Statistical Physics of Particles

Mehran Kardar

Chapter 7 Ideal quantum gases - all with Video Answers

Educators

Chapter Questions

Problem 1

Identical particle pair: let $Z_1(m)$ denote the partition function for a single quantum particle of mass $m$ in a volume $V$.
(a) Calculate the partition function of two such particles, if they are bosons, and also if they are (spinless) fermions.
(b) Use the classical approximation $Z_1(m)=V / \lambda^3$ with $\lambda=h / \sqrt{2 \pi m k_B T}$. Calculate the corrections to the energy $E$, and the heat capacity $C$, due to bose or fermi statistics.
(c) At what temperature does the approximation used above break down?

Sheh Lit Chang

University of Washington

Problem 2

Generalized ideal gas: consider a gas of non-interacting identical (spinless) quantum particles with an energy spectrum $\varepsilon=|\vec{p} / \hbar|^s$, contained in a box of "volume" $V$ in $d$ dimensions.
(a) Calculate the grand potential $\mathcal{G}_\eta=-k_B T \ln Q_\eta$, and the density $n=N / V$, at a chemical potential $\mu$. Express your answers in terms of $s, d$, and $f_m^N(z)$, where $z=\mathrm{e}^{\beta \mu}$, and
$$
f_m^\eta(z)=\frac{1}{\Gamma(m)} \int_0^{\infty} \frac{\mathrm{d} x x^{m-1}}{z^{-1} \mathrm{e}^x-\eta}
$$
(Hint. Use integration by parts on the expression for $\ln Q_\eta$.)
(b) Find the ratio $P V / E$, and compare it with the classical result obtained previously.
(c) For fermions, calculate the dependence of $E / N$, and $P$, on the density $n=N / V$, at zero temperature. (Hint. $f_m(z) \rightarrow(\ln z)^m / m !$ as $z \rightarrow \infty$.)
(d) For bosons, find the dimension $d_t(s)$, below which there is no bose condensation. Is there condensation for $s=2$ at $d=2$ ?

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26:30

Problem 3

Pauli paramagnetism: calculate the contribution of electron spin to its magnetic susceptibility as follows. Consider non-interacting electrons, each subject to a Hamiltonian
$$
\mathcal{H}_1=\frac{\vec{p}^2}{2 m}-\mu_0 \vec{\sigma} \cdot \vec{B}
$$
where $\mu_0=e \hbar / 2 m c$, and the eigenvalues of $\vec{\sigma} \cdot \vec{B}$ are $\pm B$.
(The orbital effect, $\vec{p} \rightarrow \vec{p}-e \vec{A}$, has been ignored.)
(a) Calculate the grand potential $\mathcal{G}_{-}=-k_B T \ln Q_{-}$, at a chemical potential $\mu$.
(b) Calculate the densities $n_{+}=N_{+} / V$, and $n_{-}=N_{-} / V$, of electrons pointing parallel and anti-parallel to the field.
(c) Obtain the expression for the magnetization $M=\mu_0\left(N_{+}-N_{-}\right)$, and expand the result for small $B$.
(d) Sketch the zero-field susceptibility $\chi(T)=\partial M /\left.\partial B\right|_{B=0}$, and indicate its behavior at low and high temperatures.
(e) Estimate the magnitude of $\chi / N$ for a typical metal at room temperature.

