Principles of Neutron Diffraction

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In general all the nuclei of atoms consist of protons and neutrons. The proton being positively charged, the neutron having zero net charge. The neutron though does have a magnetic moment and a half spin component both of which can be exploited to be used in experiments with their own merit. Neutrons are beneficial as unlike X-Rays they do not permanently damage inorganic matter and most samples, apart from being hot for a period of time later are reusable. When using neutrons in diffraction experiments, it is taken into account that neutrons also possess wave-like properties in addition to their particle like nature; this is described by using the De Broglie relationship given in equation 4.2.3.1.

λ

p

mv

Where λ is the De Broglie wavelength of the neutron, h is the Planck constant, p is the momentum, v is the velocity of the neutron and m is the particle rest mass which is given as 1.67495 x 10-27 kg for the neutron. Also here we define the wavevector kwhich is defined as

/λ. It is possible therefore to change the perceived wavelength of the neutron by altering its velocity, for example to achieve a de Broglie wavelength of 1 Å, then a neutron velocity of 4000 ms-1 is required or we can formulate an equation for the required energy, E, of the scattering experiment as shown in equation 4.2.3.2.

( mλ ) m

Where (h-bar) . On the basis of the amount of energy they possess, neutrons are classified (in increasing energy) as ultra-cold, very cold, cold, thermal, epithermal and fast. In the case of thermal neutrons as are used in the diffraction experiments carried out in this thesis, the energy per neutron is ~0.0322 eV (32meV), which equated to a mean velocity of 2482 ms-1 or a de Broglie wavelength of ~1.594Å, this corresponded to a neutron temperature of 290 K which was calculated by the temperature of the moderator used for the experiment. [4.2.3.1]

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The neutrons start off as fast neutrons from the core of the reactor but are then moderated by a series of materials, for the D2B machine, this is D2O (Heavy water), H2O (light water) and

then graphite respectively.

Neutrons are a useful tool in the probing of materials as they can interact in different ways with the target lattice. A target nucleus gives rise to a nuclear scattering event, but also there can be interaction via a dipole-dipole interaction which is considered a magnetic scattering event (although this usually requires polarised or spin-flipped neutrons). In this work the diffraction studies undertaken just take into account what is termed Bragg scattering, which is not dissimilar to a Young’s slit experiment but utilising a much smaller wavelength of photon (or wave-like properties of a particle). In order for constructive interference to be achieved, the interfering beams must be an integer number of wavelengths different in order to be parallel, coherent and collimated and the scattering is considered elastic.

The Bragg-Law for a scattering event from a neutron colliding with a nucleus is not dissimilar to that of X-Ray diffraction (XRD) or transmission electron diffraction from a crystal and as such the Bragg equation is given in equation 4.2.3.2 and a simple schematic of this can be found in figure 4.2.3.1.

nλ d sin θ

Here λ is the wavelength (or de Broglie wavelength) d is the interplanar spacing and θ is the angle of incidence the incoming wave makes with the lattice alternatively known as the grazing angle, n is a constant relating to an integer value of 1, 2, 3 etc which is the order of the diffraction. There are other forms of scattering such as small-angle, wide-angle, inelastic and diffuse but with this particular experimental work the elastic scattering was of main importance. The Bragg diffraction angle will give all orders of refraction up to the finite limit which is when nλ d.

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Figure 4.2.3.1 - Schematic of the Bragg Diffraction Diagram [4.2.3.2]

The uniform repetitive pattern of a crystal lattice allows for certain values of the relevant miller indices termed [hkl], in 3d-space to either be constructive or destructive which gives the relevant diffraction pattern, but the limits of this must be ascertained.

If there is an NaCl crystal with the [111] reflection peak, then the [222], [333] and [444] will also be allowed all having different angles of θ; yet the value of d must be the same for all of these reflections as the spacing is a constant and the order value n increasing. The concept implicated here is that not all lattice planes will produce a measurable diffraction and that as higher numbered planes are reached, it is much harder to represent these in normal 3-d space, even if using vectors removes some dimensional effects, for example a plane with the index [257] would be extremely difficult to represent due to the density of vectors representing the other planes within a diagram. Here, the reciprocal lattice (also called reciprocal space) is used alongside a concept known as the Ewald Sphere designed by P.P. Ewald [4.2.3.3] and makes use of the reciprocal of dhkl to allow a 3-dimensional geometrical construction of the

data set and serves as a good visual representation of understanding the diffraction results. Ewald suggested that by using a sphere of radius /λ and instead of drawing the dhkl vectors in

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real-space; that plotting the reciprocal lattice of these points would greatly help; as this would allow for the relevant indices to be plotted as a simple lattice, this equation is given in 4.2.3.3.