Mixed Quasielastic & Superelastic Channels

• Mixed quasielastic and superelastic channels are electron-lattice scattering mechanisms that combine minor phonon excitations with lattice center-of-mass recoil to conserve momentum.
• The mechanism, derived from a density–density interaction Hamiltonian, distributes the electron’s momentum between phonon energy shifts and macroscopic recoil, reconciling experimental anomalies.
• These channels dominate momentum relaxation in clean and strange metals, explaining robust quantum oscillations and diffusive, Planckian-scale transport despite minimal energy dissipation.

Mixed quasielastic and superelastic channels refer to electron-lattice scattering processes in perfect crystals where phonon occupations change but do not fully account for the electron's momentum transfer. Unlike purely inelastic (phonon-mediated) scattering, these channels distribute the electron’s transferred momentum between low-energy phonon modes and the rigid translation (center-of-mass recoil) of the entire lattice. This mechanism, established in the density–density form of the electron-lattice interaction Hamiltonian, provides a microscopic basis for momentum relaxation in regimes where local energy dissipation is negligible and reconciles anomalies in experimental transport phenomena at finite temperature (Heller et al., 31 Jan 2026).

1. Electron–Lattice Interaction Hamiltonian

The coupling between conduction electrons and the crystal lattice is formulated through a density–density interaction that explicitly incorporates both the electron's coordinate $\hat{\mathbf{r}}$ and the collective lattice center-of-mass coordinate $\hat{\mathbf{R}}$: $$H_{\mathrm{int}} = \sum_{\mathbf{q}} v(\mathbf{q})\, e^{i \mathbf{q}\cdot (\hat{\mathbf{r}} - \hat{\mathbf{R}})}\, \hat{\rho}^{\mathrm{int}}_{-\mathbf{q}},$$ where $v(\mathbf{q})$ denotes the screened Fourier-space electron-ion (deformation-potential) interaction. The internal lattice density operator, $$\hat{\rho}^{\mathrm{int}}_{\mathbf{q}} = \sum_j e^{-i \mathbf{q}\cdot(\boldsymbol{\xi}^{(0)}_j + \hat{\mathbf{u}}_j)}$$, captures all vibrational modes except the zero mode (center of mass). This formalism rigorously enforces conservation of total mechanical momentum, with the total momentum operator $$\hat{\mathbf{Q}} = \hat{\mathbf{p}} + \hat{\mathbf{P}}$$, where $\hat{\mathbf{p}}$ and $\hat{\mathbf{P}}$ are the pseudomomentum of the electron and the lattice, respectively. The exponential factor $$e^{i\mathbf{q}\cdot(\hat{\mathbf{r}} - \hat{\mathbf{R}})}$$ ensures the exchange of pseudomomentum $\hbar\mathbf{q}$ between the electron and the lattice zero mode.

2. Kinematic Structure and Channel Classification

Momentum conservation in a scattering event ensures that the electron’s momentum transfer $\hbar\mathbf{q}$ is balanced by an equal and opposite change in lattice center-of-mass momentum, with no net change in the conserved total momentum. Simultaneously, energy conservation must be satisfied by distributing the electron’s energy change among the center-of-mass recoil and internal phonon excitations: $$\varepsilon_{k'} - \varepsilon_k + \Delta E_{\text{recoil}} + \Delta E_{\text{ph}} = 0,$$ where $\Delta E_{\text{recoil}}$ is the kinetic energy of the lattice’s zero mode, and $$\Delta E_{\text{ph}} = \sum_{\mathbf{k}\lambda} \hbar\omega_{\mathbf{k}\lambda} \Delta n_{\mathbf{k}\lambda}$$ is the energy gained (or lost) by the phonon bath. Three distinct cases emerge:

• Elastic (phonon-diagonal): No change in phonon occupation ($\Delta n = 0$), negligible recoil for macroscopic $M$, so the scattering is exactly energy- and momentum-diagonal.
• Quasielastic (mixed channel): A small number of phonons are created or annihilated, offsetting residual energy differences while the zero mode covers most of the momentum.
• Superelastic (mixed channel): Phonon annihilation ($\Delta E_{\mathrm{ph}} < 0$) “super-kicks” the electron, with the remainder of the momentum transfer absorbed by the zero mode.

3. Scattering Amplitudes and Rates

The first-order Born approximation yields the transition amplitude: $$T_{i \rightarrow f} = v(\mathbf{q})\, \langle k'|e^{i\mathbf{q}\cdot\hat{\mathbf{r}}}|k\rangle\, \langle n'| e^{-i\mathbf{q}\cdot\hat{\mathbf{R}}} \hat{\rho}^{\mathrm{int}}_{-\mathbf{q}}|n\rangle,$$ where $|k,\{n\}\rangle$ represent electron and phonon states. The corresponding Golden-rule rate is given by

$$W_{i \rightarrow f} = \frac{2\pi}{\hbar} |T_{i \rightarrow f}|^2\, \delta(\varepsilon_{k'}-\varepsilon_k + \Delta E_{\mathrm{recoil}} + \Delta E_{\mathrm{ph}}).$$

Final states are grouped by the phonon occupation difference $\Delta n$.