Dr. Rajveer Singh

Numerade Educator

09:34

Problem 4

Freezing of $\mathrm{He}^3$ : at low temperatures $\mathrm{He}^3$ can be converted from liquid to solid by application of pressure. A peculiar feature of its phase boundary is that $(\mathrm{d} P / \mathrm{d} T)$ melking is negative at temperatures below $0.3 \mathrm{~K}\left[(\mathrm{~d} P / \mathrm{d} T)_m \approx-30 \mathrm{~atm} \mathrm{~K}^{-1}\right.$ at $\left.T \approx 0.1 \mathrm{~K}\right]$. We will use a simple model of liquid and solid phases of $\mathrm{He}^3$ to account for this feature.
(a) In the solid phase, the $\mathrm{He}^3$ atoms form a crystal lattice. Each atom has nuclear spin of $1 / 2$. Ignoring the interaction between spins, what is the entropy per particle $s_s$, due to the spin degrees of freedom?
(b) Liquid $\mathrm{He}^3$ is modeled as an ideal fermi gas, with a volume of $46 \mathrm{~A}^3$ per atom. What is its fermi temperature $T_F$, in degrees Kelvin?
(c) How does the heat capacity of liquid $\mathrm{He}^3$ behave at low temperatures? Write down an expression for $C_V$ in terms of $N, T, k_B, T_F$, up to a numerical constant, that is valid for $T \ll T_F$.
(d) Using the result in (c), calculate the entropy per particle $s_{\ell}$, in the liquid at low temperatures. For $T \ll T_F$, which phase (solid or liquid) has the higher entropy?
(e) By equating chemical potentials, or by any other technique, prove the ClausiusClapeyron equation $(\mathrm{d} P / \mathrm{d} T)_{\text {melling }}=\left(s_l-s_s\right) /\left(v_l-v_s\right)$, where $v_l$ and $v_x$ are the volumes per particle in the liquid and solid phases, respectively.
(f) It is found experimentally that $v_{\ell}-v_x=3 \AA^3$ per atom. Using this information, plus the results obtained in previous parts, estimate $(\mathrm{d} P / \mathrm{d} T)_{\text {melling }}$ at $T \ll T_F$.
$* * * * * * * *$

Eduard Sanchez

Numerade Educator

01:20

Problem 5

Non-interacting fermions: consider a grand canonical ensemble of non-interacting fermions with chemical potential $\mu$. The one-particle states are labeled by a wave vector $\vec{k}$, and have energies $\varepsilon(\vec{k})$.
(a) What is the joint probability $P\left(\left\{n_k\right\}\right)$ of finding a set of occupation numbers $\left\{n_k\right\}$ of the one-particle states?
(b) Express your answer to part (a) in terms of the average occupation numbers $\left\{\left\langle n_{-}^{-}\right\rangle_{-}\right\}$.
(c) A random variable has a set of $\ell$ discrete outcomes with probabilities $p_n$, where $n=1,2, \cdots, \ell$. What is the entropy of this probability distribution? What is the maximum possible entropy?
(d) Calculate the entropy of the probability distribution for fermion occupation numbers in part (b), and comment on its zero-temperature limit.
(c) Calculate the variance of the total number of particles $\left\langle N^2\right\rangle_c$, and comment on its zero-temperature behavior.
(f) The number fluctuations of a gas are related to its compressibility $\kappa_T$, and number density $n=N / V$, by
$$
\left(N^2\right\rangle_c=N n k_B T \kappa_T
$$
Give a numerical estinate of the compressibility of the fermi gas in a metal at $T=0$ in units of $\mathrm{A}^3 \mathrm{eV}^{-1}$.

Dominador Tan

Numerade Educator

03:02

Problem 6

Stoner ferromagnetism: the conduction electrons in a metal can be treated as a gas of fermions of spin $1 / 2$ (with up/down degeneracy), and density $n=N / V$. The Coulomb repulsion favors wave functions that are anti-symmetric in position coordinates, thus keeping the electrons apart. Because of the full (position and spin) anti-symmetry of fermionic wave functions, this interaction may be approximated by an effective spinspin coupling that favors states with parallel spins. In this simple approximation, the net effect is described by an interaction energy
$$
U=\alpha \frac{N_{+} N_{-}}{V}
$$
where $N_{+}$and $N_{-}=N-N_{+}$are the numbers of electrons with up and down spins, and $V$ is the volume. (The parameter $\alpha$ is related to the scattering length $a$ by $\alpha=4 \pi \hbar^2 a / \mathrm{m}$.)
(a) The ground state has two fermi seas filled by the spin-up and spin-down electrons. Express the corresponding fermi wavevectors $k_{F \pm}$ in terms of the densities $n_{ \pm}=N_{ \pm} / V$.
(b) Calculate the kinetic energy density of the ground state as a function of the densities $n_{ \pm}$, and fundamental constants.
(c) Assuming small deviations $n_{ \pm}=n / 2 \pm \delta$ from the symmetric state, expand the kinetic energy to fourth order in 8 .
(d) Express the spin-spin interaction density $U / V$ in terms of $n$ and $\delta$. Find the critical value of $\alpha_c$, such that for $\alpha>\alpha_c$ the electron gas can lower its total energy by spontaneously developing a magnetization. (This is known as the Stoner instability.)
(e) Explain qualitatively, and sketch the behavior of, the spontaneous magnetization as a function of $\alpha$.

