9  Anharmonicity and Thermal Expansion

NoteLearning Objectives
  • Explain why a strictly harmonic crystal has no thermal expansion.
  • Describe how anharmonic terms modify the lattice potential.
  • Connect anharmonicity to phonon-phonon interactions.
  • Derive the mean displacement of an atom in an asymmetric potential.
  • Define the Grüneisen parameter and relate it to thermal expansion.
  • Interpret the thermodynamic relation between \(C_p\) and \(C_V\).

9.1 Beyond the Harmonic Approximation

The harmonic approximation gives independent normal modes, but it cannot describe several basic properties of real crystals.

In the harmonic approximation, the restoring force on an atom is linear in its displacement, \[ F=-ku, \] and the potential energy is quadratic, \[ U(u)=\frac{1}{2}ku^2. \]

The potential is symmetric around its minimum. Heating the oscillator increases the vibration amplitude, but the average position remains at the minimum. Thus, a purely harmonic oscillator has \[ \langle u\rangle =0 \] at all temperatures.

A harmonic crystal can vibrate more strongly when heated, but it cannot shift its average lattice spacing.

The strict harmonic approximation therefore implies:

  • no thermal expansion;
  • pressure- and temperature-independent elastic constants;
  • equality of \(C_p\) and \(C_V\);
  • no interaction between different phonon modes.

These statements are all violated in real crystals. Anharmonicity is therefore not a small correction to every question: it is essential for expansion, finite phonon lifetimes, and thermal resistance.

Tip

A useful analogy is a ball in a bowl. In a perfectly parabolic bowl, the ball explores larger amplitudes at higher temperature, but the center of the motion remains fixed. In an asymmetric bowl, the ball spends more time on the shallow side, so its average position shifts.

Figure 9.1: Figure placeholder. Equilibrium position for different excitations of a harmonic oscillator. In a harmonic potential, the center of the oscillator state does not shift with excitation energy.

9.2 Anharmonic Lattice Potential

Anharmonicity enters when the lattice potential is expanded beyond second order in the displacement.

Consider a one-dimensional displacement \[ u=x-x_0, \] where \(x_0\) is the equilibrium position. Expanding the potential about \(x_0\) gives \[ \begin{aligned} U &= U(x_0) + \frac{1}{2} \left. \frac{\partial^2 U}{\partial x^2} \right|_{x_0} u^2 + \frac{1}{6} \left. \frac{\partial^3 U}{\partial x^3} \right|_{x_0} u^3 + \frac{1}{24} \left. \frac{\partial^4 U}{\partial x^4} \right|_{x_0} u^4 +\cdots . \end{aligned} \]

The linear term is absent because \(x_0\) is an equilibrium position: \[ \left. \frac{\partial U}{\partial x} \right|_{x_0} =0. \]

We separate the expansion into \[ U=U_{\mathrm{eq}}+U_{\mathrm{harm}}+U_{\mathrm{anh}}. \]

Here \(U_{\mathrm{harm}}\) contains the quadratic term, while \(U_{\mathrm{anh}}\) contains the cubic and higher-order terms.

A simple model potential is, for example, \[ U=U_0+a u^2-bu^3-cu^4, \qquad a,b,c\ge 0. \]

The cubic term describes the asymmetry of the interatomic potential. This asymmetry is physically expected: atoms repel each other very strongly at short distances, while the potential becomes flatter at larger separations.

The quartic term represents the next correction at larger amplitudes. In a realistic potential, higher-order terms ultimately stabilize the potential; the truncated expression is only meant as a local expansion around the equilibrium position.

Figure 9.2: Interactive local anharmonic potential. The sliders vary the anharmonic parameters in \(U(u)=U_0+a u^2-bu^3-cu^4\), with fixed \(U_0=0\). The dashed curve shows the harmonic reference \(U=U_0+a u^2\).

The cubic term is the simplest term that can shift the thermal average position. A purely symmetric potential cannot produce ordinary positive thermal expansion in this simple picture.

9.3 Harmonic Motion and the Superposition Principle

In a harmonic potential, the equations of motion are linear.

For a harmonic oscillator, \[ m\ddot u = -\left. \frac{\partial^2 U}{\partial x^2} \right|_{x_0} u = -ku. \]

A plane-wave solution has the form \[ u=u_0 e^{i(qx-\omega t)}. \]

Because the differential equation is linear, the superposition principle applies: if \(u_1(x,t)\) and \(u_2(x,t)\) are solutions, then any linear combination of them is also a solution.

