Problem Set #3

Ground Rules for Problem Sets

Problem Set 3

Articulation Questions

  1. The phase diagram of hadronic matter on the ``Nuclear Science'' wall chart (there is a link from the web page) has temperature and mass density (kg/m$^3$) as the variables. Please comment.
  2. What is the difference between a current quark and a constituent quark?
  3. What keeps a bag in the bag model in equilibrium?
  4. At a phase boundary between hadron matter and a quark-gluon plasma the chemical potentials of the phases are equal. Do the baryon densities have to be equal?

Checkpoint Questions

  1. Convert the estimates of the bag pressure $B$ given in the text to GeV/fm$^3$. Is the number reasonable?
  2. Find the energies of the lowest two s-states in the bag model, using the value of $B$ in the text. Please comment.
  3. Plot the running coupling in QCD as a function of $Q^2$. [Note: You'll need to look up a reference value; try the Particle Data Group web page.] For approximately what range of $Q^2$ is QCD perturbative?
  4. The pions are often assumed massless in order to do the integral for the pion energy density analytically. Make a graph showing how good an approximation this is as a function of $m_\pi/T$. How does this affect the estimate for the transition temperature (at zero baryon density)? How much does the massless pion estimate change if the Boltzmann occupancy weight is used instead of the Bose-Einstein one?

Project: Model of the Hadronic Matter Phase Diagram

Here you will model the hadronic and quark-gluon plasma phases of ``hadronic matter'' and study the thermodynamics of the phase transition between them. You will use a separate model for each phase and use thermodynamics to decide which phase is favored. You will need to evaluate the thermodynamic quantities numerically.

Model for quark-gluon plasma phase:

Model for hadronic phase:

  1. At $T=0$, plot the ``saturation curves'' for each phase. This is a plot of the energy per baryon as a function of the baryon density (or the Fermi momentum $k_{\scriptscriptstyle F}$). (Note that at $T=0$ you can analytically eliminate the chemical potential $\mu$ in favor of the baryon density.) At what baryon density and binding energy per nucleon is the hadronic phase in equilibrium? How does this relate to the binding energy and density of ordinary nuclei? At what baryon density and binding energy per nucleon is the quark-gluon phase in equilibrium?
  2. Plot pressure vs.\ temperature for each phase at zero baryon chemical potential. What do you learn from the plot?
  3. Generate phase equilibrium points for the two-phase system, and plot the phase boundary on a temperature--chemical potential plot. [Hint: What are the conditions for phase equilibrium?] Make an analogous plot but with temperature and baryon density. Mark on your plots where the interior of lead lies under ordinary conditions. Comment on your plots.

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Copyright © 1997,1998 Richard Furnstahl and James Steele.