Calculation of the Correlation Function of Physical Observables

Physical observables are related to operators such as the magnetic moment components or the scattering operators. We now introduce a series of observables ( ) for each subsystem and define a corresponding correlation function and dynamical susceptibility. The correlation function we denote as

Note that the observables may have some dependence (for example if we consider the Fourier transform of a magnetisation density of a magnetic ion in a crystal [42]). We omit this dependence to make notation easier but will come back to it when considering the calculation of the neutron scattering cross section.

Using the fluctuation-dissipation theorem can be related to a dynamical susceptibility :

However, in contrast to the dynamical susceptibility , based on the operators , the susceptibility cannot be obtained from solving the MF-RPA equation (191). The reason is that the derivation of (191) makes use of the dynamical evolution of the operators and not of the observables .

Nonetheless, general properties of dynamical susceptibilities may be used to get a relation between and . In general a dynamical susceptibility describes the response of a physical observable to a perturbation of the system described by an operator . In the case of the dynamical susceptibility corresponding to the observables we therefore have to set with the definitions

(195) | |||

(196) |

From linear response theory it can be shown [1, page 143], that the dynamical susceptibilities have poles at the excitation energies of the system: In equation (181) the denominator, the eigenstates and the difference in thermal population are the same for any susceptibility. The energy eigenstates of the system will be a linear combination of direct products involving single ion states. Therefore, the numerator in equation (181) will be a (usually not known) linear combination of products of the form and for and , respectively.

These terms can be related by a similar procedure to that outlined in equations (195) ff. We define the matrices,

similar to the of equation (195), however, omitting the thermal population factors and expectation values. These matricescan be diagonalised using the unitary transformations ( ) and ( ), respectively.

Again, all eigenvalues are zero except for and , respectively.

In [61] these transformations are used to derive the following expression for the dynamical susceptibility

Equation (212) shows how knowledge of the dynamical susceptibility calculated on the basis of the interaction operators between subsystems ( ) may be used to obtain the dynamical susceptibility for any set of observables of the system.

The standard procedure to avoid divergences is to substitute with and take the limit for . Using Dirac's formula

the absorptive part of the dynamical susceptibility (205) becomes

and the correlation function can be evaluated by applying the fluctuation dissipation theorem (204):

To keep notation simple, the dependence of the eigenvectors and the energies has been omitted. If the observable depends explicitly on , then also and thus will depend on . This very fundamental result will be applied to the neutron scattering cross section in section M.

The elastic contribution to equations (214) and (215) has to be evaluated taking into account a small but finite value for the energy shift introduced in the discussion of equation (192). It turns out, that and and thus also the dynamical matrix and its eigenvalues are proportional to . Making use of the normalisation for the eigenvectors we find that the dynamical susceptibility (214) is proportional to and thus zero in the limit of . However, in the correlation function (215) the denominator is proportional to leading to a finite result for the quasielastic response.

Martin Rotter 2017-01-10