= Solution
Phonon paths are limited by boundaries, defects, isotopes and phonon–phonon interactions; electrons scatter from impurities, defects, boundaries and lattice vibrations. At low temperature a clean sample can be boundary-limited, while residual electronic impurity scattering survives as phonons freeze out. In a quasiparticle relaxation-time approximation, independent resistive rates are sometimes added, $\tau^{-1}\simeq\sum_j\tau_j^{-1}$, but this is not an exact rule for every coupled process.
<Umklapp scattering> transfers crystal momentum by a reciprocal lattice vector and can relax a heat current. The <normal phonon collisions and heat-current relaxation> distinction is that a normal phonon collision conserves the total crystal momentum of the phonon distribution; it is not automatically a resistive thermal-scattering event in the same way. Normal collisions can redistribute populations or establish hydrodynamic flow, with boundaries and other momentum-relaxing processes then setting resistance. Likewise, different angular scattering kernels distinguish a single-particle lifetime from a transport lifetime. These qualifications explain why estimating ell requires physical scattering information, not simply a collision count.
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