LATTICE VIBRATIONS: DIATOMIC LINEAR CHAIN

Detailed Derivation of Acoustic and Optical Phonon Branches

1. Physical Model: Two Atoms Per Primitive Basis

We consider a 1D Bravais lattice with lattice constant $ a $. Each primitive cell contains two different atoms:

Mass $ M_1 $
Mass $ M_2 $

They are arranged alternately along a line:

            ... -- M1 -- M2 -- M1 -- M2 -- ...
        

We assume nearest-neighbor interaction only, with an identical spring constant $ C $.

Geometry of the Diatomic Chain

Let:

$ u_n $ = displacement of $ M_1 $ in the $ n $-th cell.
$ v_n $ = displacement of $ M_2 $ in the $ n $-th cell.

The equilibrium positions are:

$$ x_{M_1, n} = na, \quad x_{M_2, n} = na + \frac{a}{2} $$

2. Equations of Motion

Forces on $ M_1 $

Mass $ M_1 $ in cell $ n $ is connected to $ M_2 $ in the same cell and $ M_2 $ in the previous cell ($ n-1 $). The net restoring force is:

\[ F_{u_n} = -C(u_n - v_n) - C(u_n - v_{n-1}) \]

Using Newton’s second law:

$$ M_1 \ddot{u}_n = -C(2u_n - v_n - v_{n-1}) $$

Forces on $ M_2 $

Mass $ M_2 $ in cell $ n $ is connected to $ M_1 $ in the same cell and $ M_1 $ in the next cell ($ n+1 $).

\[ F_{v_n} = -C(v_n - u_n) - C(v_n - u_{n+1}) \]

$$ M_2 \ddot{v}_n = -C(2v_n - u_n - u_{n+1}) $$

3. Plane Wave Solution (Normal Mode Ansatz)

We seek traveling wave solutions of the form:

\[ u_n = U e^{i(kna - \omega t)} \] \[ v_n = V e^{i(kna - \omega t)} \]

Substituting neighbor terms:

\[ v_{n-1} = V e^{i(k(n-1)a - \omega t)} = V e^{-ika} e^{i(kna - \omega t)} \] \[ u_{n+1} = U e^{i(k(n+1)a - \omega t)} = U e^{ika} e^{i(kna - \omega t)} \]

4. Matrix Formulation

Substituting the solutions into the equations of motion:

For $ M_1 $:

\[ -\omega^2 M_1 U = -C [ 2U - V(1 + e^{-ika}) ] \] \[ \omega^2 M_1 U = C [ 2U - V(1 + e^{-ika}) ] \]

For $ M_2 $:

\[ -\omega^2 M_2 V = -C [ 2V - U(1 + e^{ika}) ] \] \[ \omega^2 M_2 V = C [ 2V - U(1 + e^{ika}) ] \]

This system of linear homogeneous equations can be written in matrix form:

$$ \begin{pmatrix} 2C - \omega^2 M_1 & -C(1+e^{-ika}) \\ -C(1+e^{ika}) & 2C - \omega^2 M_2 \end{pmatrix} \begin{pmatrix} U \\ V \end{pmatrix} = 0 $$

For a non-trivial solution, the determinant of the matrix must be zero.

5. Dispersion Relation

Calculating the determinant:

\[ (2C - \omega^2 M_1)(2C - \omega^2 M_2) - C^2 (1+e^{-ika})(1+e^{ika}) = 0 \]

Simplifying the cross term:

\[ (1+e^{-ika})(1+e^{ika}) = 1 + e^{ika} + e^{-ika} + 1 = 2 + 2\cos(ka) = 4\cos^2\left(\frac{ka}{2}\right) \]

The characteristic equation becomes:

\[ 4C^2 - 2C\omega^2(M_1+M_2) + \omega^4 M_1 M_2 - 4C^2 \cos^2\left(\frac{ka}{2}\right) = 0 \]

Using the identity $ 1 - \cos^2 x = \sin^2 x $, we get a quadratic equation in $ \omega^2 $:

$$ \omega^4 M_1 M_2 - 2C(M_1+M_2)\omega^2 + 4C^2 \sin^2\left(\frac{ka}{2}\right) = 0 $$

Solving for $ \omega^2 $:

$$ \omega^2 = C \left( \frac{M_1+M_2}{M_1 M_2} \right) \pm C \sqrt{ \left( \frac{M_1+M_2}{M_1 M_2} \right)^2 - \frac{4}{M_1 M_2}\sin^2\left(\frac{ka}{2}\right) } $$

6. Two Branches of Vibration

The $ \pm $ sign gives two distinct solutions for $ \omega(k) $:

Acoustic Branch ($-$ sign): Corresponds to sound waves where atoms move in phase.
Optical Branch ($+$ sign): Corresponds to high-frequency vibrations where atoms move out of phase.

7. Limiting Cases

A. Long Wavelength Limit ($ k \to 0 $)

Using the approximation $ \sin(x) \approx x $:

Acoustic Branch: $$ \omega_- \approx v_s k $$ Where sound velocity $ v_s = a \sqrt{\frac{C}{2(M_1+M_2)}} $.
Interpretation: The lattice behaves like an elastic continuum.

Optical Branch: $$ \omega_+ \approx \sqrt{2C \left( \frac{1}{M_1} + \frac{1}{M_2} \right)} $$ Interpretation: The two sub-lattices vibrate against each other even at infinite wavelength.

B. Zone Boundary ($ k = \pi/a $)

At the edge of the Brillouin zone, $ \sin(ka/2) = 1 $. The square root simplifies significantly:

$$ \omega_+^2 = \frac{2C}{M_1} $$ (if $ M_1 < M_2 $)
$$ \omega_-^2 = \frac{2C}{M_2} $$ (if $ M_1 < M_2 $)

Frequency Gap: There is a range of frequencies between $ \sqrt{2C/M_2} $ and $ \sqrt{2C/M_1} $ for which no wave can propagate. This is the forbidden band gap.

8. Amplitude Ratio (Polarization)

The ratio of amplitudes $ U/V $ describes the relative motion of the two atoms:

$$ \frac{U}{V} = \frac{C(1+e^{-ika})}{2C - \omega^2 M_1} $$

Acoustic Mode: $ U \approx V $. Atoms move in the same direction.
Optical Mode: $ U \approx -\frac{M_2}{M_1} V $. Atoms move in opposite directions, keeping the center of mass fixed.

9. Deep Physical Insights

Origin of Frequency Gap: The gap arises because the two masses are different ($ M_1 \neq M_2 $), breaking the symmetry. If $ M_1 = M_2 $, the gap disappears, reducing to the monatomic case.
Energy Storage: Acoustic modes store energy primarily in the kinetic energy of the center of mass motion. Optical modes store energy in the internal vibration of the basis.
Optical Phonons: If the atoms are charged (e.g., Na+ and Cl-), the out-of-phase motion creates an oscillating dipole moment that can couple to electromagnetic radiation (light), hence the name "Optical Branch."

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