After thermal averaging, the differential rates for key channels take the following forms:

Channel	Differential Rate $dW/d\Omega$	Distinguishing Feature
Elastic	$$\frac{m^2}{4\pi^2\hbar^4}|v(\mathbf{q})|^2 e^{-2W(\mathbf{q},T)} \delta(\varepsilon_{k'}-\varepsilon_k-\Delta E_{\rm recoil})$$	Zero-phonon events; Debye-Waller suppression
Quasielastic	$$\frac{m^2}{4\pi^2\hbar^4} |v(\mathbf{q})|^2 \sum_{\lambda} \frac{\hbar}{2M\omega_{\mathbf{q}\lambda}} [n(\omega_{\mathbf{q}\lambda})+1] \delta(\cdots)$$	Single-phonon emission, momentum mostly absorbed by zero mode
Superelastic	$$\frac{m^2}{4\pi^2\hbar^4} |v(\mathbf{q})|^2 \sum_{\lambda} \frac{\hbar}{2M\omega_{\mathbf{q}\lambda}} n(\omega_{\mathbf{q}\lambda}) \delta(\cdots)$$	Single-phonon absorption, momentum mostly absorbed by zero mode

All channels exhibit transport momentum transfer via the lattice center of mass, with only a fraction attributed to the phonon sector.

4. Physical Origin and Dominant Regimes

Conventional theories posit that phonon emission or absorption must supply the entire momentum exchange required for electronic scattering. However, the inclusion of the lattice zero mode reveals that the bulk of the momentum can be transferred via center-of-mass recoil, which costs negligible energy for a macroscopic lattice. The phonon bath therefore only needs to supply the minimum energy allowed kinematically, with the zero mode absorbing the remainder of momentum. Mixed quasielastic and superelastic channels arise whenever phonon emission or absorption is possible, even if the energy and momentum exchange in the phonon sector is small compared to the total transfer.

These channels dominate the electron momentum relaxation landscape under two primary conditions:

• The Debye-Waller factor $e^{-2W(\mathbf{q},T)}$ remains close to unity (so elastic and quasi-superelastic processes are not exponentially suppressed).
• Phonon occupation numbers $n(\omega) \sim O(1)$ ensure single-phonon events are probable.

These conditions are characteristic of clean metals at intermediate temperatures (e.g., for copper at 300 K, $f_{\mathrm{el}}(2k_F) \approx 0.95$) and in strange metals with large deformation potentials and low phonon energies (Heller et al., 31 Jan 2026).

5. Momentum Relaxation and Diffusive Transport

The electron momentum-relaxation rate $1/\tau_p$ relevant for transport emerges from summing all elastic, quasielastic, and superelastic contributions, weighted by $(1-\cos\theta)$: $$\frac{1}{\tau_p} = \int d\Omega\, (1-\cos\theta)\, \left[\frac{dW_{\mathrm{el}}}{d\Omega} + \frac{dW_{+}}{d\Omega} + \frac{dW_{-}}{d\Omega}\right].$$ Because purely elastic ($dW_{\mathrm{el}}/d\Omega$) and mixed-channel rates scale similarly with respect to $v(\mathbf{q})$, but only the former requires no phonon energy, elastic and mixed events provide the dominant contribution to $\tau_p^{-1}$. In contrast, energy relaxation times $\tau_E$ are determined by rare multi-phonon and higher-order processes and are significantly longer, $\tau_p \ll \tau_E$. This separation accounts for:

• Observed persistence of weak localization and quantum oscillations at finite temperature,
• Experimental evidence for momentum randomization without significant energy dissipation in hot-electron setups,
• Emergence of diffusive, Planckian-scale transport ($D \sim \hbar/m$) from essentially elastic dynamics in a time-dependent lattice background.

6. Significance for Experimental Phenomena

The identification of mixed quasielastic and superelastic channels resolves longstanding inconsistencies between the dominance of elastic scattering processes and the observed separation of momentum and energy relaxation times in clean crystals. Key experimental signatures explained by this mechanism include ultrasonic attenuation, robust quantum oscillations at finite temperatures, and rapid momentum dissipation without commensurate energy loss. The framework provides a unified microscopic basis for diffusive transport in clean and strange metallic phases where Planckian diffusion is observed, and where standard phonon-limited pictures fail to reconcile momentum- and energy-relaxation timescales (Heller et al., 31 Jan 2026).