- Derive the dispersion relation of a crystal lattice with a one-atomic basis for longitudinal vibrations.
- Explain the meaning of group velocity, the long-wavelength sound limit, and the first Brillouin zone.
- Derive the acoustic and optical branches for a lattice with a two-atomic basis.
- Interpret acoustic and optical modes in terms of in-phase and out-of-phase atomic motion.
- Generalize the branch counting from one-dimensional model systems to three-dimensional crystals.
6.1 Roadmap
- Start from the dynamical-matrix formulation and specialize it to explicit model systems.
- Treat longitudinal vibrations of a lattice with a one-atomic basis.
- Discuss group velocity, sound waves, and the restriction to the first Brillouin zone.
- Derive acoustic and optical branches for a two-atomic basis.
- Generalize to three-dimensional crystals and connect to phonon density of states.
| Symbol | Meaning |
|---|---|
| \(n,m\) | indices of lattice planes or unit cells |
| \(p=m-n\) | relative cell or plane index |
| \(a\) | distance between neighboring equivalent lattice planes |
| \(M\) | mass of an atom in a one-atomic basis |
| \(M_1,M_2\) | masses of the two atoms in a two-atomic basis |
| \(u_n\) | displacement of plane or atom \(n\) in a one-atomic basis |
| \(u_n,v_n\) | displacements of the two atom types in cell \(n\) |
| \(C_p\) | coupling constant between planes separated by \(p a\) |
| \(C_1\) | nearest-neighbor coupling constant for a one-atomic basis |
| \(f\) | nearest-neighbor coupling constant for a two-atomic basis |
| \(\mathbf q\) | phonon wavevector |
| \(q\) | one-dimensional component of \(\mathbf q\) |
| \(\mathbf G\) | reciprocal-lattice vector |
| \(\omega(\mathbf q)\) | angular frequency of a lattice vibration |
| \(v_g\) | group velocity |
| \(v_s\) | sound velocity in the long-wavelength limit |
| \(r'\) | number of atoms in the basis |
| \(D_{\alpha i}^{\beta j}(\mathbf q)\) | dynamical matrix |
| \(D(\omega)\) | phonon density of states in frequency space |
6.2 From the Dynamical Matrix to Explicit Models
For a crystal with basis atoms labeled by \(\alpha,\beta\) and Cartesian components \(i,j\), the harmonic equations of motion can be written as \[ M_{\alpha}\frac{\partial^2 u_{n\alpha i}}{\partial t^2} + \sum_{m,\beta,j} C_{n\alpha i}^{m\beta j}u_{m\beta j} =0. \]
The plane-wave ansatz is mass normalized: \[ u_{n\alpha i} = \frac{1}{\sqrt{M_{\alpha}}} A_{\alpha i}(\mathbf q) e^{\mathrm i(\mathbf q\cdot \mathbf R_n-\omega t)}. \]
Inserting this into the equations of motion gives \[ -\omega^2 A_{\alpha i}(\mathbf q) + \sum_{\beta,j} D_{\alpha i}^{\beta j}(\mathbf q) A_{\beta j}(\mathbf q) =0, \] where \[ D_{\alpha i}^{\beta j}(\mathbf q) = \sum_m \frac{1}{\sqrt{M_{\alpha}M_{\beta}}} C_{n\alpha i}^{m\beta j} e^{\mathrm i\mathbf q\cdot(\mathbf R_m-\mathbf R_n)}. \]
The normal-mode frequencies follow from \[ \det\left[D_{\alpha i}^{\beta j}(\mathbf q)-\omega^2\mathbf 1\right]=0. \]
For a basis with \(r'\) atoms, there are \(3r'\) branches in a three-dimensional crystal. The following sections show how this general result becomes explicit in simple models.
6.3 Crystal Lattice With a One-Atomic Basis
We first consider a crystal with a one-atomic basis and a wave propagating along a high-symmetry direction. If entire lattice planes move parallel to their normal direction, symmetry eliminates transverse force components. The problem then reduces to a one-dimensional longitudinal vibration of lattice planes.
