  Numerical simulations are important when assessing the many characteristics
of field emission related phenomena. In small simulation domains, the
electrostatic effect from the boundaries is known to influence the calculated
apex field enhancement factor (FEF) of the emitter, but no established
dependence has been reported at present. In this work, we report the dependence
of the lateral size, $L$, and the height, $H$, of the simulation domain on the
apex-FEF of a single conducting ellipsoidal emitter. Firstly, we analyze the
error, $\varepsilon$, in the calculation of the apex-FEF as a function of $H$
and $L$. Importantly, our results show that the effects of $H$ and $L$ on
$\varepsilon$ are scale invariant, allowing one to predict $\varepsilon$ for
ratios $L/h$ and $H/h$, where $h$ is the height of the emitter. Next, we
analyze the fractional change of the apex-FEF, $\delta$, from a single emitter,
$\gamma_1$, and a pair, $\gamma_2$. We show that small relative errors in
$\gamma_1$ (i.e., $\varepsilon\approx0.5\%$), due to the finite domain size,
are sufficient to alter the functional dependence $\delta(c)$, where $c$ is the
distance from the emitters in the pair. We show that $\delta(c)$ obeys a
recently proposed power law decay in the limit of infinite domain size
($\varepsilon=0$, say), in contrast to a long time established exponential
decay. Thus, power law functional dependence, $-\delta \sim c^{-n}$, with
$n=3$, is suggested to be a universal signature of the charge-blunting effect,
at sufficient large distances between similar emitters with any shape. These
results explain the origin of the discrepancies in the literature and improves
the scientific understanding of the field electron emission theory, for
accurate characterization of emitters in small clusters or arrays.

Mesoscale and Nanoscale Physics