Collision Characteristics

The spectator-participant model [7] of a heavy-ion collision is illustrated in fig. 1.3. The participating nucleons create a volume of high temperature and density, while the spectators move undisturbed through the collision. The impact parameter $b$ determines the centrality of the collision.

Figure 1.3: A collision between two heavy nuclei in the spectator-participant model.
a) The two Lorentz contracted nuclei before the collision. The centrality is determined by the impact parameter $b$.
b) After the collision a participant region with high temperature and density is created.

The impact parameter is however not directly measurable in the collisions. To determine the collision geometry, measurements of quantities which are strongly correlated to the number of participants are used, such as the transverse and forward energy and the number of produced particles. The transverse energy $E_{T}$ is defined as

$$E_T = c^2\sum_{i=1}^N(m_T)_i$$ (1)

where $i$ runs over all $N$ particles and the transverse mass $m_{T}$ is given by

$$m_T = \sqrt{m^2+(p_T/c)^2}$$ (2)

where $p_T$ is the momentum component perpendicular to the beam direction. In practice, $E_T$ is measured with a segmented calorimeter, and calculated as the sum of the energy $E_i$ at polar angle $\theta_i$ in each segment:

$$E_T = \sum_{i=1}^N E_i\cdot\sin\theta_i$$ (3)

Instead of the velocity $v$ of a particle, it is often more convenient to use a quantity called rapidity, defined by

$$y = \mathrm{arctanh} \mathit{\left(\frac{v}{c}\right)}$$ (4)

The rapidity is additive also in the relativistic case. In heavy-ion collision terminology the rapidity is measured along the beam direction. An equivalent definition is then given by

$$y = \frac{1}{2}\ln\frac{E+p_zc}{E-p_zc}$$ (5)

where $E$ is the energy of the particle and $p_z$ its momentum along the beam direction. A frequently used approximation to the rapidity is the pseudorapidity,

$$\eta = -\ln\left(\tan\frac{\theta}{2}\right)$$ (6)

Here $\theta$ is the polar emission angle, i.e. the angle between the particle momentum $\vec{p}$ and the beam axis. Expressed in terms of momentum, the pseudorapidity is

$$\eta = \frac{1}{2} \ln\left(\frac{\vert\vec{p}\vert+p_z}{\vert\vec{p}\vert-p_z}\right)$$ (7)

and the exact relation between the rapidity and the pseudorapidity is

$$\sinh \eta = \frac{m_T \cdot c}{p_T}\sinh y$$ (8)

The pseudorapidity approximation works well at large emission angles, i.e. large transverse momenta ( $p_T \gg m\cdot c$).