Atoms to Solids - II
Midterm 1
2026
,Chapter 1
Week 1
1.1 Chapter 2 - Specific Heat in Solids
The law of Dulong-Petit states that the heat capacity of a solid is C = 3R or 3kB per atom, derived
from the average kinetic energy < Ekin >= 12 m < v 2 >= 23 kB T , with the equipartition theorem
stating that 3 independent quadratic modes would give 3 times 12 kb T . In solid matter, this is 3
vibrations, and no translational motion, which makes for 6 quadratic terms. It is a good approxi-
mation, but not quite correct, with the exception of very low temperatures. Boltzmann constructed
a model that fit this - as an atom in a harmonic well due to its neighbouring atoms, where the heat
capacity of the vibration is 3kB per atom. However, quantum mechanics proved important at low
temperatures, and for diamond, room temperatures. Einstein continued by setting an oscillation
frequency ωE , the single Einstein frequency at which every atom in an identical harmonic well os-
cillated. Key assumption: atoms in a solid behave like independent quantum mechanical
harmonic oscillators.
1.1.1 Einstein
The eigenstates of P
a harmonic oscillator are En = ℏω(n + 1/2) in 1D, such that the partition
−βE −βℏω(n+1/2) , with β = kb1T . By taking: < E >= −1 δZ
P
function is Z1D = α e α = n≥− e Z δβ ,
we get the expectation energy, which is: ℏω(nB (βℏω) + 12 ), where n is the Bose occupation factor
nB = ex1−1 . The mode with the Einstein frequency is the excitation up to level nB . Then to
βℏω
obtain the heat capacity, we take: C = d<E>
dT
e
= kb (βℏω)2 (eβℏω −1)2
. In the high-temperature limit,
it tends to kB . To generalize to 3D, we need to take multiple degrees of freedom, leading to
eβℏω
C = 3kB (βℏω)2 (eβℏω −1)2
. At low temperatures, we get a freeze out, where the system gets stuck in
only the ground state and the heat capacity vanishes rapidly.
Diamond does not fit this model, since most materials will have ω to be under-fitting whereas for
diamond it is relatively high for room temperature, due to its strong bonding between patoms and
a relatively low atomic mass, such that the Einstein frequency must be very high: ω = κ/m.
2
, 1.1.2 Debye
However, at small temperatures we see a dependence of T 3 , not agreeing with Einstein. Debye
used quantum oscillations to explain this, quantizing the vibrations as waves, reasoning they are
the same as sound. To then define this, he used Planck’s reasoning for light, with the adjustment
that sounds as three polarizations of the wave-vectors k, since it has additionally motion in the
longitudinal direction, whereas light only has two transverse modes. For simplicity, they all have
the same speed.
To compute this, we use periodicity, where any wave needs to be of the same value for both position
r, and after some periodic length L, such that eikr = eikr+L P, such that k = 2πnL . This is a sum
L
for integer n. To generalize, we replace it with an integral: k = 2π
R∞
dk
. For three dimensions,
−∞
it would simply be the product of this for three directions. Debye also set that the waves now
had oscillation modes ω(k) = v|k|, where v is the speed of sound. So, for each k there are three
oscillation modes, for each direction. The rest is the same as Einstein did, where the expectation
energy was computed based on the Z-function, which gives us an integral that can beRsolved using
∞
a conversion to a one-dimensional integral based on spherical symmetry ( dk → 4π 0 k 2 dk and
R
4πL3 R ∞ 2 3 1 3
k = ω/v, gives us: < E >= 3 (2π) 3 0 ω dω(1/v )(ℏω)(nB (βℏω) + 2 ), where nL = N , where n is
2
the density of atoms, and N the total atoms. The density of states is given by g(ω) = N 9ω
ω3
, where
d
we got the Debye frequency ωd = 6π 2 nv 3 . The density of states is the total number of oscillation
modes with frequencies in a certain interval (g(ω)dω). When integrated over all frequencies will
give the expected energy, alongside a temperature-independent constant. The additional half in
the equation is the zero-point energy of each oscillator, independent of temperature, unimportant
3 4
for the heat capacity. This eventually leads to the Debye temperature ratio: C = N kB T 3T 12π 5 ,
Debye
ℏωD
where the Debye temperature is TD = kB .
The problem now is that the Debye C does not level off for high temperatures up to an infinite
number of sound wave modes, with a finite number of atoms. However, there should be the
same amount of modes as degrees of freedom, such that we should only consider modes up to a
certainR cut-off frequency, which should give 3N sound modes (3 degrees of freedom per particle):
ω
3N = 0 c dωg(ω). This cut-off does not matter for low temperatures, as the Bose factor will very
rapidly go to zero at frequencies well below the cut-off. At high temperatures, it will then give
the correct Dulong-Petit maximum, with the cut-off frequency being exactly equal to the Debye
frequency.
Flaws of Debye The cut-off is not physically reasoned but more of a succesful cheat, the sound
speed of vk does not work for very high frequencies, at intermediate temperatures it proves inac-
curate, and metals have a different C than we would have computed using Debye: C = γT + αT 3 .
It has an additional term that becomes dominant at low temperatures. These can be investigated
using crystal structures or the behaviour of electrons.
