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

中文翻译：

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通过包含有限的核厚度来计算重离子碰撞中的初始能量密度

选择适当的形成时间后，可以用Bjorken能量密度公式估算在重离子碰撞中产生的初始能量密度 $\tau {}_{\mathrm{F}}$。但是，Bjorken公式在低能量时会分解，因为它忽略了有限的核厚度。这里我们包括初始能量产生的有限持续时间和有限纵向延伸。什么时候$\tau {}_{\mathrm{F}}$比两个核的穿越时间小很多，我们的结果与先前的研究相似，后者仅考虑了有限的持续时间。特别是，我们发现，在低能量时，初始能量密度的最大值要低得多，但其演化要比Bjorken公式长得多，而在足够大的情况下$\tau {}_{\mathrm{F}}$和/或足够高的能量，我们的结果接近Bjorken公式。我们在最大能量密度上也发现了质的区别${\epsilon }^{\mathrm{最大限度}}$ 在 $\tau {}_{\mathrm{F}}=0$ 是有限的，而Bjorken公式发散为 $\mathrm{1个}/\tau {}_{\mathrm{F}}$ 和以前的结果分歧为 $ln（\mathrm{1个}/\tau {}_{\mathrm{F}}）$ 在低能量但 $\mathrm{1个}/\tau {}_{\mathrm{F}}$在高能量下。此外，我们对能量密度的解决方案近似满足比例关系。结果，$\tau {}_{\mathrm{F}}$ 的依赖 ${\epsilon }^{\mathrm{最大限度}}$ 确定 $\mathrm{一种}$ 依赖和弱者 $\tau {}_{\mathrm{F}}$ 的依赖 ${\epsilon }^{\mathrm{最大限度}}$ 在低能量的结果中，意味着 ${\epsilon }^{\mathrm{最大限度}}$ 和 $\mathrm{一种}$。