Example

Consider two plane-wave trial solutions, \[ u_1(x,t)=A e^{i(q_1x-\omega t)} \] and \[ u_2(x,t)=B e^{i(q_2x-\omega t)}. \]

For the local harmonic equation of motion \[ m\ddot u=-ku, \] only the time dependence matters. Therefore \[ \ddot u_1(x,t)=-\omega^2 A e^{i(q_1x-\omega t)} =-\omega^2 u_1(x,t), \] and similarly \[ \ddot u_2(x,t)=-\omega^2 u_2(x,t). \]

Substitution into the equation of motion gives \[ m\ddot u_1=-m\omega^2 u_1. \]

Thus \(u_1\) is a solution if \[ m\omega^2=k, \] or equivalently \[ \omega=\sqrt{\frac{k}{m}}. \]

The same condition makes \(u_2\) a solution.

Now form a linear combination \[ u(x,t)=c_1u_1(x,t)+c_2u_2(x,t), \] where \(c_1\) and \(c_2\) are arbitrary constants. Taking the second time derivative gives \[ \ddot u = c_1\ddot u_1+c_2\ddot u_2. \]

Since both \(u_1\) and \(u_2\) satisfy the same harmonic equation, \[ m\ddot u_1=-ku_1, \qquad m\ddot u_2=-ku_2. \]

Therefore \[ m\ddot u = c_1m\ddot u_1+c_2m\ddot u_2 = -c_1ku_1-c_2ku_2. \]

Factoring out \(-k\) gives \[ m\ddot u = -k(c_1u_1+c_2u_2) = -ku. \]

Hence the superposition \[ u(x,t)=c_1 A e^{i(q_1x-\omega t)} + c_2 B e^{i(q_2x-\omega t)} \] is again a solution of the harmonic equation of motion, provided the frequency satisfies \[ \omega^2=\frac{k}{m}. \]

The important point is that the equation of motion is linear in \(u\). No terms such as \(u^2\), \(u^3\), or products of different waves appear. Therefore the two waves do not mix: each component evolves independently.

For a harmonic crystal this has a powerful consequence. The normal modes are independent. Once a mode is excited, it does not exchange energy with other modes. In this idealized limit, phonons have infinite lifetime.

9.4 Three-phonon processes

Consider the linear monatomic chain from Lecture 6, but now include a small cubic anharmonic correction to the nearest-neighbor potential. Let \(u_p(t)\) be the displacement of atom \(p\) from equilibrium. We write the potential energy as

\[ U = \sum_p \left[ \frac{K}{2}(u_{p+1}-u_p)^2 + \frac{\alpha}{3}(u_{p+1}-u_p)^3 \right]. \]

The first term is harmonic. The second term is cubic and therefore anharmonic. The resulting equation of motion is

\[ m\ddot u_p = K(u_{p+1}+u_{p-1}-2u_p) + \alpha \left[ (u_{p+1}-u_p)^2 - (u_p-u_{p-1})^2 \right]. \]

The harmonic part is linear in \(u_p\), while the anharmonic part is quadratic in displacement differences. This quadratic term is the origin of three-phonon processes.

First recall the harmonic plane-wave solution

\[ u_p(t) = u_0 e^{i(qpa-\omega t)}. \]

Substitution into the harmonic part gives the usual monatomic-chain dispersion relation

\[ \omega^2(q) = \frac{4K}{m}\sin^2\left(\frac{qa}{2}\right). \]

Now consider a displacement that is the sum of two harmonic plane waves,

\[ u_p(t) = u_{01}e^{i(q_1pa-\omega_1t)} + u_{02}e^{i(q_2pa-\omega_2t)}. \]

For the first bond difference, we obtain

\[ u_{p+1}-u_p = u_{01} \left(e^{iq_1a}-1\right) e^{i(q_1pa-\omega_1t)} + u_{02} \left(e^{iq_2a}-1\right) e^{i(q_2pa-\omega_2t)}. \]