Ren Jie Tuieng

Numerade Educator

Problem 7

Boson magnetism: consider a gas of non-interacting spin 1 bosons, each subject to a Hamiltonian
$$
\mathcal{H}_1\left(\vec{p}, s_z\right)=\frac{\vec{p}^2}{2 m}-\mu_0 s_z B,
$$
where $\mu_0=e \hbar / m c$, and $s_z$ takes three possible values of $(-1,0,+1)$. (The orbital effect, $\vec{p} \rightarrow \vec{p}-e \vec{A}$, has been ignored.)
(a) In a grand canonical ensemble of chemical potential $\mu$, what are the average occupation numbers $\left\{\left(n_{+}(\vec{k})\right\rangle,\left\langle n_0(\vec{k})\right\rangle,\left\langle n_{-}(\vec{k})\right\rangle\right\}$ of one-particle states of wavenumber $\vec{k}=\vec{p} / \hbar$ ?
(b) Calculate the average total numbers $\left\{N_{+}, N_0, N_{-}\right\}$of bosons with the three possible values of $s_z$ in terms of the functions $f_m^{+}(z)$.
(c) Write down the expression for the magnetization $M(T, \mu)=\mu_0\left(N_{+}-N_{-}\right)$, and by expanding the result for small $B$ find the zero-field susceptibility $\chi(T, \mu)=$ $\partial M /\left.\partial B\right|_{B=0}$.

To find the behavior of $\chi(T, n)$, where $n=N / V$ is the total density, proceed as follows:
(d) For $B=0$, find the high-temperature expansion for $z(\beta, n)=\mathrm{e}^{\beta \mu}$, correct to second order in $n$. Hence obtain the first correction from quantum statistics to $\chi(T, n)$ at high temperatures.
(c) Find the temperature $T_c(n, B=0)$ of Bose-Einstein condensation. What happens to $\chi(T, n)$ on approaching $T_c(n)$ from the high-temperature side?
(f) What is the chemical potential $\mu$ for $T<T_c(n)$, at a small but finite value of $B$ ? Which one-particle state has a macroscopic occupation number?
(g) Using the result in (f), find the spontaneous magnetization
$$
\bar{M}(T, n)=\lim _{B \rightarrow 0} M(T, n, B)
$$
$* * * * * * * *$

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Problem 8

Dirac fermions are non-interacting particles of spin $1 / 2$. The one-particle states come in pairs of positive and negative energies,
$$
\varepsilon_{ \pm}(\vec{k})= \pm \sqrt{m^2 c^4+\hbar^2 k^2 c^2}
$$
independent of spin.
(a) For any fermionic system of chemical potential $\mu$, show that the probability of finding an occupied state of energy $\mu+\delta$ is the same as that of finding an unoccupied state of energy $\mu-\delta$. ( $\delta$ is any constant energy.)
(b) At zero temperature all negative energy Dirac states are occupied and all positive energy ones are empty, that is, $\mu(T=0)=0$. Using the result in (a) find the chemical potential at finite temperature $T$.
(c) Show that the mean excitation energy of this system at finite temperature satisfies
$$
E(T)-E(0)=4 V \int \frac{\mathrm{d}^3 \vec{k}}{(2 \pi)^3} \frac{\mathcal{E}_{+}(\vec{k})}{\exp \left(\beta \mathcal{E}_{+}(\vec{k})\right)+1}
$$
(d) Evaluate the integral in part (c) for massless Dirac particles (i.e., for $m=0$ ).
(e) Calculate the heat capacity, $C_V$, of such massless Dirac particles.
(f) Describe the qualitative dependence of the heat capacity at low temperature if the particles are massive.

