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In the case of a particle-phonon coupling linear in
we write the Hamiltonian of the excitations of the lattice
plus the interaction between the lattice atoms and
the interstitial muonium (which, in the adiabatic approximation
is at the centre of the well, so this Hamiltonian depends on
displacements of the lattice atoms only)

where the sum is over all phonons present.
The coupling constant governs the strength of the interaction with the interstitial.
[54,53]
The effect of the polaron deformation and the origin of the
associated trapping potential become apparent in
calculating .We start by using a normal oscillator shift in which

| |
(5) |

to define
new operators ,

Substituting these operators into the Hamiltonian,
Eq. (5.9) becomes
(noting that

| |
(6) |

which simply corresponds to a new set of harmonic oscillators
with a shift in energy, but the one-phonon interaction has now
been eliminated.
The wave functions we need to construct are then

We are now able to write the matrix elements of
| |
(7) |

The *polaron effect* corresponds to the diagonal elements ,

| |
(8) |

which, with the number of phonons of frequency now a function of temperature
| |
(9) |

reduces to
(suppressing an overall constant)
| |
(10) |

It is usual in the literature to define the ``polaron exponent"
as the sum

| |
(11) |

so that
the polaron effect manifests itself as a renormalization
(reduction) of the raw tunnelling bandwidth
| |
(12) |

** Next:** 5.4 Barrier Fluctuations
** Up:** 5 Quantum Diffusion I:
** Previous:** 5.2 Renormalization of the