Squaring this expression gives

\[\begin{align} (u_{p+1}-u_p)^2 & = u_{01}^2 \left(e^{iq_1a}-1\right)^2 e^{i(2q_1pa-2\omega_1t)}\\ & + u_{02}^2 \left(e^{iq_2a}-1\right)^2 e^{i(2q_2pa-2\omega_2t)}\\ & + 2u_{01}u_{02} \left(e^{iq_1a}-1\right) \left(e^{iq_2a}-1\right) e^{i[(q_1+q_2)pa-(\omega_1+\omega_2)t]}. \end{align}\]

The important contribution is the cross term,

\[ 2u_{01}u_{02} \left(e^{iq_1a}-1\right) \left(e^{iq_2a}-1\right) e^{i[(q_1+q_2)pa-(\omega_1+\omega_2)t]}. \]

This term has the form of a new plane wave with wave vector

\[ q_3=q_1+q_2 \]

and angular frequency

\[ \omega_3=\omega_1+\omega_2. \]

The second bond difference gives an analogous contribution. Since

\[ u_p-u_{p-1} = u_{01} \left(1-e^{-iq_1a}\right) e^{i(q_1pa-\omega_1t)} + u_{02} \left(1-e^{-iq_2a}\right) e^{i(q_2pa-\omega_2t)}, \]

its square also contains a cross term proportional to

\[ e^{i[(q_1+q_2)pa-(\omega_1+\omega_2)t]}. \]

Therefore, the anharmonic force contains a contribution of the form

\[ F_p^{(3)} \propto u_{01}u_{02} e^{i[(q_1+q_2)pa-(\omega_1+\omega_2)t]}. \]

This explicitly shows that the cubic anharmonicity couples two waves \((q_1,\omega_1)\) and \((q_2,\omega_2)\) and generates a third component with

\[ q_3=q_1+q_2, \qquad \omega_3=\omega_1+\omega_2. \]

This is the microscopic origin of a three-phonon combination process, often written schematically as

\[ (q_1,\omega_1)+(q_2,\omega_2) \longrightarrow (q_3,\omega_3). \]

In a crystal, wave vector conservation is understood modulo a reciprocal lattice vector \(G\). More generally,

\[ q_3=q_1+q_2+G. \]

For \(G=0\), the process is a normal process.

For \(G\neq 0\), it is an Umklapp process.

The essential point is that the harmonic equation of motion is linear and does not mix plane waves. The cubic anharmonic term, however, contains products of displacements. Products of two plane waves generate sum and difference frequencies and wave vectors. Therefore, anharmonicity couples phonons and allows three-phonon processes.

Normal processes conserve the total phonon crystal momentum inside the first Brillouin zone. Umklapp processes transfer crystal momentum to the lattice.

This distinction is central for heat transport. Normal processes redistribute energy among phonons, but they do not by themselves relax the net phonon momentum associated with heat flow. Umklapp processes can reverse the direction of the phonon group velocity and therefore contribute directly to thermal resistance.

Figure 9.3: Figure placeholder. Three-phonon normal and umklapp processes in a two-dimensional square lattice. The figure should show how adding a reciprocal-lattice vector returns the final wave vector to the first Brillouin zone.

9.5 Thermal Expansion: Definitions

Thermal expansion describes the change of a crystal dimension with temperature under constant pressure.

The linear thermal expansion coefficient is \[ \alpha_L \equiv \frac{1}{L} \left. \frac{\partial L}{\partial T} \right|_p . \]

Here \(L\) is a sample length and the derivative is taken at constant pressure.

The volume thermal expansion coefficient is \[ \alpha_V \equiv \frac{1}{V} \left. \frac{\partial V}{\partial T} \right|_p . \]

For an isotropic solid, \[ \alpha_V=3\alpha_L. \]

Typical room-temperature values of \(\alpha_L\) for solids are of order \[ 10^{-5}\ \mathrm{K}^{-1}. \]

This small number is important: lattice spacings change only weakly with temperature, but the effect is measurable and technologically important.

9.6 Mean Displacement in an Anharmonic Potential

Thermal expansion can be understood microscopically from the thermal average position of an atom in an asymmetric potential.

For a one-dimensional oscillator, the thermal mean displacement is \[ \langle u\rangle = \frac{ \int_{-\infty}^{\infty}du\, u\, e^{-\beta U(u)} }{ \int_{-\infty}^{\infty}du\, e^{-\beta U(u)} }, \] where \(\beta=1/(k_B T)\).

