Thermodynamic Properties

Since heavy-ion collisions produce a very high multiplicity of particles, a statistical treatment of the system can be adopted. If thermal equilibrium is reached, the system is characterized by thermodynamic observables, such as volume, lifetime, temperature and energy density.

The expected space-time evolution of a heavy-ion collision - with and without QGP formation - is shown in fig. 1.4.

Figure 1.4: A collision between two heavy nuclei takes place at $(z,t) = (0,0)$ where $z$ is the space coordinate along the beam axis and $t$ is the time coordinate. The figures show the evolution of the system with QGP formation (left), and without QGP formation (right). The hyperbolae are constant proper-time curves, where the proper-time $\tau =\sqrt {t^2-z^2}$ ($c$=1). In the hydrodynamic model described in [3] the energy density, entropy density and temperature are constant on each curve. The time-scale is on the order of a few fm/c.

=9cm images/spacetimeA.eps

=9cm images/spacetimeB.eps

Identical-particle interferometry, also referred to as Hanbury-Brown-Twiss (HBT) correlations [8], e.g. $\pi\pi$ or KK correlations, can be used to study the space-time dynamics of nuclear collisions. From such two-particle correlations it is possible to obtain information on the transverse and longitudinal size, on the lifetime and on flow patterns of the source at the freeze-out time. For instance, different particle species may freeze out at different times and give different source sizes, due to the expansion of the source.

With a hydrodynamic description of the colliding system, the energy density $\epsilon$ can be estimated for central collisions from the transverse energy per unit rapidity as

$$\epsilon = \frac{1}{V}\frac{dE_T}{dy} = \frac{1}{\tau_0\pi R^2}\frac{dE_T}{dy}$$ (9)

Here $R$ is the radius of the participant zone and $\tau_0$ is the proper formation time. Usually a value of $\tau_0$ = 1 fm/c is used.

The particle emission from a source in thermal equilibrium is Maxwell-Boltzmann distributed according to [9]

$$\frac{dN}{m_Tdm_T} \propto m_TK_1\left(\frac{m_T}{T}\right)$$ (10)

where $K_1$ is the modified Bessel function

$$K_1(m_T/T) = \int \cosh(y)e^{-m_T\cosh(y)/T}dy$$ (11)

With the assumption that $m_T \gg T$ eq. 1.10 can be approximated as

$$\frac{dN}{m_Tdm_T} \propto \sqrt{m_T}e^{-m_T/T}$$ (12)

The temperature would then be given by the inverse slope of a semilogarithmic plot of $\frac{dN}{m_T^{3/2}dm_T}$ vs. $m_T$.