit agrees with
D Using the partition function, calculate the heat ca- the law of Dulong of Petit.
pacity 3kB . D Sketch the heat capacity as a function of tempera-
D Conclude that if you can consider a solid to consist ture.
of N atoms all in harmonic wells, then the heat capac- (See also exercise 2.7 for more on the same topic)
(a)
p2 k
H=+ x2
2m 2
∫ dp ∫
Z = dx e—βH(p,x)
(2πk)3
Since,
∫ ∞ 2
√
dy e—ay = π/a
—∞
in three dimensions, we get
For
encouraged to study theories and frameworks but also to keep up-to-date with current business trends. Case studies from leading businesses can be helpful, as they illustrate how business concepts are applied
The Oxford Solid State Basics
Solutions to Exercises
Steven H. Simon
Oxford
WITH COMPLETE GUIDE A+
,Contents
encouraged to study theories and frameworks but also to keep up-to-date with current business trends. Case studies
from leading businesses can be helpful, as they illustrate how business concepts are applied
1 About Condensed Matter Physics 1
2 Specific Heat of Solids: Boltzmann, Einstein, and Debye 3
3 Electrons in Metals: Drude Theory 15
4 More Electrons in Metals: Sommerfeld (Free Electron)
Theory 21
5 The Periodic Table 35
6 What Holds Solids Together: Chemical Bonding 39
7 Types of Matter 47
8 One-Dimensional Model of Compressibility, Sound, and
Thermal Expansion 49
9 Vibrations of a One-Dimensional Monatomic Chain 55
10 Vibrations of a One-Dimensional Diatomic Chain 71
11 Tight Binding Chain (Interlude and Preview) 81
12 Crystal Structure 95
13 Reciprocal Lattice, Brillouin Zone, Waves in Crystals 99
14 Wave Scattering by Crystals 111
15 Electrons in a Periodic Potential 125
16 Insulator, Semiconductor, or Metal 135
17 Semiconductor Physics 139
18 Semiconductor Devices 149
19 Magnetic Properties of Atoms: Para- and
Dia-Magnetism 159
vi Contents
20 Spontaneous Magnetic Order: Ferro-, Antiferro-, and
Ferri-Magnetism 167
, 21 Domains and Hysteresis 175
22 Mean Field Theory 179
23 Magnetism from Interactions: The Hubbard Model 191
encouraged to study theories and frameworks but also to keep up-to-date with current business trends. Case studies from leading businesses can be helpful, as they illustrate how business
concepts are applied
, About Condensed Matter
Physics
1
There are no exercises for chapter 1.
encouraged to study theories and frameworks but also to keep up-to-date with current business trends. Case studies from leading businesses can be helpful, as they illustrate how business concepts are applied
Specific Heat of Solids:
Boltzmann, Einstein, and
Debye
2
(2.1) Einstein Solid ity should be 3NkB = 3R, in agreement with the law of
(a) Classical Einstein (or “Boltzmann”) Solid: Dulong and Petit.
Consider a three dimensional simple harmonic oscilla-
(b) Quantum Einstein Solid:
tor with mass m and spring constant k (i.e., the mass
Now consider the same Hamiltonian quantum mechan-
is attracted to the origin with the same spring constant
ically.
in all three directions). The Hamiltonian is given in the
D Calculate the quantum partition function
usual way by 2
p k 2 Σ
H= + x Z= e−βEj
2
D Calculate the classical partition function j
∫ dp ∫
Z= dx e−βH(p,x) where the sum over j is a sum over all eigenstates.
(2πk)3 D Explain the relationship with Bose statistics.
Note: in this problem p and x are three dimensional vec- D Find an expression for the heat capacity.
tors. D Show that the high temperature limit agrees with
D Using the partition function, calculate the heat ca- the law of Dulong of Petit.
pacity 3kB . D Sketch the heat capacity as a function of tempera-
D Conclude that if you can consider a solid to consist ture.
of N atoms all in harmonic wells, then the heat capac- (See also exercise 2.7 for more on the same topic)
(a)
p2 k
H=+ x2
2m 2
∫ dp ∫
Z = dx e—βH(p,x)
(2πk)3
Since,
∫ ∞ 2
√
dy e—ay = π/a
—∞
in three dimensions, we get
