## 4 Ruddlesden-Popper Layered Oxides  
Ruddlesden-Popper series oxides have been considered for intermediate temperature oxide fuel cells and in particular cathodes for a number of years [114; 115; 116; 117]. They can be described by the formula A\({}_{n+1}\)B\({}_{n}\)O\({}_{3n+1}\). The first members (n = 1) of the series include the widely studied La\({}_{2}\)NiO\({}_{4}\). La\({}_{2}\)NiO\({}_{4}\) is tetragonal (space group I4/mmm) [117] at a wide temperature range (423-1073 K) including the temperature that are important of Intermediate temperature solid oxide fuel cells (IT-SOFC). For completeness, La\({}_{2}\)NiO\({}_{4}\) is orthorhombic (Cmca) at room temperature [118]. As a side line it should be mentioned La\({}_{2}\)NiO\({}_{4}\) is in the K\({}_{2}\)NiF\({}_{4}\) structure that is widely known as it is the structure of La\({}_{2-x}\)Ba\({}_{x}\)CuO\({}_{4}\) the archetypal material for high temperature superconductivity [119].  
The intricacy in these materials is that oxygen concentration can have a significant impact on the oxygen diffusion properties. This is because self-diffusion is dominated by oxygen and oxygen vacancies (in oxygen deficient or hypostoichiometric oxides) or oxygen interstitials (in oxygen excess or hyperstoichiometric oxides) act as the vehicles for diffusion. More formally and in Kroger-Vink notation [120] this can be described by:  
\[D_{O}=D_{V}\big{[}V^{\prime\prime}_{O}\big{]}+D_{I}\big{[}O^{\prime\prime}_{i }\big{]} \tag{5}\]  
where \(D_{0}\) is the oxygen diffusion coefficient, \(D_{V}\) is the vacancy diffusion coefficient, \(D_{i}\) is the interstitial diffusion coefficient, \(\big{[}V^{\prime\prime}_{O}\big{]}\) is the oxygen vacancy concentration and \(\big{[}O^{\prime\prime}_{i}\big{]}\) is the oxygen interstitial concentration.  
Considering La\({}_{2}\)NiO\({}_{4}\) is oxygen hypostoichiometric, the excess oxygen is typically related to the experimental conditions of the samples [70; 121; 122]. For numerous years the experimentally determined activation energies of diffusion in lanthanum mikelate were in a wide range 0.19-0.90 eV [40; 115; 122; 123; 124; 125], whereas there was no agreement on the diffusion mechanism. Classical molecular dynamics work [124] calculated an activation energy of diffusion of 0.51 eV in excellent agreement with the time of flight secondary ion mass spectrometry (ToF-SIMS) study of [PERSON] et al. [122] (0.54 eV). Additionally, the MD study proposed the interstitialcy in the a-b plane diffusion mechanism for oxygen self-diffusion (refer to Figure 2) [125]. An analogous interstitialcy mechanism in the a-b plane was also proposed in the experimental (neutron scattering experiments with analysis based on the maximum entropy method) work by [PERSON] et al. [74] in the related tetragonal (Pr\({}_{0.9}\)La\({}_{0.1}\))\({}_{2}\)(Ni\({}_{0.74}\)Cu\({}_{0.21}\)Ga\({}_{0.05}\))O\({}_{4+\delta}\).  
It should be noted that research on the Ruddlesden-Popper series oxides continues to attract the interest of the research community and although self-diffusion is well characterised theoretically and experimentally, the focus is on proton diffusivity, doping and structural/electronic/magnetic properties [126; 127; 128; 129; 130; 131; 132; 133; 134; 135].