The purpose of this post is to build up to an explanation of the Mössbauer effect in quantum mechanics. At an intuitive level, the Mössbauer effect simply asserts that when an atom absorbs or emits a \(\gamma\)-ray photon, if it’s part of a large solid lattice, then this absorption/emission will be essentially recoil-free (due to the large solid lattice) and thus the energy \(E_{\gamma}\) of the \(\gamma\)-ray photon can be known very precisely \(\Delta E_{\gamma}\approx 0\). This is what underlies the basic technique of Mössbauer spectroscopy for high-precision experiments with atoms.
Monatomic Circular Chain
Consider a classical system of \(N\) atoms (each of identical mass \(m\)) which at equilibrium are pairwise separated by \(\Delta x\) in an \(S^1\) topology. Let \(\eta_i(t)\) denote each atom’s position relative to where it would rest at equilibrium. Then the Lagrangian \(\mathcal L\) for this system is:
\[\mathcal L=\frac{m}{2}\sum_{i=1}^N\dot{\eta}_i^2-\sum_{i=1}^NV(\eta_{i+1}-\eta_i+\Delta x)\]
where the interatomic potential energy \(V(r)\) is the same for each pair of adjacent atoms. Assuming the \(\eta_i(t)\) are small, we can Taylor expand the interatomic potential energy \(V(r)\) about \(r=\Delta x\) to obtain the harmonic oscillator form:
\[V(r)\approx V(\Delta x)+\frac{1}{2}m\omega_0^2(r-\Delta x)^2+O_{r\to\Delta x}(r-\Delta x)^3\]
So that \(V(\eta_{i+1}-\eta_i+\Delta x)\approx\frac{1}{2}m\omega_0^2(\eta_{i+1}-\eta_i)^2\) ignoring the overall constant. Chugging this through the Euler-Lagrange equations then produces the system of coupled harmonic oscillators:
\[\ddot{\eta}_i=-\omega_0^2(2\eta_i-\eta_{i-1}-\eta_{i+1})\]
The Born-von Karmen periodic boundary conditions mean that \(\eta_i=\eta_{i+N}\), so defining \(\boldsymbol{\eta}(t):=(\eta_1(t),\eta_2(t),…,\eta_N(t))^T\in\textbf R^N\), we have an \(N\times N\) circulant matrix \(\Omega^2=\omega_0^2\text{diag}_{\circ}(-1,2,-1)\) for which \(\ddot{\boldsymbol{\eta}}(t)=-\Omega^2\boldsymbol{\eta}(t)\) has the simple harmonic oscillator form with general solution \(\boldsymbol{\eta}(t)=(\alpha e^{i\Omega t}+\beta e^{-i\Omega t})\boldsymbol{\eta}(0)\) a superposition of normal modes. The \(N\) orthonormal eigenvectors of \(\Omega^2\) are given by the usual \(N\) columns of the \(N\times N\) DFT matrix \(\hat{\boldsymbol{\omega}}^2_j(t)=\frac{1}{\sqrt N}(1,e^{-2\pi ij/N},…,e^{-2\pi ij(N-1)/N})^T\) for \(j=1,…,N\) with corresponding normal mode eigenfrequencies:
\[\omega_j=2\omega_0\left|\sin\frac{j\pi}{N}\right|\]
And indeed, this is basically the dispersion relation for the monatomic circular chain of atoms. More precisely, if we recall that the Born-von Karmen periodic boundary conditions quantizes \(k_j=2\pi j/N\Delta x\), and then ignore the \(j\)-label, then we do arrive at the proper dispersion relation:
\[\omega(k)=2\omega_0\left|\sin\frac{k\Delta x}{2}\right|\]
The phase velocity and group velocity are respectively:
\[v_p(k)=\frac{\omega(k)}{k}=\frac{2\omega_0\left|\sin\frac{k\Delta x}{2}\right|}{k}\]
\[v_g(k)=\frac{\partial\omega}{\partial k}=\text{sgn}(k)\omega_0\Delta x\cos\frac{k\Delta x}{2}\]
Or non-dimensionalizing:
\[\frac{v_p(k)}{\omega_0\Delta x}=\text{sgn}(k)\left|\text{sinc}\frac{k\Delta x}{2}\right|\]
\[\frac{v_g(k)}{\omega_0\Delta x}=\text{sgn}(k)\cos\frac{k\Delta x}{2}\]
It is therefore apparent that, in the low-momentum \(k\to 0\) limit, the dispersion relation can be linearized as \(\omega(k)\approx \omega_0\Delta x k\) which gives the speed of sound at low \(k\), i.e. \(c_s=\omega_0\Delta x\) as evident in the plots too. This stands in sharp contrast with the dispersion relation of electrons in a tight-binding lattice where instead of a linear dispersion relation as \(k\to 0\), it was quadratic, indicating a free non-relativistic massive quantum particle where “massive” referred to an effective mass emergent from the physical properties of the lattice. Finally, notice that we haven’t done everything quantum mechanical in the discussion above; everything has been classical so far!