Using \[ U(u)=U_0+a u^2-bu^3-cu^4, \] and assuming that the anharmonic terms are small compared with \(a u^2\), we expand \[ e^{-\beta U(u)} = e^{-\beta U_0} e^{-\beta a u^2} e^{\beta(bu^3+cu^4)} \simeq e^{-\beta U_0} e^{-\beta a u^2} \left(1+\beta b u^3+\beta c u^4+\cdots\right). \]

The common factor \(e^{-\beta U_0}\) cancels between numerator and denominator.

The numerator becomes \[ \int_{-\infty}^{\infty}du\, u\, e^{-\beta U(u)} \simeq \int_{-\infty}^{\infty}du\, e^{-\beta a u^2} \left( u+\beta b u^4+\beta c u^5+\cdots \right). \]

Terms that are odd functions of \(u\) vanish when integrated over symmetric limits. Thus \[ \int_{-\infty}^{\infty}du\, u\, e^{-\beta a u^2}=0, \] and \[ \int_{-\infty}^{\infty}du\, \beta\, c\, u^5 e^{-\beta a u^2}=0. \]

The leading nonzero contribution is \[ \beta b \int_{-\infty}^{\infty}du\, u^4\, e^{-\beta a u^2}. \]

Using \[ \int_{-\infty}^{\infty}du\, e^{-\beta a u^2} = \sqrt{\frac{\pi}{\beta a}} \] and \[ \int_{-\infty}^{\infty}du\, u^4 e^{-\beta a u^2} = \frac{3\sqrt{\pi}}{4(\beta a)^{5/2}}, \] we obtain \[ \langle u\rangle = \frac{ \beta b \frac{3\sqrt{\pi}}{4(\beta a)^{5/2}} }{ \sqrt{\frac{\pi}{\beta a}} } = \frac{3b}{4a^2}k_B T. \]

Therefore, \[ \boxed{ \langle u\rangle = \frac{3b}{4a^2}k_BT } \]

The result is linear in temperature within this simple approximation. It also shows explicitly that the shift vanishes if the cubic asymmetry parameter \(b\) is zero.

The relative length change is approximately \[ \frac{\Delta L}{L} \simeq \frac{\langle u\rangle}{R_0}, \] where \(R_0\) is the equilibrium distance of the atoms.

Thus the linear thermal expansion coefficient is \[ \alpha_L = \frac{1}{R_0} \left. \frac{\partial \langle u\rangle}{\partial T} \right|_p = \frac{3b}{4a^2} \frac{k_B}{R_0}. \]

In this microscopic picture, thermal expansion is the temperature-dependent shift of the thermal average position in an asymmetric interatomic potential.

Figure 9.4: Figure placeholder. Thermal expansion caused by an anharmonic potential. At higher temperature, oscillator states with larger amplitudes are occupied, and their centers shift toward larger interatomic separation.

9.7 Thermodynamic View: Free Energy and Pressure

Thermal expansion can also be derived from the volume dependence of the phonon free energy.

The pressure is obtained from the Helmholtz free energy \(F\): \[ p = - \left. \frac{\partial F}{\partial V} \right|_T . \]

For a crystal with phonon frequencies \(\omega_r(q)\), the internal energy is \[ \langle U\rangle = U_{\mathrm{eq}} + \sum_{q,r}\frac{1}{2}\hbar\omega_r(q) + \sum_{q,r} \frac{\hbar\omega_r(q)} {e^{\hbar\omega_r(q)/(k_BT)}-1}. \]

The first term is the static lattice energy. The second term is the zero-point energy. The third term is the thermal phonon energy.

The important point is that, in an anharmonic crystal, the phonon frequencies depend on volume: \[ \omega_r(q)=\omega_r(q,V). \]

This volume dependence produces a temperature-dependent pressure. Gross and Marx write the pressure in the form \[ p = -B\frac{\delta V}{V} \frac{\partial}{\partial V} - \sum_{q,r} \frac{1}{2}\hbar\omega_r(q) - \hbar \sum_{q,r} \frac{\partial\omega_r(q)}{\partial V} \frac{1}{e^{\hbar\omega_r(q)/(k_BT)}-1}. \]

The first term is the elastic response to a volume change. The second term is the zero-point contribution. The third term is the thermal phonon contribution.

At fixed volume, only the last term has explicit temperature dependence. Therefore, thermal expansion is controlled by how the phonon frequencies change when the crystal is compressed or expanded.

