Including Phonons and Crystal-Field Phonon interactions

In this section we discuss, how lattice dynamics may be considered in the framework of
the Hamiltonian (178). We will see that this corresponds
to a system of coupled Einstein oscillators. One such oscillator can be modelled
by setting up a *sipf* file with the module *phonon*. Coupling has
to be done in *mcdisp.j*. Rephrasing lattice dynamics in this way allows
to couple phonons to the crystal field.

A three dimensional Einstein oscillator (for atom ) in a solid can be described by the following Hamiltonian

Here is the dimensionless displacement vector ( , with the Bohr radius Å), the mass of the atom , the conjugate momentum to and the Matrix describing the restoring force.

Coupling such oscillators leads to the Hamiltonian

(66) |

Note that our coupling constants , where are the second-order derivatives of the potenatial energy as defined e.g. in [26, page 99].

In a mean field type of theory the phonon single ion module has thus to solve the Hamiltonian

Here the force corresponds to the exchange field and to the general operator and to of equation (178), respectively. The single ion Hamiltonian (70) can be solved by transforming it to normal coordinates (main axis of the Einstein oscillator) using the transformation matrix , which diagonalises :

(68) |

Due to the action of the force the equilibrium position of the oscillator
is
(it is the task of the function
*Icalc* to return this equilibrium position), the energies correspond to the three elements
of the diagonal matrix
, i.e.
,
,
. In order to run *mcdisp* we
have to calculate the transition matrix elements:

The single ion susceptibility for such a transition, e.g. - corresponds to

(70) | |||

(71) |

Because the different components of commute and the Hamiltonian (72) is separable, for the transition only the terms with in the nominator contribute:

(72) | |||

(73) |

In order to derive the last result we had to express in terms of ladder operators and apply , and , . This shows that the single ion susceptibility of our atom can be written as a sum of three effective transitions (with temperature independent susceptibility)

(74) |

Thus the module *phonon* has to provide in it's function *du1calc* these three
transitions (=number of transitions).