Dispersion Relations for Acoustic & Optical Branches of Diatomic Chain
If one now considers an alternating chain \(m,M,m,M,…\) of two types of atoms, then the equations of motion would look something like (no need to go back to the Lagrangian again, just think Newtonian):
\[m\ddot{\eta}_{2j}=k_s(\eta_{2j+1}-\eta_{2j})+k_s(\eta_{2j-1}-\eta_{2j})=-k_s(2\eta_{2j}-\eta_{2j+1}-\eta_{2j-1})\]
\[M\ddot{\eta}_{2j+1}=k_s(\eta_{2j+2}-\eta_{2j+1})+k_s(\eta_{2j}-\eta_{2j+1})=-k_s(2\eta_{2j+1}-\eta_{2j+2}-\eta_{2j})\]
Basically (and this is an approach we could have used earlier for the monatomic chain as well), the idea is that there are two distinct forms of periodicity in this system; one is the spatial periodicity of the chain (although now, if the atoms are separated by \(\Delta x\), the spatial period is \(2\Delta x\) because of the chain’s diatomic nature whereas the periodicity was just \(\Delta x\) for the monatomic chain). The second form of periodicity is temporal periodicity since we expect each atom to oscillate about its equilibrium position. Of course, whenever there is periodicity involved we expect that Fourier-analytic techniques will be useful. We can first exploit the temporal periodicity of each atomic oscillator by taking the temporal Fourier transform of the equations of motion to “algebraize” the time derivatives:
\[-m\omega^2\hat{\eta}_{2j}=-k_s(2\hat{\eta}_{2j}-\hat{\eta}_{2j+1}-\hat{\eta}_{2j-1})\]
\[-M\omega^2\hat{\eta}_{2j+1}=-k_s(2\hat{\eta}_{2j+1}-\hat{\eta}_{2j}-\hat{\eta}_{2j+2})\]
So now it’s no longer a system of differential equations, but a system of difference equations. Now we exploit the spatial periodicity (both in the microscopic lattice but also the macroscopic Born-von Karmen periodicity) by taking the discrete Fourier transform \(\hat{\eta}_j\mapsto\text{DFT}\{\hat{\eta}_j\}(k)=\sum_{j=1}^N\hat{\eta}_j e^{-ik\Delta xj}\) which obeys \(\text{DFT}\{\hat{\eta}_{j\pm 1}\}(k)=e^{ik\Delta x}\text{DFT}\{\hat{\eta}_j\}(k)+\hat{\eta}_1(e^{-ik\Delta x N}-1)\) and \(\text{DFT}\{\eta_{j-1}\}(k)=e^{-ik\Delta x}(\text{DFT}\{\eta_j\}+\eta_0(1-e^{-ik\Delta xN}))\):
\[\]