Chapter 2 Experimental
2.2. Characterisation methods
2.2.2. Neutron scattering
Neutron scattering is a complementary technique to x-ray diffraction. In addition to the diffraction spectra it provides information about the diffuse components which makes it a suitable technique for the investigation of the structure of disordered, amorphous and liquid materials.98 The basic working principle of neutron scattering is very similar to that of X-ray diffraction and the Bragg’s law (Eq 2.3) can still be applied. A beam is sent to the sample and after it has been scattered of an angle 2θ it is detected (Figure 2.4). The difference between the two techniques is the source of radiation employed. In a neutron experiment, as the term says, neutrons are used in place of x-ray.
The advantage of using neutrons is that they are more penetrating, allowing for sample containers or special experimental set-ups, and are scattered by the nuclei rather than the electrons of the atoms.17 This remove the dependence from the atomic number typical of X-ray sources.17 Additionally, it is possible to distinguish between neighbouring atoms on the periodic table and isotopes, making neutron diffraction a more sensitive technique.
Figure 2.4 Schematic of a Neutron diffractometer. Basic schematic representation of a neutron diffractometer. Image modified from ref. 99.
2θ Incident beam Sample Scattered irradiation Detector
Two ways are in use to produce neutron beams: nuclear reactor and pulsed sources.17, 99 In the first case the neutrons are produced by nuclear fission, where a nuclei is split in two and releases at least two highly energetic neutrons which will be passed through a moderator (usually water) to slow them down. The second option is the spallation source, where the neutrons are produces by an accelerated beam of protons colliding with a heavy metal nucleus. This can be uranium, tantalum or tungsten.99 The generated neutrons are passed through a moderator and then concentrated by reflecting them into the target by mean of a beryllium reflector. The advantages of this second source is that it is safer and it can be pulsed. This allows to have the neutrons leaving the source all at the same time. Therefore, by measuring the time required to reach the detector (time of flight, TOF) it is possible to calculate the energy and the wavelength, λ of the neutrons. As a consequence, a pulsed spallation source does not require the use of a monochromator.
The two type of neutron source differ in terms of measured variables in the experiment. In the case of a nuclear reactor the wavelength is constant and the angles is varied, in the same way as an x-ray diffraction experiment. In the case of a spallation source the angle is fixed and the wavelength is the variable.17, 99 This is determined by measuring the TOF (1/v, μs·m-1) of the neutron and applying the de Broglie equation (Eq. 2.6):17, 99
mv h
λ Eq. 2.6
where h is the Plank constant (6.626×10-34 J·s), m the mass of the neutron (1.675×10-27 kg) and v is its speed.
In a typical neutron diffraction output the data are expressed in Q and not 2θ as in an x-ray diffraction experiment. Q represents the magnitude of the momentum transfer vector of
Figure 2.5 Physical meaning of Q. Representation of Q as a momentum transfer vector, the difference between the wave vector of the incident beam, ki, and the wave vector of the scattered
beam, ks.
ks
ki
Q
the neutron after it has been scattered by the sample. More simply, it represents the difference between the wave vectors of the beam before and after the collision (Figure 2.5). With trigonometric equations and given the wave vector as k = 2π/λ, its magnitude is given by:100
sin 4 Q Eq. 2.7
where λ is the wavelength of the incident beam and θ the scattering angle.
The measured values of Q are then used to calculate the total structure factor, F(Q), which mathematically is defined as in Eq. 2.8 which can be seen as a weighted sum of partial structure factors, Sαβ(Q).101
c c b b S Q 1 2 ) Q ( F Eq. 2.8This equation shows that the structure factor of a material depends on the concentration of the different atoms and their neutron scattering lengths, expressed as cα, cβ, and bα, bβ
respectively. To avoid counting the same kind of interaction twice the Kronecker function, δαβ, is employed.101
The application of a Fourier transform on the total structure factors, yields the total radial distribution function, g(r).99, 101 This is a measure of the number of atoms that are at a distance r from an atom of reference.99 This can be more usefully converted into a differential pair correlation function, D(r), where the correlation happening at longer distances, r, are weighted by their length scale (Eq. 2.9), graphically enhancing them without decreasing the correlations happening at shorter distances.
r 4 r
g
r 1
D Eq. 2.9
An example of outputs of g(r) and D(r) for melamine are illustrated in Figure 2.6. The intensities are plotted against the atomic distances, r (Å). The effect of weighting the g(r) by the length at which the interaction occurs, is evident and results in an increased intensity for the peaks at higher r. This does not compromise the features at lower distances. This is very useful to obtain information for interatomic distances which are longer than simple bond lengths.
There are however some difficulties in a neutron scattering measurement. Light atoms, especially hydrogen where the nucleus is close in size to the neutron, can cause gain or loss in the energy of the incident beam (ki ≠ ks). This is the inelastic scattering as opposed
to the elastic scattering described in Figure 2.5 where ki = ks.99 The measured data will
have to be corrected for the incoherent scattering by using complex equations.99, 102 This process is more accurate for heavy atoms, therefore, in the case of hydrogen or deuterium the correction will introduce some uncertainty in the g(r) and consequently the D(r).99, 102 The diffractometer used for the current investigation is the Near and InterMediate Range Order Diffractometer (NIMROD) at the ISIS facility of the Rutherford Appleton Laboratory.101 It is specifically designed to study amorphous materials and its lengths scale goes from below 1 Å to above 300 Å with a resolution of ~ 0.1 Å.101 It is specifically designed to minimise inelastic scattering, therefore it is particularly suitable for organic materials containing light elements (H, C, N).101
For a powder neutron scattering experiment few extra measurements other than the sample have to be taken. These are: the empty instrument as a background, the empty sample container and vanadium. The latter is used to calibrate the instrument and has to be scanned in the same beam conditions of the sample, therefore its neutron scattering data will be acquired each time. Vanadium is chosen because of its small coherent scattering length, which means weak Braggs diffraction, and due to its size will have low inelasticity effect which will be easy to correct for.99 The container can be made out of vanadium or TiZr alloy. The reason for choosing vanadium has already been explained. TiZr is chosen because the two metals scatter neutrons with opposite phase which will cancel out.99
Figure 2.6 Examples of g(r) and D(r) for melamine. Comparison between the total radial distribution function, g(r) (blue) and the differential pair correlation function, D(r), (black).
4 8 12 16 20 r / Å g( r) / a rb. unit D r / a rb. unit
In the experiments for this study the sample holder was a flat plate TiZr can with sample thickness of 1 and 2 mm (Figure 2.7). The beam area was 30 mm × 30 mm. Each sample was scanned five times to average the data. Data analysis was carried out using the software GudronN and modelling was performed with PDFGui.103