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09:34

Problem 9

Numerical estimates: the following table provides typical values for the fermi energy and fermi temperature for (i) electrons in a typical metal; (ii) nucleons in a heavy nucleus; and (iii) $\mathrm{He}^3$ atoms in liquid $\mathrm{He}^3$ (atomic volume $=46.2 \mathrm{~A}^3$ per atom).
(a) Estimate the ratio of the electron and phonon heat capacities at room temperature for a typical metal.
(b) Compare the thermal wavelength of a neutron at room temperature to the minimum wavelength of a phonon in a typical crystal.
(c) Estimate the degeneracy discriminant, $n \lambda^3$, for hydrogen, helium, and oxygen gases at room temperature and pressure. At what temperatures do quantum mechanical effects become important for these gases?
$$
\begin{array}{lllll}
\hline \hline & n\left(1 / \mathrm{m}^3\right) & m(\mathrm{~kg}) & \varepsilon_F(\mathrm{eV}) & T_F(\mathrm{~K}) \\
\hline \text { Electron } & 10^{29} & 9 \times 10^{-31} & 4.4 & 5 \times 10^4 \\
\text { Nucleons } & 10^{44} & 1.6 \times 10^{-27} & 1.0 \times 10^8 & 1.1 \times 10^{12} \\
\text { Liquid } \mathrm{He}^3 & 2.6 \times 10^{28} & 4.6 \times 10^{-27} & 10^{-3} & 10^1 \\
\hline \hline
\end{array}
$$
(d) Experiments on $\mathrm{He}^4$ indicate that at temperatures below $1 \mathrm{~K}$, the heat capacity is given by $C_V=20.4 T^3 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$. Find the low-energy excitation spectrum, $\varepsilon(k)$, of $\mathrm{He}^4$.
(Hint. There is only one non-degenerate branch of such excitations.)

Eduard Sanchez

Numerade Educator

01:37

Problem 10

Solar interior: according to astrophysical data, the plasma at the center of the Sun has the following properties:
Temperature: $\quad T=1.6 \times 10^7 \mathrm{~K}$
Hydrogen density: $\rho_H=6 \times 10^4 \mathrm{~kg} \mathrm{~m}^{-3}$
Helium density: $\quad \rho_{H e}=1 \times 10^5 \mathrm{~kg} \mathrm{~m}^{-3}$.
(a) Obtain the thermal wavelengths for electrons, protons, and $\alpha$-particles (nuclei of He).
(b) Assuming that the gas is ideal, determine whether the electron, proton, or $\alpha$-particle gases are degenerate in the quantum mechanical sense.
(c) Estimate the total gas pressure due to these gas particles near the center of the Sun.
(d) Estimate the total radiation pressure close to the center of the Sun. Is it matter, or radiation pressure, that prevents the gravitational collapse of the Sun?

Donald Albin

Numerade Educator

05:33

Problem 11

Bose condensation in $d$ dimensions: consider a gas of non-interacting (spinless) bosons with an energy spectrum $\epsilon=p^2 / 2 m$, contained in a box of "volume" $V=L^d$ in $d$ dimensions.
(a) Calculate the grand potential $\mathcal{G}=-k_B T \ln Q$, and the density $n=N / V$, at a chemical potential $\mu$. Express your answers in terms of $d$ and $f_m^{+}(z)$, where $z=\mathrm{e}^{\beta \mu}$, and
$$
f_m^{+}(z)=\frac{1}{\Gamma(m)} \int_0^{\infty} \frac{x^{m-1}}{z^{-1} \mathrm{e}^x-1} \mathrm{~d} x
$$
(Hint. Use integration by parts on the expression for $\ln Q$.)
(b) Calculate the ratio $P V / E$, and compare it with the classical value.
(c) Find the critical temperature, $T_c(n)$, for Bose-Einstein condensation.
(d) Calculate the heat capacity $C(T)$ for $T<T_c(n)$.
(e) Sketch the heat capacity at all temperatures.
(f) Find the ratio, $C_{\operatorname{mas}} / C(T \rightarrow \infty)$, of the maximum heat capacity to its classical limit, and evaluate it in $d=3$.
(g) How does the above calculated ratio behave as $d \rightarrow 2$ ? In what dimensions are your results valid? Explain.