6.3.1 Force Law for Longitudinal Plane Vibrations
Let \(u_n\) denote the displacement of plane \(n\). The force exerted on plane \(n\) by plane \(n+p\) is proportional to the relative displacement \(u_{n+p}-u_n\). Summing over all planes gives \[ F_n = \sum_p C_p(u_{n+p}-u_n). \]
Newton’s equation is therefore \[ M\frac{\partial^2u_n}{\partial t^2} - \sum_p C_p(u_{n+p}-u_n) =0. \]
We see that a uniform translation \(u_n=\mathrm{const.}\) produces no restoring force.
6.3.2 Dispersion Relation for General Couplings
Use the plane-wave ansatz \[ u_n = A e^{\mathrm i(qna-\omega t)}. \]
Then \[ u_{n+p} = u_n e^{\mathrm i qpa}. \]
Substitution gives \[ -\omega^2 M - \sum_p C_p\left(e^{\mathrm i qpa}-1\right) =0. \]
Using inversion symmetry, \(C_{-p}=C_p\), this becomes \[ -\omega^2 M = \sum_{p=1}^{\infty} C_p \left(e^{\mathrm i qpa}+e^{-\mathrm i qpa}-2\right) = 2\sum_{p=1}^{\infty} C_p\left(\cos qpa-1\right). \]
Hence
This is the longitudinal dispersion relation for a one-atomic basis with arbitrary plane-plane couplings \(C_p\).
6.3.3 Nearest-Neighbor Approximation
If only nearest-neighbor planes are coupled, then \(C_1\neq 0\) and all other \(C_p\) vanish. The dispersion reduces to \[ \omega^2(q) = \frac{2C_1}{M}(1-\cos qa) =\frac{4C_1}{M}\sin^2\frac{qa}{2}. \]
For \(0\le q\le \pi/a\),
The frequency is periodic in reciprocal space and even in \(q\): \[ \omega(q)=\omega(q+G), \qquad \omega(q)=\omega(-q), \] with \(G=2\pi/a\) in one dimension.
6.3.4 Interpretation and Limiting Cases
At the zone boundary, \[ q=\frac{\pi}{a}, \] the frequency reaches \[ \omega_{\max} = \sqrt{\frac{4C_1}{M}}. \]
The group velocity is \[ v_g = \nabla_q\omega(q). \]
For the one-dimensional nearest-neighbor dispersion, \[ v_g(q) = \sqrt{\frac{C_1a^2}{M}}\cos\frac{qa}{2} \qquad (0\le q\le \pi/a). \]
Thus \[ v_g\left(\frac{\pi}{a}\right)=0. \]
At the Brillouin-zone boundary the wave is a standing wave, not a propagating wave packet.
6.3.5 Long-Wavelength Limit
For \(qa\ll 1\), \[ \sin\frac{qa}{2} \simeq \frac{qa}{2}, \] and therefore \[ \omega(q) \simeq \sqrt{\frac{C_1a^2}{M}} q. \]
This is a linear acoustic dispersion, \[ \omega(q)=v_s q, \] with
In this limit the discrete lattice behaves like an elastic continuum. More generally, the linear relation \(\omega\propto q\) at small \(q\) does not depend on keeping only nearest-neighbor couplings; longer-range couplings merely change the numerical value of the sound velocity.
6.3.6 Continuum Check
In the long-wavelength limit, write \[ u_n(t)\rightarrow u(x,t), \qquad u_{n\pm1}(t)\rightarrow u(x\pm a,t). \]
A Taylor expansion gives \[ u(x\pm a,t) = u(x,t) \pm a\frac{\partial u}{\partial x} + \frac{a^2}{2}\frac{\partial^2u}{\partial x^2} +\cdots. \]
For nearest-neighbor coupling, \[ M\frac{\partial^2u}{\partial t^2} = C_1\left[u(x+a,t)+u(x-a,t)-2u(x,t)\right], \] so \[ \frac{\partial^2u}{\partial t^2} = \frac{C_1a^2}{M} \frac{\partial^2u}{\partial x^2} = v_s^2 \frac{\partial^2u}{\partial x^2}. \]
The lattice model therefore reproduces the standard one-dimensional sound-wave equation when \(\lambda\gg a\).
