  We report a first-principle theoretical study of the adiabatic decoherence
undergone by a nuclear spin system in a solid, coupled to the phonon field
through the dipolar interaction. The calculations are performed for a chain of
weakly interacting 1/2-spin pairs, considered as an open quantum system in
contact with a bosonic heat bath. By incorporating to the whole system
Hamiltonian the fluctuations of the local dipolar energy produced by low
frequency phonons, and assuming that this low energy fluctuations are
adiabatic, we find that the spin dynamics can be described in closed form
through a spin-boson model. The obtained results show that the coupling with
the phonons destroy the spin coherence, and the efficiency of the process
significantly depends on the complexity of the involved spin states. By using
realistic values for the various parameters of the model, we conclude that this
mechanism can be particularly efficient to degrade multi-spin coherences, when
the number of `active' spins involved in a given coherence is high. In this
way, we show that the spin coherence in the adiabatic regime can be noticeably
affected by this mechanism.
