TIPOS DE ENTORNOS VIRTUALES DE APRENDIZAJE MÁS UTILIZADOS

Group theory is a powerful mathematical tool which allows for the study of not only mathematical problems but for the formulation of many complex problems in both physics and chemistry. This thesis is primarily concerned with the use of group theory and symmetry rules for their ability to classify crystal structures and the symmetry of defects within these structures. Many publications deal with the group theory in a range of degrees of mathematical rigour [169][170][171]. This section will not discuss group theory in detail but will instead focus on its application in determining the structures of defects in a diamond crystal.

In this thesis the Schoenflies notation will be used. The diamond lattice, comprised of two interlocking face-centred cubic cells, has Td crystal symmetry which has 24 symmetry operations. This means that in order to simulate an EPR spectrum for a defect with Td symmetry it is only necessary to simulate one of these sites as the other 23 are the same. A table of the possible defect symmetries which are possible for a defect in a Td crystal are listed in Table 3.2.

An EPR active defect which does not have Td symmetry will have an EPR spec- trum which is dependent upon the orientation ofBwith respect to the crystal axis. It is easiest to describe the effects on the EPR spectrum of a defect with symmetry lower than Td by using an example: in this instance the single substitutional ni-

340.0 345.0 350.0 355.0 Bk〈001〉 Bk〈110〉 Magnetic Field (mT) Bk〈111〉 〈001〉 〈111〉 〈110〉 344.0 346.0 348.0 350.0 352.0 0 20 40 60 80 100 Angle from [001] (Degrees) Magnetic Field (mT) (b) (a)

Figure 3.3: Figure (a) shows the EPR signals observed for the Ns0 defect

centre with the magnetic field along three different crystallographic directions. Figure (b) shows a roadmap of this centre rotated from [001] to [110] in a [1¯10] plane. The red lines have been used to denote where two lattice sites are

degenerate throughout the whole rotation.

in an anti-bonding orbital along one of the〈111〉bond directions which elongates the bond by 28% [172]. This change in bond length reduces the symmetry to C3v due to the three fold rotation axis which is now about the〈111〉direction between the nitrogen and the unique carbon atom. In this case the electron can lie along one of the four nitrogen-carbon bonds. In a system with multiple Ns0 defects

(and no additional factors such as in-situ heating or uniaxial stress) the electrons will be statistically distributed amongst the four possible bond directions. When simulating a single crystal spectrum of the defect it is necessary to consider the orientation which the magnetic field makes with the principal axis of the defect for these four possible orientations.

Ns0 has an electron spin of S = 1/2 and a nuclear spin of I = 1 for a 14N

isotope. A simulation of this centre for a single site would lead to S(2S+ 1)I(2I+ 1) = 3 allowed transitions as shown in Figure 3.3. When B is placed along the [001] crystallographic direction the magnetic field makes an equal angle with each of the principal axes. This results in each of the four sites having degenerate energies and so only three lines are observed. With the magnetic field along the [111] direction, B lies along one of the four possible orientations and makes an equal angle with each of the other three. This results in the outer peaks having relative intensity of one compared to the peaks from the other three sites which

sum to give peaks of relative intensity three.

With knowledge of the Hamiltonian parameters of the system it is possible to generate a simulation of all possible orientations where the magnetic field is plotted along the x-axis and the angle of the magnetic field from [001] is plotted along the y-axis. It is common, in diamond, to rotate about the [1¯10] direction as this means that the magnetic field will run from [001] through [111] to [110] allowing for three high symmetry directions to be observed. This is the case for the sample roadmap of Ns0 in Figure 3.3. No information is given about the intensity of the

lines observed in the plot displayed but it is possible to provide such information using a colour gradient on the plotted data. This is more commonly used for experimental data and such data will be presented in Chapter 5.

With an understanding of how the symmetry of a defect affects the EPR spec- trum it is then possible to determine the symmetry of an unknown defect from experimental results. This has been successful in a number of instances, however, caution must be exercised when performing this analysis. Some factors can distort the observed symmetry of the defect:

• Preferential Orientation: In which the defects in the sample are not randomly orientated. This leads to preferential population of some sites compared to others making the symmetry difficult to determine

• Rapid Reorientation: The rapid reorientation of the defect or certain species in the defect can lead to a different symmetry being observed in the EPR spectrum compared to the symmetry observed from stressed optical spectra. This will now be discussed in further detail.

3.5

Electronic configuration of the vacancy in di-