6.4 The First Brillouin Zone
The dynamical matrix contains phase factors of the form \[ e^{\mathrm i\mathbf q\cdot(\mathbf R_m-\mathbf R_n)}. \]
Since \(\mathbf R_m-\mathbf R_n\) is a Bravais-lattice vector, adding a reciprocal-lattice vector \(\mathbf G\) leaves the phase factor unchanged: \[ e^{\mathrm i(\mathbf q+\mathbf G)\cdot(\mathbf R_m-\mathbf R_n)} = e^{\mathrm i\mathbf q\cdot(\mathbf R_m-\mathbf R_n)}. \]
Therefore \[ D_{\alpha i}^{\beta j}(\mathbf q) = D_{\alpha i}^{\beta j}(\mathbf q+\mathbf G), \] and the phonon frequencies satisfy \[ \omega(\mathbf q)=\omega(\mathbf q+\mathbf G). \]
Time-reversal symmetry further gives \[ \omega(-\mathbf q)=\omega(\mathbf q). \]
For the one-dimensional lattice it is therefore sufficient to specify the dispersion in
Physically, wavevectors that differ by a reciprocal-lattice vector describe the same displacements at the lattice sites. The apparent shape of the wave between atoms is not an observable of the discrete lattice model.
6.4.1 Transverse and Mixed Polarizations
For transverse vibrations of a one-atomic basis, the derivation is analogous, but the coupling constants \(C_p\) are generally different from the longitudinal ones. Purely longitudinal and purely transverse waves occur only for propagation along suitable symmetry directions. In a cubic crystal these include the \([100]\), \([110]\), and \([111]\) directions. For general propagation directions, the atomic displacements need not be exactly parallel or perpendicular to \(\mathbf q\), and the polarization is mixed.
6.5 Crystal Lattice With a Two-Atomic Basis
We now consider a lattice with two atoms per basis, with masses \(M_1\) and \(M_2\). The displacement of the \(M_1\) plane in cell \(n\) is \(u_n\), and the displacement of the \(M_2\) plane is \(v_n\). The distance between equivalent planes is \(a\). Only nearest-neighbor interactions are retained, and their coupling constant is denoted by \(f\).
6.5.1 Equations of Motion
The restoring force on \(u_n\) comes from its two neighboring \(v\) planes: \[ M_1\frac{\partial^2u_n}{\partial t^2} = f(v_n-u_n)+f(v_{n-1}-u_n). \]
Similarly, \[ M_2\frac{\partial^2v_n}{\partial t^2} = f(u_n-v_n)+f(u_{n+1}-v_n). \]
Equivalently, \[ M_1\frac{\partial^2u_n}{\partial t^2} + f(2u_n-v_n-v_{n-1}) =0, \] \[ M_2\frac{\partial^2v_n}{\partial t^2} + f(2v_n-u_n-u_{n+1}) =0. \]
6.5.2 Mass-Normalized Plane-Wave Ansatz
As an ansatz for the solutions, we use \[ u_n(q) = \frac{1}{\sqrt{M_1}} A_1(q)e^{\mathrm i(qan-\omega t)}, \] \[ v_n(q) = \frac{1}{\sqrt{M_2}} A_2(q)e^{\mathrm i(qan-\omega t)}. \]
Substitution gives \[ \left(\frac{2f}{M_1}-\omega^2\right)A_1 - \frac{f}{\sqrt{M_1M_2}} \left(1+e^{-\mathrm iqa}\right)A_2 =0, \] \[ - \frac{f}{\sqrt{M_1M_2}} \left(1+e^{+\mathrm iqa}\right)A_1 + \left(\frac{2f}{M_2}-\omega^2\right)A_2 =0. \]
The corresponding dynamical matrix is \[ D(q) = \begin{pmatrix} \dfrac{2f}{M_1} & -\dfrac{f}{\sqrt{M_1M_2}}(1+e^{-\mathrm iqa})\\ -\dfrac{f}{\sqrt{M_1M_2}}(1+e^{+\mathrm iqa}) & \dfrac{2f}{M_2} \end{pmatrix}. \]
The allowed frequencies follow from \[ \det[D(q)-\omega^2\mathbf 1]=0. \]
6.5.3 Acoustic and Optical Branches
The determinant condition gives
There are two branches because there are two independent longitudinal degrees of freedom per cell. The lower branch \(\omega_-(q)\) is the acoustic branch; the upper branch \(\omega_+(q)\) is the optical branch.
