Calculating the initial energy density in heavy ion collisions by including the finite nuclear thickness

The initial energy density produced in heavy ion collisions can be estimated with the Bjorken formula after choosing a proper formation time $\ensuremath{\tau}{}_{\mathrm{F}}$. However, breaks down at low energies because it neglects finite nuclear thickness. Here we include both duration and longitudinal extension of production. When $\ensuremath{\tau}{}_{\mathrm{F}}$ is not too much smaller than crossing two nuclei, our results are similar to those from previous study that only considers duration. In particular, find has lower maximum value but evolves longer formula, while large-enough and/or high-enough result approaches formula. We also qualitative difference ${\ensuremath{\epsilon}}^{\mathrm{max}}$ $\ensuremath{\tau}{}_{\mathrm{F}}=0$ finite, diverges as $1/\ensuremath{\tau}{}_{\mathrm{F}}$ $ln(1/\ensuremath{\tau}{}_{\mathrm{F}})$ high energies. Furthermore, solution approximately satisfies scaling relation. As result, dependence determines $A$ dependence, weaker means slower increase $A$.