Past exam of the condensed matter physics major option course of the University of Oxford / 2019 / 1

Infrared spectroscopy measures absorption as the incident photon frequency is swept through molecular normal mode frequencies. In the electric-dipole approximation the interaction is $-\mathbf{p}\cdot \mathbf{E}$. Expanding the molecular electric dipole moment in a normal coordinate $Q$ givesThe fundamental one-quantum vibrational transition dipole moment is proportional to the derivative multiplying $Q$. Thusis the criterion for vibrational infrared activity; the derivative must also have a component along the incident polarization of an electromagnetic wave. A permanent molecular electric dipole moment is not necessary. Absorption occurs near $\hslash \omega =E_{v+1}-E_v$, with broadened lines reflecting finite lifetimes and the molecular environment.

Use the indicated out-of-plane displacements and the equivalence of the four oxygen sites and of the two carbon sites in the isolated planar oxalate ion. In $V_1$, the two oxygen atoms on one side move oppositely to those on the other side, and the two carbon atoms also move oppositely. Their dipole changes cancel. The mode is even under molecular inversion symmetry, which independently forbids an electric-dipole transition.

In $V_2$, all four oxygen atoms move together out of the plane while both carbon atoms move in the opposite direction. The positive and negative charge distributions acquire a relative out-of-plane displacement, producing a nonzero derivative of the molecular electric dipole moment.

In $V_3$, two oxygen atoms move out and two move in, in a checkerboard arrangement, while the carbon atoms are stationary. The equivalent oxygen contributions cancel. This mode illustrates that being odd under inversion symmetry is necessary but not sufficient for vibrational infrared activity: its full symmetry does not transform like any component of an electric dipole moment.These conclusions concern the isolated-molecule displacements drawn in the source; a different crystal environment can alter selection rules through additional symmetry breaking.

There are six atoms in the nonlinear oxalate ion, hence $3\cdot 6=18$ positional degrees of freedom. Subtract three rigid translations and three rigid rotations to obtain $3N-6=12$ vibrational normal modes. Planarity does not remove out-of-plane motions. Since three modes have been displayed,

The primitive unit cell contains two carbon atoms and four edge-shared oxygen atoms, each contributing one half: $n=2+4/2=4$ atoms. Equivalently, one molecular repeat contributes its two carbon atoms but only half of each of its four shared oxygen atoms. There are therefore $3n=12$ phonon branches.

At the Brillouin zone centre, three rigid translations give three acoustic phonon branches; the remaining $3n-3$ are optical phonon branches:A free layer has two in-plane acoustic modes and an out-of-plane flexural acoustic mode. A two-dimensional lattice does not imply that its atoms can move only in two directions. Artificially suppressing all out-of-plane motion would instead define a different model with two acoustic and six optical branches.

The primitive translations connect the two centred-rectangular sublattices. In the orientations marked in the source, ${\mathbf{a}}_p$ points right and down, whereas ${\mathbf{b}}_p$ points right and up. ThusTheir parallelogram has half the area of the rectangular cell, confirming that the latter contains two primitive unit cells.

Let the reciprocal lattice vector be $\mathbf{G}=h_r{\mathbf{a}}_r^{\ast }+k_r{\mathbf{b}}_r^{\ast }=h_p{\mathbf{a}}_p^{\ast }+k_p{\mathbf{b}}_p^{\ast }$, with each dual pair obeying $\mathbf{a}\cdot {\mathbf{a}}^{\ast }=\mathbf{b}\cdot {\mathbf{b}}^{\ast }=2\pi$ and zero cross products. Its coefficients are obtained by dotting with the corresponding direct-space primitive lattice translation vectors:HenceFor actual integer-indexed reciprocal lattice vectors, this also explains the centred-rectangular condition that $h_r$ and $k_r$ have the same parity. General phonon wave vectors need not have integer coefficients.

Comparing the two coefficients in the reciprocal-coordinate transformation givesThe inverse transformation is $h_r=h_p+k_p$, $k_r=k_p-h_p$. This provides an independent check on the signs associated with the two primitive arrows in the source.

Apply the same change of reciprocal lattice coordinates to the phonon wave vectors:Although $(1,0{)}_r$ is an integer coordinate of the rectangular reciprocal grid, it is not a reciprocal lattice vector of the actual primitive crystal. Thus $P_1$ is not equivalent to the Brillouin zone centre.

For ordinary one-phonon infrared optical absorption, both an energy and a momentum selection rule must hold. The optical photon has negligible crystal momentum on the scale of the Brillouin zone, so crystal momentum conservation selects $\mathbf{q}\simeq 0$ modulo a genuine reciprocal lattice vector. In addition, the mode must change the macroscopic electric polarization, with a component along the incident polarization of an electromagnetic wave.

In terms of atomic displacements ${\mathbf{u}}_j$ and Born effective charge tensors $Z_j^{\ast }$, the relevant derivative is proportional to ${\sum }_j{Z}_j^{\ast }{\mathbf{u}}_j$ per primitive unit cell. Thus an optical phonon at $\mathbf{q}=0$ is not automatically infrared active: its dipole contributions may cancel. Disorder-assisted and multiphonon processes relax these simple first-order selection rules.

$P_1$ has primitive coordinates $(1/2,1/2)$ and therefore fails the first-order optical momentum condition; its local $V_1$ pattern also has zero dipole derivative. $P_2$ is a zone-centre mode with the dipole-active $V_2$ pattern, so its molecular dipole changes add coherently.The allowed absorption for $P_2$ uses an polarization of an electromagnetic wave with an out-of-plane component. At normal incidence on an ideal free-standing layer the electric field is in the plane, so that specific measurement geometry would suppress it despite the mode's intrinsic vibrational infrared activity.