6.5.4 Long-Wavelength Limits
For \(qa\ll1\), expand the square root in the dispersion relation. The two branches become \[ \omega_+^2(q) \simeq 2f\left(\frac{1}{M_1}+\frac{1}{M_2}\right) - \frac{a^2f}{2(M_1+M_2)}q^2, \] and \[ \omega_-^2(q) \simeq \frac{a^2f}{2(M_1+M_2)}q^2. \]
Thus the acoustic branch is linear at small \(q\): \[ \omega_-(q) \simeq v_s q, \] with \[ v_s = \sqrt{\frac{a^2f}{2(M_1+M_2)}}. \]
The optical branch has a nonzero zone-center frequency:
6.5.5 Mode Character at the Zone Center
At \(q\to0\), the acoustic branch has neighboring atoms moving in phase: \[ \frac{u}{v}=1 \qquad (\mathrm{for}~~ \omega_-). \]
The optical branch has neighboring atoms moving out of phase: \[ \frac{u}{v} = -\frac{M_2}{M_1} \qquad (\mathrm{for}~~ \omega_+). \]
For the optical mode, \[ M_1u+M_2v=0, \] so the center of mass of the two-atom basis remains at rest.
This distinction is the origin of the terms acoustic and optical. Acoustic modes reduce to ordinary sound waves in the long-wavelength limit. Optical modes describe relative motion of the atoms in the basis; in ionic crystals this motion can produce an oscillating electric dipole moment and therefore couple strongly to light.
6.5.6 Zone-Boundary Limits and Frequency Gap
At the one-dimensional zone boundary, \[ q=\frac{\pi}{a}, \] the two branches have frequencies \[ \omega_+\left(\frac{\pi}{a}\right) = \sqrt{\frac{2f}{M_2}}, \] \[ \omega_-\left(\frac{\pi}{a}\right) = \sqrt{\frac{2f}{M_1}}, \] assuming \(M_1>M_2\).
A frequency gap opens between the upper edge of the acoustic branch and the lower edge of the optical branch. The gap grows as the mass ratio \(M_1/M_2\) becomes larger. In the gap the wavevector becomes imaginary, corresponding to a spatially damped wave rather than a propagating normal mode.
6.5.7 Limiting Mass Ratios
If \(M_1\gg M_2\), the acoustic branch is governed mainly by the heavy atoms: \[ \omega_-^2 \simeq \frac{2f}{M_1} \sin^2\frac{qa}{2}. \]
The optical branch is nearly flat and is controlled mainly by the light atoms: \[ \omega_+^2 \simeq \frac{2f}{M_2} - \frac{2f}{M_1} \sin^2\frac{qa}{2}. \]
If \(M_1=M_2=M\), the chain is physically a one-atomic lattice with half the spacing. The apparent separation into acoustic and optical branches is then a consequence of using a doubled unit cell; the dispersion has been folded into the smaller Brillouin zone.
6.6 Lattice Vibrations in Three Dimensions
The one-dimensional models are pedagogical reductions. In a real three-dimensional crystal, each atom has three displacement components, and the polarization structure becomes central.
6.6.1 One-Atomic Basis in Three Dimensions
For a three-dimensional crystal with a one-atomic basis, \(r'=1\), there are \[ 3r'=3 \] branches for each wavevector \(\mathbf q\).
They are classified as:
- one longitudinal acoustic branch, usually labeled LA;
- two transverse acoustic branches, usually labeled TA.
In high-symmetry directions, the longitudinal branch has displacement parallel to \(\mathbf q\), while the transverse branches have displacement perpendicular to \(\mathbf q\). In general directions, the eigenvectors need not be exactly longitudinal or transverse; the polarization may be mixed.
Usually the longitudinal branch lies at higher frequency than the transverse branches for the same small wavevector magnitude, reflecting the larger stiffness associated with compression compared with shear.
The periodicity and evenness of the dispersion remain valid: \[ \omega(\mathbf q)=\omega(\mathbf q+\mathbf G), \] \[ \omega(-\mathbf q)=\omega(\mathbf q). \]
Therefore phonon dispersions are plotted only within the first Brillouin zone, typically along high-symmetry paths.
6.6.2 Multiatomic Basis in Three Dimensions
For a basis with \(r'\) atoms, the determinant equation has \[ 3r' \] solutions for each \(\mathbf q\). In a three-dimensional crystal these separate into
The three acoustic branches correspond to long-wavelength translations of the basis. The optical branches correspond to internal relative motion of atoms within the basis.
