1.5 Scope of the thesis
2.1.1 Neutron cross section
The basic quantity to be measured in the scattering process is the partial (or double) differ-ential cross-section, which gives the number of incident neutrons with energy Ei scattered into a solid angle dΩ, with an energy in the interval[E0, E0+dE0]. This quantity is denoted by
d2σ
dΩdE0. (2.2)
The complete derivation of double differential scattering cross section can be simplified as-suming the scattering from single bound nuclei. As mentioned before, the neutron is a weak probe and it does not interact strongly with solids (except for strong absorbing materials).
Thus, the total scattering amplitude of a sample containingN scatterers is the sum of single scattering amplitudes. The expression is given first for the elastic case, then will be gener-alized for the inelastic one.
Denoting the initial state of the incident neutron as ψkand the final state of the scattered neutron as ψk0, the differential cross-section is calculated starting from the probability of a transition between the state |ki and |k0i, with the same energy E. The probability follows the Fermi’s Golden rule,
The potential Vˆ describes the interaction that allows a transition from a state |ki to a state |k0i, and ρk0(E) is the density of final states. The final expression for the scattering cross section becomes:
In order to get the partial differential cross-section, we must consider inelastic events. In case of inelastic scattering, the neutron energy change is a quanta of ~ω, given or taken by the sample in the scattering process. Assuming the target in a state|αi, the state describing the incident neutron and the initial state of target is the product of the two ket vectors, namely|k, αi.The target energy is denoted by the eigenvectorEα. In case there is an energy transfer between target and neutron, the conservation of energy in this process leads to
~ω=Eα−Eα0, (2.5)
and the cross section reads
dσ
The partial differential cross section is obtained from the conservation of energy equation as written above: The previous equation relates the scattering process from a state |αi to a final state |α0i.
Depending on the target, only certain states can be accessible; therefore it is necessary to include a weight pα in the double differential cross-section:
dσ
Figure 2.3: Scattering of a incident plane wave into a target (green sphere). The outgoing wave is spherical (s-wave).
where the average is performed on all possible and accessible states. The expression for the differential cross section2.8is obtained in the first Born approximation (or first perturbation series), under the hypothesis that the interaction neutron-target is weak. In all previous expressions for differential and partial differential cross-sections the unknown variable is the potential Vˆ. The interaction between neutrons and nuclei is in the order 10−15m, at least four orders of magnitude smaller than the typical wavelength for slow neutrons(∼10−10m), and therefore the neutron-nuclei scattering is isotropic and contains only plane waves. The outgoing wave is characterized by a parameterb, calledscattering length. This quantity can be complex and depends on the energy of incident neutron, on the isotope and on the relative orientation between the incident neutron spin and the nuclei spin.
If the scattering potential Vˆ is a δ-function (i.e. a point-like interaction), the outgoing wave is spherical. Fermi’s pseudo-potential satisfies the requirement. Assuming a rigid array of N nuclei fixed in positions denoted by Rl and scattering length bl, the expression of Vˆ is
Vˆ(r) = 2π~2 m
X
l
blδ(r−Rl). (2.9)
The lengthbl can be interpreted as a radius of an impenetrable sphere centred at the nuclei l and it has dimensions of a [length]. Given this expression for the Fermi’s potential, it is possible, in eq. 2.8, to evaluate:
hk0|Vˆ |ki=X
l
blexp(iQ·Rl) (2.10)
and then the eq. 2.6 becomes:
dσ
In the last equation, the scattering length (that in general is a complex number) has to be averaged all over random nuclear spin orientations and random isotope distributions, and it is generally dependent on the position Rl. Assuming that there is no correlation between different sitesl and l0, then
b∗l0bl = In this way the contribution in the average has been separated in two terms, one coming from different pair of atoms, and one from the same atoms. Therefore, the scattering cross-section can be written as
dσ is the coherentcross-section, and
dσ
The first one, the coherent cross-section, is dependent on the interference between waves scattered from different lattice sites and it is weighted with an average scattering length. The incoherent cross-section, instead, is lattice independent and it is weighted with a mean-square scattering length deviation. In a scattering experiment, both contributions are detected: the coherent cross-section is angle (or momentum transfer) dependent, whereas the incoherent one does not depend on Q and therefore leads to an isotropic signal.
The scattering cross-sections are given usually in units of area, 1 barn =10−24cm2, and they vary randomly from element to element, and from isotope to isotope. The most extreme case is the scattering cross-section of hydrogen: the most abundant isotope, 1H, has a coherent cross section of 1.8 barn, whereas the incoherent cross-section is 81.7 barn. Deuterium, instead, has σi= 2.05 barn and σc = 5.59 barn. A list of all scattering lengths and cross sections can be found in literature [108]. The next section will deal with the formalism of inelastic neutron scattering and how it is related with positions of particles in space and time.