Tip

The quasi-harmonic viewpoint keeps the phonons harmonic at each fixed volume, but allows their frequencies to shift when the equilibrium volume changes. It is a practical way to include the part of anharmonicity responsible for thermal expansion.

9.8 The Grüneisen Parameter

The Grüneisen parameter measures the volume sensitivity of phonon frequencies.

For each phonon mode, define \[ \gamma_{q,r} \equiv - \frac{V}{\omega_r(q)} \frac{\partial \omega_r(q)}{\partial V} = - \frac{\partial \ln\omega_r(q)}{\partial\ln V}. \]

If \(\gamma_{q,r}>0\), the mode frequency decreases when the crystal expands. This is common because larger interatomic spacing usually weakens the effective restoring force.

If \(\gamma_{q,r}<0\), the mode frequency increases when the crystal expands. Such modes can contribute to negative thermal expansion.

The modal heat-capacity contribution per unit volume is \[ c_{V,r}(q) = \frac{1}{V} \hbar\omega_r(q) \frac{\partial n_r(q)}{\partial T}, \] where \[ n_r(q) = \frac{1}{e^{\hbar\omega_r(q)/(k_BT)}-1} \] is the Bose-Einstein occupation number.

The total heat capacity per unit volume is \[ c_V = \sum_{q,r}c_{V,r}(q). \]

The heat-capacity-weighted Grüneisen parameter is \[ \gamma = \frac{ \sum_{q,r}\gamma_{q,r}c_{V,r}(q) }{ \sum_{q,r}c_{V,r}(q) }. \]

Thus, \(\gamma\) is not a simple arithmetic average over all phonons. It is weighted by the modes that actually contribute to the heat capacity at temperature \(T\).

9.9 Grüneisen Relation for Thermal Expansion

The Grüneisen relation connects thermal expansion, heat capacity, elastic stiffness, and anharmonicity.

For an isotropic solid, \[ \alpha_L = \frac{1}{3V} \left. \frac{\partial V}{\partial T} \right|_p . \]

Using the thermodynamic identity \[ \left. \frac{\partial V}{\partial T} \right|_p = - \left. \frac{\partial V}{\partial p} \right|_T \left. \frac{\partial p}{\partial T} \right|_V \] and the bulk modulus relation \[ B = -V \left. \frac{\partial p}{\partial V} \right|_T, \] we obtain \[ \alpha_L = \frac{1}{3B} \left. \frac{\partial p}{\partial T} \right|_V . \]

From the pressure expression above, \[ \left. \frac{\partial p}{\partial T} \right|_V = - \hbar \sum_{q,r} \frac{\partial\omega_r(q)}{\partial V} \frac{\partial n_r(q)}{\partial T}. \]

Using \[ \frac{\partial\omega_r(q)}{\partial V} = - \frac{\gamma_{q,r}\omega_r(q)}{V}, \] we find \[ \left. \frac{\partial p}{\partial T} \right|_V = \sum_{q,r} \gamma_{q,r} \frac{1}{V} \hbar\omega_r(q) \frac{\partial n_r(q)}{\partial T}. \]

The term on the right is just \[ \sum_{q,r}\gamma_{q,r}c_{V,r}(q). \]

Therefore, \[ \alpha_L = \frac{1}{3B} \sum_{q,r}\gamma_{q,r}c_{V,r}(q). \]

Using the weighted Grüneisen parameter gives the compact result \[ \boxed{ \alpha_L = \frac{\gamma c_V}{3B} } \]

and, for the volume expansion coefficient, \[ \boxed{ \alpha_V = \frac{\gamma c_V}{B} }. \]

These are the central relations of this lecture.

Thermal expansion is large when the phonon frequencies are strongly volume dependent, the heat capacity is large, and the material is mechanically soft.

Since \(B\) and \(\gamma\) often vary only weakly with temperature, the temperature dependence of \(\alpha_L\) frequently follows that of \(c_V\).

At low temperature, the Debye heat capacity behaves as \[ c_V\propto T^3. \]

Therefore, for approximately constant \(B\) and \(\gamma\), \[ \alpha_L\propto T^3. \]

At high temperature, the lattice heat capacity approaches a nearly constant Dulong-Petit value, and the thermal expansion coefficient often becomes approximately constant.

