Dan Valente's pictureDan Valente

Hope College

Dr. Catherine Mader

Supported by the NSF-REU


The natural world is probably the most complex entity known to man. By modeling physical systems we can obtain a simplified explanation of the way nature works. We make models so that we can understand and predict the behavior of the world around us. Models are especially useful in nuclear systems, where the scale is so small that it is difficult to directly observe and measure everything about the system. These models help us to understand interactions such as the nuclear force. The most common way to study the behavior of nuclei is to model nuclear collisions. Looking at the particles that are involved in the collision and their products can provide much information about nuclear interactions.

This summer, I used the Boltzmann-Uehling-Uhlenbeck (BUU) transport model for nuclear collisions to study the 139La + 139La system at both 400 MeV/A and 1200 MeV/A. BUU is a computer program which simulates a nuclear collision by solving a form of Newton’s Second Law for each particle involved in the collision, tracking the trajectories of the particles as a function of time. The model allows "billiard ball" type collisions and accounts for the potentials in the interactions, the Pauli exclusion principle, and other quantum mechanical effects.

When two nuclei collide, an extremely dense region of nuclear matter is created around the area of the impact. The nucleons in this region are very energetic, colliding frequently with each other. If conditions in this “hot” region are acceptable, two colliding nucleons can create a D particle—an excited form of a nucleon. Like anything in an excited state, the D particle quickly decays back into a ground-state nucleon by emitting a pion. A pion is a type of particle known as a meson, consisting of a quark/anti-quark pair. There are three types of pions that can come from a decay—p+, p- , p0. After the nuclear collision there are usually two residual nuclei and a stream of other particles which follow trajectories depending on the parameters of the collision. Low energy collisions will tend to be pulled around each other by their nuclear interactions; high energy nuclei tend to scatter much like billiard balls colliding.

My focus this summer was on the flow of the pions created in the La + La collision. Flow is a description of where particles go after the collision, and can be pictured as the direction that billiard balls would tend to go after scattering. If particles have a positive velocity and a negative transverse momenturm, they are said to exhibit flow with the axes we have chosen (positive x is in the vertical direction). Particles are said to exhibit anti-flow if they have a postive velocity and a positive transverse momentum. In the billiard ball scattering of high energy collisions, nucleons always exhibit flow. Flow and anti-flow both arise from the nuclear interactions that act on the particles in the collision. To see flow, the average transverse momentum of the particles is plotted vs. their rapidity. Rapidity is a relativistic generalization of velocity, used mainly because it transforms easily. In can be inferred from the definition above for flow that these plots have a characteristic S shape. With the axes that BUU has chosen, plots that I made show flow if the are negatively sloped at the origin and show anti-flow if the are positively sloped.

This summer, I was concerned with several questions of pion flow. Firstly, we wanted to know if pions flow with the nucleons, opposite the nucleons, or not at all. In the literature there is some experimental evidence that at high energies (1150 MeV) pion flow is dependent upon impact parameter and that both positive and negative pions exhibit anti-flow in peripheral collisions. Knowing this, we also wanted to know what causes pion flow and, most importantly, what factors affect pion flow. To test this we tried two different beam energies (400 MeV, 1200 MeV), two different impact parameters (b=2 fm, b=7 fm), and two different equations of state (stiff, soft). We also examined what affects the Coulomb interaction had on the flow of the pions since pions are charged particles.

For the 400 MeV data, it was found that pion flow does change with impact parameter. Central collisions always exhibit flow, where peripheral collisions exhibit anti-flow. To understand this I looked at the average x momentum of forward moving particles as a function of time. I found that at both impact parameters, pions emitted at early times in the collision exhibited flow, but as the collision progressed the peripheral collisions show a transition to anti-flow, with a peak in anti-flow around 35 fm/c. Both positive and negative pions exhibited the same behavior, leading us to conclude that this could be a geometrical effect. When two nuclei collide in a central collision, the hot zone occurs somewhere in the middle of the large area of nuclear matter that is formed. The pions created here have a difficult time escaping this large "blob". Therefore, they tend to get reabsorbed by the nucleons and exhibit flow. But in peripheral collisions, the pions aren’t caught up in the center of this area—they are created in a narrow neck region where the collision took place. As the residual nuclei pass each other, the pions are forced in the direction of flow because the nuclei block the anti-flow path. When the nuclei pass each other, the nuclei block any flow movement and force anti-flow.

When we move to 1200 MeV/A, we see the same differences between central and peripheral as we did in 400 MeV/A. But when we look at flow as a function of time, we see that at early times there is no evidence of flow like there was in the 400 MeV/A run. This is because this collision happens much faster than the 400 MeV/A collision, so there is not enough time for the pions to be emitted when the geometry forces flow. Therefore 1200 MeV pions exhibit strong anti-flow.

In both energy cases it was also found that the Coulomb interaction has a small affect on flow. The geometry of the system influences whether the interaction adds or subtracts from the flow. For positive pions in central collisions the Coulomb repulsion adds to flow, since these pions are emitted with momentum already in the direction of flow. The interaction gives them a push in the right direction. In peripheral collisions they are now pushed further in the direction of anti-flow. For negative pions at central collisions the Coulomb interaction has the affect of adding to the flow, since these pions are attracted to the nucleons. At peripheral collisions this attraction subtracts from the anti-flow. In all cases the Coulomb interaction never drastically changed flow, in only enhanced it.

In the future I will be looking at how our results change when we look at a different equation of state which is dependent on the momentum of each particle involved in the collision.


Publications and Presentations:
“Pre-equilibrium Particle Emission in 35 and 60 MeV/A Kr + Au,Th Reactions.” B. Hughey*, G. Peaslee, C. Mader, P. DeYoung, K. Drake*, M. Kern*, D. Valente*, G. Westfall, A. VanderMolen, O. Bjarki, and D. Magestro, Paper AS.038. Division of the Nuclear Physics Fall Meeting. Williamsburg, VA, October 2000.
“Excitation Function Studies for Fission Cross Sections for the Reactions 11C+197Au, 194Pt, 196Pt, and 198Pt.” G.F. Peaslee, P.A. DeYoung, D.P. Valente*, L.W. Phair, T.S. Fan, G.J. Wozniak, M.A. McMahan, D. Wutte, J. Powell, R. Joosten, and the BEARS Collaboration. Paper P14.006. Spring Meeting of the American Physical Society. Long Beach, CA, April 2000.
“Pre-Equilibrium Particle Emission in 35 MeV/A Kr+Au, Th Reactions.” D. Valente*, P.A. DeYoung, K.L. Drake*, M.M Kern*, C.M. Mader, and G.F. Peaslee. Poster WC44.84. Centennial meeting of the American Physical Society. Atlanta, GA March 1999.