For example, silicon has the diamond structure with a two-atomic basis. Thus \(r'=2\), and there are \[ 3r'=6 \] phonon branches: three acoustic and three optical.
6.6.3 Sanity Checks for Branch Counting
- A one-atomic basis in three dimensions has only acoustic branches.
- A two-atomic basis in three dimensions has six branches in total.
- Optical branches require internal degrees of freedom inside the basis.
- Degeneracies can reduce the number of visibly distinct curves along high-symmetry directions, even though the number of branches is unchanged.
6.7 Bridge to the Next Topic: From Dispersion to a Phonon Density of States
The central output of lattice dynamics is the set of branches \[ \omega_r(\mathbf q), \] where \(r\) labels the branch. The next step is to count how many modes lie in a frequency interval. This leads to the phonon density of states.
For a macroscopic three-dimensional crystal with periodic Born–von Karman boundary conditions, the allowed wavevectors become very dense in reciprocal space. The density of allowed \(\mathbf q\)-points is \[ Z(\mathbf q) = \frac{V}{(2\pi)^3}. \]
Sums over phonon modes can then be replaced by integrals: \[ \sum_{\mathbf q,r}F[\omega_r(\mathbf q)] = \sum_r \int_{\mathrm{1.\ BZ}} d^3q Z(\mathbf q) F[\omega_r(\mathbf q)], \] where \(F[\omega_r(\mathbf q)]\) is an arbitrary function of the frequency.
Equivalently, one defines a frequency-space density of states \[ D(\omega) = \sum_r \int_{\mathrm{1.\ BZ}} d^3q Z(\mathbf q) \delta \left(\omega-\omega_r(\mathbf q)\right). \]
The physical point is simple: a flat dispersion produces many states in a small frequency interval, while a steep dispersion produces fewer. This is why zone-boundary flattening and optical branches are important for thermal properties.
- Lattice vibrations in a periodic crystal reduce to eigenvalue problems for the dynamical matrix.
- A one-atomic basis gives acoustic branches because uniform translation costs no restoring energy.
- The long-wavelength limit of an acoustic branch reproduces continuum sound propagation.
- The first Brillouin zone contains all independent phonon wavevectors.
- A two-atomic basis produces acoustic and optical branches.
- Acoustic modes involve in-phase motion of neighboring basis atoms at small wavevector.
- Optical modes involve relative motion inside the basis and can couple to light in ionic crystals.
- In three dimensions, a basis with multiple atoms produces both acoustic and optical branches.
6.8 Problem Set
One-Atomic Basis With General Couplings. Starting from \[ M\frac{\partial^2u_n}{\partial t^2} - \sum_p C_p(u_{n+p}-u_n) =0, \] insert \(u_n=Ae^{\mathrm i(qna-\omega t)}\) and derive \[ \omega^2(q) = \frac{2}{M} \sum_{p=1}^{\infty} C_p(1-\cos qpa). \] Then specialize to nearest-neighbor coupling.
Group Velocity and the Sound Limit. For the nearest-neighbor one-atomic basis, \[ \omega(q)=2\sqrt{\frac{C_1}{M}}\sin\frac{qa}{2} \qquad (0\le q\le \pi/a), \] compute \(v_g(q)\). Evaluate the limits \(q\to0\) and \(q\to\pi/a\), and interpret the results.
First Brillouin Zone. Show that the displacement pattern produced by \(q\) is identical to that produced by \(q+2\pi/a\) on the lattice sites. Explain why this allows the dispersion relation to be restricted to \(-\pi/a\le q\le \pi/a\).
Two-Atomic Basis. Starting from \[ M_1\frac{\partial^2u_n}{\partial t^2} + f(2u_n-v_n-v_{n-1})=0, \] \[ M_2\frac{\partial^2v_n}{\partial t^2} + f(2v_n-u_n-u_{n+1})=0, \] derive the \(2\times2\) dynamical matrix using the mass-normalized ansatz and obtain the determinant equation for \(\omega^2\).
Branch Counting in Three Dimensions. A three-dimensional crystal has \(r'\) atoms in its basis. Determine the total number of phonon branches, the number of acoustic branches, and the number of optical branches. Apply the result to silicon.