Mahnoor Amin

Numerade Educator

Problem 12

Exciton dissociation in a semiconductor: shining an intense laser beam on a semiconductor can create a metastable collection of electrons (charge $-e$, and effective mass $m_e$ ) and holes (charge $+e$, and effective mass $m_{\mathrm{L}}$ ) in the bulk. The oppositely charged particles may pair up (as in a hydrogen atom) to form a gas of excitons, or they may dissociate into a plasma. We shall examine a much simplified model of this process.
(a) Calculate the free energy of a gas composed of $N_{\mathrm{e}}$ electrons and $N_{\mathrm{h}}$ holes, at temperature $T$, treating them as classical non-interacting particles of masses $m_e$ and $m_{\mathrm{h} \mathrm{h}}$.
(b) By pairing into an excition, the electron hole pair lowers its energy by $\varepsilon$. (The binding energy of a hydrogen-like exciton is $\varepsilon \approx m e^4 /\left(2 \hbar^2 \epsilon^2\right)$, where $\epsilon$ is the dielectric constant, and $m^{-1}=m_e^{-1}+m_{\mathrm{h}}^{-1}$ ) Calculate the free energy of a gas of $N_{\mathrm{p}}$ excitons, treating them as classical non-interacting particles of mass $m=m_c+m_{\mathrm{h}}$ -
(c) Calculate the chemical potentials $\mu_{\mathrm{e}}, \mu_{\mathrm{h}}$, and $\mu_{\mathrm{p}}$ of the electron, hole, and exciton states, respectively.
(d) Express the equilibrium condition between excitons and electrons/holes in terms of their chemical potentials.
(e) At a high temperature $T$, find the density $n_p$ of excitons, as a function of the total density of excitations $n \approx n_{\mathrm{e}}+n_{\mathrm{a}}$.
$* * * * * * * *$

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02:51

Problem 13

Freezing of $\mathrm{He}^4$ : at low temperatures $\mathrm{He}^4$ can be converted from liquid to solid by application of pressure. An interesting feature of the phase boundary is that the melting pressure is reduced slightly from its $T=0 \mathrm{~K}$ value, by approximately $20 \mathrm{~N} \mathrm{~m}^{-2}$ at its minimum at $T=0.8 \mathrm{~K}$. We will use a simple model of liquid and solid phases of $\mathrm{He}^4$ to account for this feature.
(a) The important excitations in liquid $\mathrm{He}^4$ at $T<1 \mathrm{~K}$ are phonons of velocity $c$. Calculate the contribution of these modes to the heat capacity per particle, $C_V^{\ell} / N$, of the liquid.
(b) Calculate the low-temperature heat capacity per particle, $C_V^s / N$, of solid $\mathrm{He}^4$ in terms of longitudinal and transverse sound velocities $c_L$ and $c_T$.
(c) Using the above results calculate the entropy difference $\left(s_L-s_s\right)$, assuming a single sound velocity $c \approx c_L \approx c_T$, and approximately equal volumes per particle $v_{\ell} \approx v_s \approx v$. Which phase (solid or liquid) has the higher entropy?
(d) Assuming a small (temperature-independent) volume difference $\delta v=v_{\ell}-v_x$, calculate the form of the melting curve. To explain the anomaly described at the beginning, which phase (solid or liquid) must have the higher density?

Sana Riaz

Numerade Educator

12:42

Problem 14

Neutron star core: Professor Rajagopal's group at MIT has proposed that a new phase of QCD matter may exist in the core of neutron stars. This phase can be viewed as a condensate of quarks in which the low-energy excitations are approximately
$$
\varepsilon(\vec{k})_{ \pm}=\hbar_{ \pm} \hbar^2 \frac{\left(|\vec{k}|-k_F\right)^2}{2 M} .
$$
The excitations are fermionic, with a degeneracy of $g=2$ from spin.
(a) At zero temperature all negative energy states are occupied and all positive energy ones are empty, that is, $\mu(T=0)=0$. By relating occupation numbers of states of energies $\mu+\delta$ and $\mu-\delta$, or otherwise, find the chemical potential at finite temperatures $T$.
(b) Assuming a constant density of states near $k=k_F$, that is, setting $\mathrm{d}^3 k \approx 4 \pi k_F^2 \mathrm{~d} q$ with $q=|\vec{k}|-k_F$, show that the mean excitation energy of this system at finite temperature is
$$
E(T)-E(0) \approx 2 g V \frac{k_F^2}{\pi^2} \int_0^{\infty} \mathrm{d} q \frac{\mathcal{E}_{+}(q)}{\exp \left(\beta \varepsilon_{+}(q)\right)+1}
$$
(c) Give a closed form answer for the excitation energy by evaluating the above integral.
(d) Calculate the heat capacity, $C_V$, of this system, and comment on its behavior at low temperature.

