Mott-Littleton method

Figure 2.9 Mott-Littleton method of dividing a crystal matrix for the purpose of calculating defect energies. Region I contains the defect and surrounding atoms, which are treated explicitly. Region lib is treated as a continuum reaching to infinity. Region Ha is a transition region interposed between regions I and lib.

The location of extra framework cations is a major problem in characterising zeolites. Simulation is becoming an increasingly powerful tool for the exploration and rationalisation of cation positions, since it not only allows atomic level models to be compared to bulk experimental behaviour, but can also make predictions about the behaviour of systems not readily accessible to experimental probing. In the first part of this work we use the Mott-Littleton method in conjunction with empirical potential energy functions to predict and explore the locations of calcium cations in chabazite. Subsequently, we have used periodic non-local density functional calculations to validate these results for some cases. 

Defect modelling with the Mott-Littleton method... 

The Mott-Littleton method has been employed to model Ca-Chabazite in the limit as the... 

A computer code, CHAOS (Duffy and Tasker, 1983), has been written to calculate the energy of defects at interfaces. It uses the structure of the relaxed interface as a starting point and then applies the Mott-Littleton method discussed above. The presence of the interface complicates the calculation. Details are given in the reference above the essential difference is that part of the polarization calculation must be done as a sum over planes. 

The Mott-Littleton method is now a routine tool in computational solid-state chemistry and physics, and is implemented, together with other static lattice modelling tools, in the GULP code [24], written by Gale. Two recent applications serve to illustrate the range and diversity of current applications. 

Acknowledgements. I am grateful to many colleagues for collaboration and discussion relating to the Mott-Littleton method, but perhaps most notably to A.B. Lidiard, A.M. Stoneham, M.J. Norgett, J.H. Harding, R.W. Grimes, M.S. Islam, R.A. Jackson, P.W.M. Jacobs, J. Corish and A.V. Chadwick. 

Sometimes, the system of interest is not the inhnite crystal, but an anomaly in the crystal, such as an extra atom adsorbed in the crystal. In this case, the inhnite symmetry of the crystal is not rigorously correct. The most widely used means for modeling defects is the Mott-Littleton defect method. It is a means for performing an energy minimization in a localized region of the lattice. The method incorporates a continuum description of the polarization for the remainder of the crystal. 

In these calculations, the isolated defect or defect cluster is embedded in the crystal, which extends to infinity, and the contrast between this approach and that used in the supercell methods is illustrated diagrammatically in Figure 1. The normal procedure in a Mott-Littleton calculation is to relax all the atoms in a region of crystal surrounding the defect, containing typically 100-300 atoms, until all are at zero force. Newton-Raphson minimization methods are generally used. The relaxation of the remainder of the crystal is then described by more approximate methods in which the polarization, P at a point r, is calculated for crystals that have dielectric isotropy, from the expression ... 

Most calculations of the internal energy of a defect process (hereafter referred to simply as the energy) use classical potential models. Two kinds of approach have been used to calculate defect processes first, that based on the Mott Littleton approximation and second, the supercell method (closely related to the lattice minimization methods discussed in Chapter 3). 

The most basic data that the Mott-Littleton and supercell methods provide are the energies and entropies of defect formation. Nevertheless, despite the fact that these techniques are essentially static approaches it can also be possible to deduce information on the dynamic processes of diffusion and conductivity. These two processes are related by the Nemst-Einstein relationship ... 

For ceramic materials, defects within the lattice are inextricably linked with transport properties. The diffusion of a cation in a ceramic, for example, involves the formation of vacancy or interstitial states within the crystal, and the migration of these species leads to a net transport of material through the lattice. These processes may be modeled by means of ion pair potentials in conjunction with the Mott-Littleton defect approach, direct molecular dynamics techniques, 24 or Monte Carlo methods to describe overall transport on the basis of calculated individual process statistics. 

The future of the field will unquestionably be in this kind of predictive and design application. We can also anticipate rapid growth in the use of embedded-cluster techniques in which, as described earlier, the core region surrounding the defect is treated by a high-level quantum mechanical method. With these and other developments, methods building on the approach established in Mott and Littleton s remarkable paper are likely to continue to play a productive role in simulating the complex solid-state chemistry of defective compounds. 

The force equilibrium group of methods are all drawn from a famous publication from Mott and Littleton [MOT 38]. This femily is therefore known as the ML method, which, over time, has undergone certain modifications which tend to increase the precision of the calculations and yield the following methods Point Polarizable Ion (PPI) model. Modified Point Polarizable Ion (MPI) model and Extended Polarizable Point Ion (EPPI) model. For example, Mott and Littleton only took account of the electrical forces and the forces of repulsion. The other forces have been added in order to improve the results. 

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