Theory for program

This is an extension of the theory published by Klaus W. Becker, Peter Fulde and
Joachim Keller in Z. Physik B 28,9-18, 1977
"Line width of crystal-field excitations in metallic rare-earth systems"
and an introduction to the computer program for the calculation of the neutron
scattering cross section. The computer program *bfk* is written by J. Keller,
University of Regensburg.

Here we present a brief outline of the theoretical concepts to calculate the dynamical susceptibility of the Re ions and the scattering cross section.

The neutron-scattering cross section is related to the dynamic susceptibility
of the RE ions

whose Fourier-Laplace transform

determines the inelastic neutron scattering crossection (Stephen W. Lovesey; "Theory of neutron scattering from condensed matter" Vol 2, equ. 11,144).

Here and denote the wave number of the neutron before and after the scattering. is the scattering wave vector, . cm is the basic scattering length, is the Landé factor, the atomic form factor of the rare earth ion.

Formal evaluation of the dynamic and static susceptiblity.

The dynamic spin-susceptibilities are correlation functions of the form

where is a Heisenberg operator

Introducing a Liouville operator (acting on operators of dynamical variables) by the Heisenberg operator can also be written formally as

With help of this definition the dynamical susceptibility of two variables can be written as

and their Laplace transform

With help of the Liouvillian these quantities can be written as

and their Laplace transform

The static isothermal susceptibilities can also formally be calculated with help of
the Liouvillian.

The static susceptibilities are used to define a scalar product between the
dynamical variables:

It fulfills the axioms of a scalar product and furthermore it has the important property

With help of this relation the dynamical susceptibility can be expressed as

and finally as

The second term is the so-called relaxation function

The model:

We calculate the spin susceptibility of a RE ion in the presence of exchange
interaction with conduction electrons. The system is described by the
Hamiltonian

The first part is the cf-Hamiltonian of the spin-system:

written in terms of the crystal field eigenstates . The second part is the Hamiltonian of the conduction electrons

The third part is the interaction between local moments and the conduction electrons

We assume, that the energies and the eigenstates expressed by angular momentum eigenstates are known.

Definition of dynamical variables

In our case we use as dynamical variable the standard-basis operators

describing a transition between CEF levels and . In the absence of the interaction with conduction electrons

In order to get the spin suceptibility we have to multiply the final expressions by the spin-matrixelements:

The idea of the projection formalism to calculate the dynamical
susceptibility of a variable is to project this variable onto a closed
set of dynamical variables and to solve approximately the coupled
equations between these variables. For this purpose a projector is defined
by

where is the -component of the inverse matrix of .

For the resolvent operator of the relaxation function

one obtains the exact equation

with the memory function

where . In components

with

and the memory function

Now we apply the formalism to the coupled spin-electron system and restrict
ourselves to the lowest order contributions of the spin electron
interaction. As dynamical variables we choose a decomposition of the
original spin-variable:

where denotes a transition performed with the standard-basis operator .

In lowest (zeroth) order in the el-cf interaction

and the scalar product is diagonal in lowest order in the transition operators,

where is the thermal occupation number. For the frequency term we then get

Neglecting the second-order energy corrections in the following we obtain the equation for the relaxation function

and it remains to calculate the memoryfunction containing the relaxation processes.

In lowest order in the electron-spin interaction
can be replaced by
.
Then we get for the memory function

with

Now

with

With help of the symmetry properties

with

we obtain

In order to calculate the relaxation functions
we use the general relation between relaxation function and dynamic
susceptibility

and calculate instead the corresponding susceptibility (using tr ):

We are interested in the imaginary part describing the relaxation processes:

Writing and we obtain

For the integrals we get

This makes

which has to be used to calculate the imaginary part of the memory function. Writing

which also be written in symmetrized form as

we obtain with

from which we get the memory function matrix in the space of dynamical variables

Summary:
For the neutron scattering cross section we need the function
, where
is the frequency dependent part of the dynamic
susceptibility
for spin components ,, which is
related to the corresponding relaxation function
by

For the full dynamical susceptibility we need the static suseptibility which in lowest order in the exchange interaction is given by

The above relaxation function is calculated with help of the Mori-Zwanzig projection formalism by

where denotes a transition from to between crystal field levels of the magnetic ion. The partial relaxation functions are obtained by solving the matrix equation

with

where is the energy difference of cf-levels.

Only terms in lowest order in the el-ion interaction are kept. We neglect frequency shifts due to the electron-ion interaction. Then the memory function is purely imaginary (with a negative sign).

Note that compared to our paper BFK, Z.Physik B28, 9-18, 1977 we have used here a different sign-convention.

For numerical reasons it is more convenient to calculate the relaxation
function in the following way:

with

From the relaxation function we get for the dynamic scattering cross section

with

Here the scattering function depends only on the scattering vector and the energy loss Note that in our formulas contains a factor and is the energy loss. If we want to have meV as energy unit and Kelvin as temperature unit, we have to write .

For the analysis of polarised neutron scattering the different
spin-components
of are needed.
These are defined by

with

The complex dynamic susceptbility is calculated from

where the static susceptibilities are diagonal in our approximation.

Martin Rotter 2017-01-10