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Introduction

The quantity of interest in neutron scattering experiments with thermal neutrons is the dynamic structure factor, , which is closely related to the double differential cross-section [7], . The double differential cross section is defined as the number of neutrons which are scattered per unit time into the solid angle interval and into the energy interval [E,E+dE]. It is normalized to , dE, and the flux of the incoming neutrons,

| (4.130)

Here N is the number of atoms, and and are the wave numbers of scattered and incident neutrons, respectively. They are related to the corresponding neutron energies by and , where is the neutron mass. The arguments of the dynamic structure factor, q and , are the momentum and energy transfer in units of , respectively:

|  |  | (4.131)
|  |  | (4.132)

The modulus of the momentum transfer can be expressed in the scattering angle , the energy transfer, and the energy of the incident neutrons:

| (4.133)

The dynamic structure factor contains information about the structure and dynamics of the scattering system [67]. It can be written as

| (4.134)

is called the intermediate scattering function and is defined as

|  |  | (4.135)
|  |  | (4.136)

The operators in Eq. (4.135) are the position operators of the nuclei in the sample. The brackets denote a quantum thermal average and the time dependence of the position operators is defined by the Heisenberg picture. The quantities are the scattering lengths of the nuclei which depend on the isotope and the relative orientation of the spin of the neutron and the spin of the scattering nucleus. If the spins of the nuclei and the neutron are not prepared in a special orientation one can assume a random relative orientation and that spin and position of the nuclei are uncorrelated. The symbol appearing in denotes an average over isotopes and relative spin orientations of neutron and nucleus.

Usually one splits the intermediate scattering function and the dynamic structure factor into their coherent and incoherent parts which describe collective and single particle motions, respectively. Defining

|  |  | (4.137)
|  |  | (4.138)

the coherent and incoherent intermediate scattering functions can be cast in the form

|  |  | (4.139)
|  |  | (4.140)

Rewriting these formulas, nMOLDYN introduces the partial terms as:

|  |  | (4.141)
|  |  | (4.142)

where:

|  |  | (4.143)
|  |  | (4.144)

where , , , and are defined in Section 4.2.1.

The corresponding dynamic structure factors are obtained by performing the Fourier transformation defined in Eq. 4.134.

An important quantity describing structural properties of liquids is the static structure factor, which is defined as

| (4.145)

In the classical framework the intermediate scattering functions are interpreted as classical time correlation functions. The position operators are replaced by time-dependent vector functions and quantum thermal averages are replaced by classical ensemble averages. It is well known that this procedure leads to a loss of the universal detailed balance relation,

| (4.146)

and also to a loss of all odd moments

| (4.147)

The odd moments vanish since the classical dynamic structure factor is even in , assuming invariance of the scattering process with respect to reflections in space. The first moment is also universal. For an atomic liquid, containing only one sort of atoms, it reads

| (4.148)

where is the mass of the atoms. Formula (4.148) shows that the first moment is given by the average kinetic energy (in units of ) of a particle which receives a momentum transfer . Therefore is called the recoil moment. A number of `recipes' has been suggested to correct classical dynamic structure factors for detailed balance and to describe recoil effects in an approximate way. The most popular one has been suggested by Schofield [68]

| (4.149)

One can easily verify that the resulting dynamic structure factor fulfills the relation of detailed balance. Formally, the correction (4.149) is correct to first order in . Therefore it cannot be used for large q-values which correspond to large momentum transfers . This is actually true for all correction methods which have suggested so far. For more details we refer to Ref. [9].

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