tokens: Photoionization is assumed to be negligible (Kozma Fransson 1998 ). Ionization balance is achieved by equating the rate of impact ionization with the recombination rate for each ion. Within each ion, level populations are computed solving the rate equations for each level under the assumption of thermal balance, i.e. equating the non-thermal heating rate and the rate of cooling via line emission. The assumption is made that the nebula is optically thin, and radiation transport in not performed. The emissivity in the various lines is used to compute the emerging spectrum. The code can be run as a simple one-zone code, in which case the line profile is a parabola with characteristic width determined by the speed of expansion of the outer boundary of the nebula (which is input to the code). Alternatively, a stratified version can use a one-dimensional explosion model (i.e. a radially dependent density/abundance profile), in which case energy deposition, ionization balance, level populations, and emissivity are computed for each shell. Each shell then contributes its own emissivity. Line profiles are built by summing truncated parabolae bounded by each shell’s inner and outer boundary velocity. In both cases, the effects of line blending are automatically taken into account in the computation of the emitted spectrum. Our code has been used for a number of SNe Ia (e.g. Mazzali et al. 2011 ; Mazzali Hachinger 2012 ) and Ib/c (e.g. Mazzali et al. 2007b ), and it can yield a description of the inner layers of the SN ejecta. When used in combination with a description of the outer layers, which can be obtained through a study of the early spectra, the mapping of the inner ejecta that results from nebular studies can be used to obtain a full description of the density and abundance distribution in an SN (e.g. Mazzali et al. 2015 ; Ashall et al. 2016 , 2018 ). One aspect our code attempts to treat is clumping of the ejecta. The simple assumption is made that all ejecta reside in clumps, which occupy a fraction of the total volume, as defined by a filling factor. The density in the clumps is then increased following the decrease in volume, and the composition of the clumps is set to be the same as that of the radial ejecta shell they are in, i.e. all clumps are homogeneously and microscopically mixed. The clumps are still assumed to be in thermal equilibrium. These are of course major simplifications, but in the inner layers of an SN Ia only few elements are present, so it is not unreasonable. Strong clumping has been shown to be very useful in order to reproduce the low ionization seen in late-time spectra of SNe Ic (e.g. Mazzali et al. 2001b ), where it may be thought of as a proxy for the non-spherically symmetric distribution of the ejecta. Clumping is used later in this paper to improve our synthetic spectra, especially in the final epoch. 4 SPECTRAL MODELS AND THE EVOLVING PROPERTIES OF SN 2014J We started our analysis by deriving some basic
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