User:Steakdeprived/Phonon bottleneck effect

The phonon bottleneck effect is a phenomenon where relaxation of paramagnetic spins by the direct process (described in the next paragraph) may be impeded when the energy released by the spins cannot escape quickly enough from the narrow bands of lattice modes.

Imagine a system of paramagnetic spins in a lattice, where the spins are allowed to either point "up" or "down". Each spin, upon flipping, can emit energy in the form of a phonon which may then propagate through the lattice. Assume the magnetic susceptibility of the system is measured by applying a magnetic field that oscillates at angular frequency $$\omega$$, the spin-spin coupling relaxation time is $$\tau_s$$, and the spin-lattice coupling relaxation time is $$\tau$$. If $$\omega$$ is small compared with $1/ &tau;_{s}$ then the spins will be in equilibrium with each other, and it is possible to define a "spin temperature" $$T_s$$. However, this spin temperature will not be the same as the temperature of the lattice if $$\omega$$ is comparable or larger than $1/ &tau;$, and we can now think of the spins as being in contact with a bath (the lattice).

The frequency can be larger, smaller or comparable to $1/ &tau;$:
 * $$\omega \ll \frac{1}{\tau}\longrightarrow$$ the flipping of spins would be isothermal, where spin and lattice are in equilibrium
 * $$\omega \gg \frac{1}{\tau}\longrightarrow$$ the flipping of spins would be adiabatic
 * $$\omega \approx \frac{1}{\tau} \longrightarrow$$ characterized by dispersion

In this third case, phonons (energy emitted from the flipping of spins) will only be allowed to propagate through the lattice at frequencies that match the modes of the lattice. This restriction creates a bottleneck through which spins are allowed to transfer energy to the bath, and places a limit on the magnetic susceptibility of the system.