List of abbreviations/Symbols
2.2 Scattering Techniques
2.2.3 Neutron Scattering
For performing neutron scattering experiments, one needs to produce the neutrons. In general, there are two best techniques are known for neutron production which are (i) nuclear fission reaction and (ii) spallation source. Fig. 2.3 presents a schematic presentation of the nuclear fission reaction where single neutron gives rise to three another neutrons. This reaction will continue like a chain reaction if the produced neutron involves in the further fission. In case of neutron production through spallation source, a high intensity proton beam having energy of
~1 GeV strikes on a heavy metal target of W or Ta. Different particles like neutron, proton etc.
are emitted during the returning of excited nucleus (happens due to interaction between high energy proton and target nucleus) to ground state. If these emitted neutrons again interact with the other nuclei, a neutron shower takes place similar to the case of fission reaction. Compare to fission reaction, spallation source produces more high energy neutrons; therefore the energy spectra of produced neutrons from these methods are also different.
As neutron is a neutral (chargeless) particle, it penetrates deep into materials with short range interaction. In contrast to x-ray atomic form factor which strongly depends on atomic
number (Z), neutron atomic scattering factor is not following any trend. Another important thing is the neutron scattering length which is different for different isotopes of the same element which is essential for different isotopes study.
Figure 2.4 Comparison of the nuclear and magnetic form factors in neutron diffraction along with X-ray form factor plotted against scattering angle.
In comparison to X-rays which is generally used for investigation of chemical structure, neutron is an excellent probe to investigate both chemical and magnetic structures of the solids.
As neutron has a magnetic moment which interacts via a long range dipole-dipole interaction with the atomic magnetic moments of sample, one can investigate the magnetic structure of the sample. Fig. 2.4 shows a comparative plot between the nuclear as well as magnetic form factors in neutron diffraction and X-ray form factor plotted against scattering angle. Unlikely, the nuclear scattering form factor, magnetic form factor f(Q) has Q dependence and decreases rapidly with Q, therefore magnetic scattering can be observed up to the low-Q region. In addition, one can also investigate the energy dynamics of sample as thermal neutrons have the energy of the order of few meV which matches with the excitation energy of the solids.
In the present thesis, NPD has been used to study the crystal and magnetic structures of the prepared oxide compounds. Before going further, one should understand the concept of the scattering cross-section as it is the main quantity which is measured in the diffraction
experiment. In the NPD experiment, one measures the differential scattering cross-section, which is defined as [118,119],
=
(2.2)
Where, is the incident flux and dΩ is the solid angle in the θ, φ direction.
Figure 2.5 Schematic diagram presents the neutron nuclear scattering during neutron scattering experiment.
2.2.3.1 Elastic Nuclear Scattering
As we know that neutron has both nuclear as well as magnetic scattering from the sample, we first discuss the scattering of the neutrons with a fix nucleus (nuclear scattering) in the present section. In case of neutron scattering experiments, neutron is supposed as a plane wave (i = eikz where k is the wave vector = 2/). In elastic neutron scattering, |ki |= |kf|, i.e. E = 0 and Q (scattering vector) has to be equal to a reciprocal lattice vector (G) of the sample. As compare to neutron wavelength (), the range of nuclear force (~ 1 fm) is very small which gives rise to a point like scattering. This elastic nuclear scattering is isotropic in nature as shown in Fig. 2.5.
In case of nuclear scattering, both coherent and incoherent scattering arises from a nucleus. Coherent scattering is the one which is of interest during crystal structure determination of a material. Along with coherent scattering, incident neutron can also scatter incoherently and such scattering can be seen as background in the data.
The total nuclear elastic differential cross-section is given by,
( )
( )
(2.3) For the crystal structures, having more than one atom per unit cell, the coherent scattering cross- section is given by,
( )
〈 〉 ∑ | |
(2.4) Here N is the total number of unit cells, and V is the volume of the unit cell. The δ-function in coherent scattering cross-section implies that coherent scattering cross-section is finite when Q = G, otherwise it is zero. We know that reciprocal lattice vector |G| = 2π/dhkl and scattering vector |Q| = 4π sinθ/λ, then the condition Q = G yields, Bragg’s law.
With nuclear structure factor,
∑ (2.5) For the elastic neutron scattering at finite temperature, the Debye-waller factor of the atom as a function of scattering vector and temperature must be included. It takes into account the small thermal fluctuation of the atom.
2.2.3.2 Magnetic Neutron Scattering
Neutron possesses a magnetic moment of -1.913 ( is nuclear Bohrmagneton) which is responsible for the magnetic scattering in the material. This magnetic moment of a neutron can interact with the existing magnetic moment of the unpaired electron in the sample via dipole- dipole interaction. The main difference between nuclear and magnetic cross-sections is that magnetic moment arising from the electron cloud of unpaired electrons has a large special distribution of its scattering potential whereas nuclear scattering taking place when the neutron is
very close to it and has a small range of scattering potential. A necessary condition for magnetic scattering to occur is that an incident neutron only scatters from a magnetic moment of a material when the present moment is perpendicular to the scattering vector Q.
The differential cross-section for elastic magnetic scattering is given as, ( )
∑ | | (2.6) Where NMis the number of magnetic unit cells, VM the volume of the magnetic cell and GM is the reciprocal lattice vector in the same magnetic unit cell.
Figure 2.6 Schematic representation of different components of magnetic moments along with the scattering vector during magnetic scattering.
Next,
̂ ̂ (2.7) Here, Fm is known as the magnetic structure factor and is defined as:
∑ (2.8) Where γ = 1.9132 is the gyromagnetic ratio, r0 = 2.8×10−15 m (classical radius of the electron), and µj the magnetic moment of the jth ion. Here f(Q) magnetic form factor which is given by:
∫ (2.9) Where M(r)is the magnetization density. This f(Q) arises because of the finite spatial extent of electron cloud of unpaired electrons in the atom.
Detailed mathematical derivations and equations used in above sections are given in many books on neutron scattering [118–123].
Figure 2.7 Block diagram of Neutron diffractometer at UGC-DAE Consortium Scientific Research, BARC, Mumbai Centre, Trombay, India.