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An Introduction to Source Imaging in Heavy-Ion Reactions |
If we want to understand the bulk properties of nuclear matter at high energy densities, then we must compress and heat nuclei. The simplest way to do this is to smash two of them together. In fact, if we do this with the right amount of force, we may be able to cause the nuclei to undergo various phase transitions, either through the liquid-gas phase transition (at temperatures around 8 MeV) or through the quark-gluon phase transition at much higher temperatures (around 170 MeV). However, when one smashes two nuclei together, one does not simply heat and compress the nuclear matter adiabatically. Rather, one induces shocks and various other forms of collective motion. The interplay of these non-equilibrium processes and any phase-transitions leads to a complicated space-time evolution of the colliding nuclear system. The complexity of the collisions requires us to use as many tools as possible to understand what is going on.
One especially useful tool is the two-particle correlation. Because of a combination of the Bose-Einstein (or Fermi-Dirac) statistics of particle pairs and the interactions between the pairs of particles, the correlations are sensitive to the space-time extent of the nuclear collisions. Schematically, this is illustrated here:
Here, the emission region emits two identical particles, with momenta
and
respectively. As the pair travels out to the detectors A and B, they become
correlated through a combination of quantum statistics and final state
interactions. The effect is called the HBT effect after R. Hanbury
Brown and Richard Twiss, who first used this effect to measure the radius of
the star Sirius.
To observe the HBT effect in an heavy-ion reaction,
one counts up the number of identical pairs of particles,
N2 (with a particular relative momentum
), and the number single
particles, N1, and makes the following ratio:
is the
correlation function.
In 1977, Steve Koonin proposed a simple formalism to explain the features in proton-proton correlations and was generalized by Scott Pratt and others to handle arbitrary types of identical-particle pairs. The resulting Koonin-Pratt equation is:
Here 
is the pair relative wave function which
describes the propagation of the pair from a relative separation of
to the detector with relative momentum
.
The source function 
is the key
as it is the probability of emitting a pair of particles a distance
apart. Hence the source function encodes the space-time information of the
emitting source.
There are several alternatives toward dealing with the Koonin-Pratt equation:
The trick to imaging is to not try to conjure up a model source to convolute with the wavefunction like CRAB, but rather to work backwords from the data itself to image the source directly. For this to work, we first convert the Koonin-Pratt equation into a matrix equation. We can not simply invert the matrix since we wish to also know the uncertainty on our reconstructed source. Instead, we must view the problem as a least-square inversion problem. The details of the formalism can be found in the references below.
Here is an example of the imaging approach at work. Below is plotted a simulated two-proton correlation function, such as one might measure in experiment (shown in black) along with the restored correlation (shown in red) obtained by imaging and then un-imaging the data.
Below, plotted on both linear and logrithmic scales, is the imaged source (shown in red) and the input source (shown in black) used to generate the simulated correlation.
The set of imaging codes that that generated these plots is called HBTprogs.
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May 30, 2002. For information about this page, contact
David Brown
UCRL-WEB-148721 |
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