Guilherme Barros

Numerade Educator

07:40

Problem 15

Non-interacting bosons: consider a grand canonical ensemble of non-interacting bosons with chemical potential $\mu$. The one-particle states are labeled by a wave vector $\vec{q}$, and have energies $\mathcal{E}(\vec{q})$.
(a) What is the joint probability $P\left(\left\{n_{\bar{q}}\right\}\right)$ of finding a set of occupation numbers $\left\{n_{\bar{q}}\right\}$ of the one-particle states, in terms of the fugacities $z_{\bar{q}} \equiv \exp [\beta(\mu-\mathcal{E}(\vec{q}))]$ ?
(b) For a particular $\vec{q}$, calculate the characteristic function $\left(\exp \left[\mathrm{i} k n_{\vec{q}}\right]\right)$.
(c) Using the result of part (b), or otherwise, give expressions for the mean and variance of $n_{\bar{q}}$ occupation number $\left\langle n_{\bar{q}}\right\rangle$.
(d) Express the variance in part (c) in terms of the mean cocupation number $\left\langle n_{\bar{p}}\right\rangle$.
(e) Express your answer to part (a) in terms of the occupation numbers $\left\{\left(n_{\bar{\psi}}\right)\right\}$.
(f) Calculate the entropy of the probability distribution for bosons, in terms of $\left\{\left\langle n_{\bar{\alpha}}\right)\right\}$, and comment on its zero-temperature limit.

Sheh Lit Chang

University of Washington

Problem 16

Relativistic bose gas in d dimensions: consider a gas of non-interacting (spinless) bosons with energy $\epsilon=c|\vec{p}|$, contained in a box of "volume" $V=L^d$ in $d$ dimensions.
(a) Calculate the grand potential $\mathcal{G}=-k_{\mathrm{B}} T \ln Q$, and the density $n=N / V$, at a chemical potential $\mu$. Express your answers in terms of $d$ and $f_{\mathrm{w}}^{+}(z)$, where $z=\mathrm{e}^{\beta \mu}$, and
$$
f_m^{+}(z)=\frac{1}{(m-1) !} \int_0^{\infty} \frac{x^{m-1}}{z^{-1} \mathrm{e}^x-1} \mathrm{~d} x
$$
(Hint. Use integration by parts on the expression for $\ln Q$.)
(b) Calculate the gas pressure $P$, its energy $E$, and compare the ratio $E /(P V)$ to the classical value.
(c) Find the critical temperature, $T_c(n)$, for Bose-Einstein condensation, indicating the dimensions where there is a transition.
(d) What is the temperature dependence of the heat capacity $C(T)$ for $T<T_{\mathrm{c}}(n)$ ?
(e) Evaluate the dimensionless heat capacity $C(T) /\left(N k_B\right)$ at the critical temperature $T=T_c$, and compare its value to the classical (high-temperature) limit.
$* * * * * * * *$

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Problem 17

Graphene is a single sheet of carbon atoms bonded into a two-dimensional hexagonal lattice. It can be obtained by exfoliation (repeated peeling) of graphite. The band structure of graphene is such that the single-particle excitations behave as relativistic Dirac fermions, with a spectrum that at low energies can be approximated by
$$
\mathcal{E}_{ \pm}(\vec{k})= \pm \hbar v|\vec{k}| .
$$
There is spin degeneracy of $g=2$, and $v \approx 10^6 \mathrm{~m} \mathrm{~s}^{-1}$. Experiments on unusual transport properties of graphene were reported in Nature 438, 197 (2005). In this problem, you shall calculate the heat capacity of this material.
(a) If at zero temperature all negative energy states are occupied and all positive energy ones are empty, find the chemical potential $\mu(T)$.
(b) Show that the mean excitation energy of this system at finite temperature satisfies
$$
E(T)-E(0)=4 A \int \frac{\mathrm{d}^2 \vec{k}}{(2 \pi)^2} \frac{\varepsilon_{+}(\vec{k})}{\exp \left(\beta \varepsilon_{+}(\vec{k})\right)+1} .
$$
(c) Give a closed form answer for the excitation energy by evaluating the above integral.
(d) Calculate the heat capacity, $C_V$, of such massless Dirac particles.
(e) Explain qualitatively the contribution of phonons (lattice vibrations) to the heat capacity of graphene. The typical sound velocity in graphite is of the order of $2 \times 10^4 \mathrm{~m} \mathrm{~s}^{-1}$. Is the low temperature heat capacity of graphene controlled by phonon or electron contributions?
$* * * * * * *$

Susan Hallstrom

Numerade Educator