3
Midterm 1
2026
,Chapter 1
Week 1
1.1 Chapter 2 - Specific Heat in Solids
The law of Dulong-Petit states that the heat capacity of a solid is C = 3R or 3kB per atom, derived
from the average kinetic energy < Ekin >= 12 m < v 2 >= 23 kB T , with the equipartition theorem
stating that 3 independent quadratic modes would give 3 times 12 kb T . In solid matter, this is 3
vibrations, and no translational motion, which makes for 6 quadratic terms. It is a good approxi-
mation, but not quite correct, with the exception of very low temperatures. Boltzmann constructed
a model that fit this - as an atom in a harmonic well due to its neighbouring atoms, where the heat
capacity of the vibration is 3kB per atom. However, quantum mechanics proved important at low
temperatures, and for diamond, room temperatures. Einstein continued by setting an oscillation
frequency ωE , the single Einstein frequency at which every atom in an identical harmonic well os-
cillated. Key assumption: atoms in a solid behave like independent quantum mechanical
harmonic oscillators.
1.1.1 Einstein
The eigenstates of P
a harmonic oscillator are En = ℏω(n + 1/2) in 1D, such that the partition
−βE −βℏω(n+1/2) , with β = kb1T . By taking: < E >= −1 δZ
P
function is Z1D = α e α = n≥− e Z δβ ,
we get the expectation energy, which is: ℏω(nB (βℏω) + 12 ), where n is the Bose occupation factor
nB = ex1−1 . The mode with the Einstein frequency is the excitation up to level nB . Then to
βℏω
obtain the heat capacity, we take: C = d<E>
dT
e
= kb (βℏω)2 (eβℏω −1)2
. In the high-temperature limit,
it tends to kB . To generalize to 3D, we need to take multiple degrees of freedom, leading to
eβℏω
C = 3kB (βℏω)2 (eβℏω −1)2
. At low temperatures, we get a freeze out, where the system gets stuck in
only the ground state and the heat capacity vanishes rapidly.
Diamond does not fit this model, since most materials will have ω to be under-fitting whereas for
diamond it is relatively high for room temperature, due to its strong bonding between patoms and
a relatively low atomic mass, such that the Einstein frequency must be very high: ω = κ/m.
2
, 1.1.2 Debye
However, at small temperatures we see a dependence of T 3 , not agreeing with Einstein. Debye
used quantum oscillations to explain this, quantizing the vibrations as waves, reasoning they are
the same as sound. To then define this, he used Planck’s reasoning for light, with the adjustment
that sounds as three polarizations of the wave-vectors k, since it has additionally motion in the
longitudinal direction, whereas light only has two transverse modes. For simplicity, they all have
the same speed.
To compute this, we use periodicity, where any wave needs to be of the same value for both position
r, and after some periodic length L, such that eikr = eikr+L P, such that k = 2πnL . This is a sum
L
for integer n. To generalize, we replace it with an integral: k = 2π
R∞
dk
. For three dimensions,
−∞
it would simply be the product of this for three directions. Debye also set that the waves now
had oscillation modes ω(k) = v|k|, where v is the speed of sound. So, for each k there are three
oscillation modes, for each direction. The rest is the same as Einstein did, where the expectation
energy was computed based on the Z-function, which gives us an integral that can beRsolved using
∞
a conversion to a one-dimensional integral based on spherical symmetry ( dk → 4π 0 k 2 dk and
R
4πL3 R ∞ 2 3 1 3
k = ω/v, gives us: < E >= 3 (2π) 3 0 ω dω(1/v )(ℏω)(nB (βℏω) + 2 ), where nL = N , where n is
2
the density of atoms, and N the total atoms. The density of states is given by g(ω) = N 9ω
ω3
, where
d
we got the Debye frequency ωd = 6π 2 nv 3 . The density of states is the total number of oscillation
modes with frequencies in a certain interval (g(ω)dω). When integrated over all frequencies will
give the expected energy, alongside a temperature-independent constant. The additional half in
the equation is the zero-point energy of each oscillator, independent of temperature, unimportant
3 4
for the heat capacity. This eventually leads to the Debye temperature ratio: C = N kB T 3T 12π 5 ,
Debye
ℏωD
where the Debye temperature is TD = kB .
The problem now is that the Debye C does not level off for high temperatures up to an infinite
number of sound wave modes, with a finite number of atoms. However, there should be the
same amount of modes as degrees of freedom, such that we should only consider modes up to a
certainR cut-off frequency, which should give 3N sound modes (3 degrees of freedom per particle):
ω
3N = 0 c dωg(ω). This cut-off does not matter for low temperatures, as the Bose factor will very
rapidly go to zero at frequencies well below the cut-off. At high temperatures, it will then give
the correct Dulong-Petit maximum, with the cut-off frequency being exactly equal to the Debye
frequency.
Flaws of Debye The cut-off is not physically reasoned but more of a succesful cheat, the sound
speed of vk does not work for very high frequencies, at intermediate temperatures it proves inac-
curate, and metals have a different C than we would have computed using Debye: C = γT + αT 3 .
It has an additional term that becomes dominant at low temperatures. These can be investigated
using crystal structures or the behaviour of electrons.
3
