6_bonds_and_spectra.md

from matplotlib import pyplot

import numpy as np

from common import draw_classic_axes, configure_plotting

configure_plotting()

pi = np.pi

!!! summary "Learning goals"

After this lecture you will be able to:

- Explain the origins of interatomic forces
- Compute vibrational spectra of small molecules in 1D
- Formulate Hamiltonians and equations of motion of bulk materials (but not yet solve them)

Adding repulsion

If bringing two atoms closer would keep increasing energy, any covalent bond would collapse; eventually the two atoms must start repelling (at least when the nuclei get close, but really already much earlier).

Van der Waals bond

Not our focus, but we will discuss it for completeness

Originates from attraction between dipole moments of two atoms:

If one atom has a dipole moment p_1, it creates electric field

E = \frac{p_1}{4\pi \varepsilon_0 r^3}

at the position of another atom. The other atom then develops a dipole moment p_2 = \chi E with \chi the polarizability of the atom.

This results in an attractive potential U(r) = -E p_2 \sim r^{-6}.

This attraction is much weaker than covalent bonds but drops slower with increasing distance. (Q: how fast does the interaction of covalent bonds drop?)

Van der Waals bonds hold layers of covalently bonded carbon atoms together when forming graphite:

(image source: Wikipedia)

First steps towards phonons

The atomic interaction is minimized when the distance between the atoms is at the minimum of the potential: U = U_0 when \delta r = a.

Near the minimum, the potential is approximately parabolic:

U = U_0 + \frac{\kappa}{2} (\delta r - a)^2 - \frac{\kappa_3}{3!} (\delta r - a)^3 + \ldots

This is a harmonic oscillator potential with the higher order term \kappa_3 providing anharmonicity.

Rigidity

If we stretch a material with length L containing N=L/a atoms by a lenght \delta L, the displacement of each atom is \delta r - a = \delta L/N.

The returning force is

F = - \frac{d U}{d(\delta r)} (\delta r - a) = \kappa a \delta L / L,

which leads us to the material compressibility

\beta \equiv -\frac{1}{L} \frac{\partial L}{\partial F} = \frac{1}{\kappa a}.

Then using compressibility we can derive the wave equation and obtain the speed of sound in the material relating it to the bond strength.

Thermal expansion

The nonlinearity \kappa_3 means that the potential grows slower when the atoms are further apart than when they are closer to each other.

Therefore the larger the energy of the atoms, the more their average distance increases \Rightarrow we have a model of thermal expansion.

Looking ahead: multiple atoms

Our aim is to understand electrons and phonons in solids containing N\to\infty atoms.

We now have a microscopic model of what happens when there are two atoms; let's try to see what happens when we take several.

Phonons

Our plan:

• Consider only harmonic potential acting between atoms
• Write down equations of motion
• Compute normal modes

For simplicity we consider 1D motion, and let's start with 3 atoms:

Newton's equations of motion:

\begin{aligned} m \ddot{x}_1 &= - \kappa (x_1 - x_2) \\ m \ddot{x}_2 &= -\kappa (x_2 - x_1) -\kappa (x_2 - x_3) \\ m \ddot{x}_3 &= - \kappa (x_3 - x_2) \end{aligned},

Or in matrix form:

m \ddot{x} = -\kappa \begin{pmatrix} 1 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 1 \end{pmatrix}x

We search for normal modes: patterns of motion that are periodic and have a fixed frequency: x(t) = x_0 e^{i\omega t}.

Substituting into the equations of motion we get an eigenvalue problem:

\omega^2 x_0 = \sqrt{\frac{\kappa}{m}} \begin{pmatrix} 1 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 1 \end{pmatrix}x_0.

The solutions of this eigenvalue problem are the phonon modes that we occupy.

Electrons

Same as a single covalent bond, only more atoms in a line. Considering 3 atoms:

E \begin{pmatrix} c_1 \\ c_2 \\ c_3 \end{pmatrix} = \begin{pmatrix} E_0 & t & 0 \\ t & E_0 & t \\ 0 & t & E_0 \end{pmatrix} \begin{pmatrix} c_1 \\ c_2 \\ c_3 \end{pmatrix}

Numerical test

Diagonalizing large matrices is unwieldy, but let's try and check it numerically to see if we notice a trend.

Eigenfrequencies of 3 atoms: [0.0 1.0 1.732050]

def DOS_finite_phonon_chain(n):
    rhs = 2 * np.eye(n) - np.eye(n, k=1) - np.eye(n, k=-1)
    rhs[0, 0] -= 1
    rhs[-1, -1] -= 1
    pyplot.figure()
    pyplot.hist(np.sqrt(np.abs(np.linalg.eigvalsh(rhs))), bins=30)
    pyplot.xlabel("$\omega$")
    pyplot.ylabel("Number of levels")

DOS_finite_phonon_chain(3)

Energies of 3 orbitals: [-1.41421356 0.0 1.41421356]

def DOS_finite_electron_chain(n):
    rhs = - np.eye(n, k=1) - np.eye(n, k=-1)
    pyplot.figure()
    pyplot.hist(np.linalg.eigvalsh(rhs), bins=30)
    pyplot.xlabel("$E$")
    pyplot.ylabel("Number of levels")

DOS_finite_electron_chain(3)

From 3 atoms to 300

Frequencies of many phonons:

DOS_finite_phonon_chain(300)

Energies of many electrons:

DOS_finite_electron_chain(300)

Summary

• Electrons in atoms occupy "shells" in the energetically favorable order
• When two atoms come close, electrons occupy molecular orbitals and bind atoms together.
• Motion of electrons makes atoms attract
• Oscillatory motion of atoms and hopping of electrons between atoms give rise to the macroscopic behavior of the materials (next week)