This simple scaling is useful, but it has limitations. In anisotropic crystals, expansion depends on direction. In some materials, different phonon modes have Grüneisen parameters with different signs, so \(\alpha_L\) can even change sign as a function of temperature.

Figure 9.5: Figure placeholder. Temperature dependence of the linear thermal expansion coefficient of silicon. The low-temperature sign change illustrates that different phonon modes can contribute with different signs.

9.10 Relation Between \(C_p\) and \(C_V\)

The difference between heat capacities at constant pressure and constant volume is another consequence of thermal expansion.

The general thermodynamic result is \[ C_p-C_V = TVB\alpha_V^2. \]

For an isotropic solid, \(\alpha_V=3\alpha_L\), and therefore \[ C_p-C_V = 9TVB\alpha_L^2. \]

Equivalently, per unit volume, \[ c_p-c_V = B\, T\, \alpha_V^2 = 9 B\, T\, \alpha_L^2. \]

Since \(B>0\) and the square of the expansion coefficient is nonnegative, \[ C_p\ge C_V. \]

The physical reason is straightforward. At constant volume, supplied heat increases the internal energy. At constant pressure, the solid also expands, so part of the supplied heat is associated with mechanical expansion work.

In the harmonic limit, \(\alpha_V=0\), and therefore \(C_p=C_V\).

For most ordinary solids the difference is small, because thermal expansion coefficients are small. Nevertheless, the difference is conceptually important because it directly reflects anharmonicity.

9.11 Connection to the Next Lecture: Thermal Conductivity

Anharmonicity also gives phonons finite lifetimes, which is essential for thermal transport.

In the harmonic approximation, phonons do not interact. A phonon would propagate indefinitely, and the system could not properly redistribute energy among modes.

Anharmonic terms allow processes such as \[ (q_1,r_1)+(q_2,r_2)\rightarrow(q_3,r_3) \] and the reverse decay process. These processes give phonons a finite lifetime \(\tau\) and a finite mean free path \[ \ell=v\tau, \] where \(v\) is a characteristic group velocity.

A kinetic estimate for the phonon thermal conductivity is \[ \kappa \simeq \frac{1}{3}c_V v\ell. \]

The next lecture will use this idea to discuss heat conduction in insulating crystals. The key new question will be which scattering processes actually relax heat current. Normal three-phonon processes redistribute phonons, while umklapp processes transfer crystal momentum to the lattice and are central for thermal resistance.

NoteTake-Home Messages
  • A strictly harmonic crystal cannot thermally expand.
  • Thermal expansion requires an asymmetric interatomic potential.
  • Cubic anharmonicity couples phonons and enables three-phonon processes.
  • Umklapp processes are important because they transfer crystal momentum to the lattice.
  • The Grüneisen parameter measures how strongly phonon frequencies respond to volume changes.
  • Thermal expansion is controlled by phonon heat capacity, elastic stiffness, and anharmonicity.
  • Constant-pressure heat capacity exceeds constant-volume heat capacity when a solid expands upon heating.
  • Anharmonicity links equilibrium properties such as expansion to transport properties such as heat conduction.

Problem Set

  1. Harmonic Crystal. Explain why a crystal with a purely harmonic potential has no thermal expansion. Use both the symmetry argument for a single oscillator and the thermodynamic identity \[ \alpha_L= \frac{1}{3B} \left. \frac{\partial p}{\partial T} \right|_V . \]

  2. Mean Displacement in an Anharmonic Potential. Consider \[ U(u)=U_0+au^2-bu^3, \qquad a,b>0. \] Treat the cubic term as a small perturbation and show that \[ \langle u\rangle = \frac{3b}{4a^2}k_BT. \] What does this imply for the sign of thermal expansion?

  3. Mode Grüneisen Parameter. Suppose a phonon mode has volume dependence \[ \omega(V)=\omega_0 \left(\frac{V}{V_0}\right)^{-\eta}, \] where \(\eta\) is a constant. Compute the mode Grüneisen parameter and state whether the mode softens or hardens upon expansion for \(\eta>0\).

  4. Low-Temperature Expansion. Assume a cubic insulating crystal has \[ c_V(T)=A T^3 \] at low temperature, while \(B\) and \(\gamma\) are temperature independent. Use the Grüneisen relation to find the low-temperature dependence of \(\alpha_L(T)\).