Properties of nuclear matter have inspired much of the theoretical work in many-body theory during the last decades. It is much richer than that of the pion gas, which it contains as a special case. The mass spectrum and its Laplace transform are used to obtain a thermodynamic description of the system.
For now, in.m/ is a function that represents our (imperfect) knowledge of the true mass spectrum.m/. 2/3.m/ : (23.4) The exact relationship between av,in and will not concern us here - indeed, if we take avDinD, we will find that we solve the equation. where T0 is the 'limit temperature' and where the values of a and T0 depend on the version of Eq. Solving the Bootstrap equation. If we were to use the IMS measure in Eq. 23.3), would be the density of states of the pion gas.
The baryon number (number of baryons minus the number of antibaryons) is preserved using the KroneckerıK.bP. The essential step now consists in the correct extraction of the mass spectrum .p2;b/from the function Q. In Figure 23-1, we briefly summarize the relationships between the functions resulting from the application of L and L transformations.
Quite apart from physical questions, we must demand a mathematical solution of the momentum equation beyond this boundary line.
Thermodynamics
Thus, with the methods of complex analysis, we will be able to study new phases in the future. Here, the four-volume Vex is an arbitrary external parameter3 [an arbitrary-volume field Vex D .VV/1=2 with an arbitrary four-velocity], whereas earlier, in Section 23.2, we took the volume to be a dynamically determined correct accompanying volume of the particle.ˇ has now the meaning of the inverse temperature four-vector, where (Lorentz invariant)T D .ˇˇ/1=2 is the temperature at rest of the thermometer frame. Z1.0/ is by construction a function of the invariants ˇ2, Vex2 and ˇVex. This means that if the mass spectrum of the interaction is known, replacing the interacting particles with an ideal phase of infinite components and weighting the different components with respect to the mass spectrum results in the same distortion of phase space as the interaction would.
Now assuming that the bootstrap statistical model has given us the correct spectrum, the corresponding statistical thermodynamics of strongly interacting particles follow from the ideal gas formulas given in Section 23.3, now generalized to include the mass spectrum. We can now proceed in the same way as in Eq. 23:39),n is now the average number of fireballs present. This result (23:43), which is exact in the framework of this model, shows once again how simple things become once the interaction is hidden in the mass spectrum.
While the small variation of T0 m from version to version is of no physical significance, the nature of the system when T !T0 depends critically on the poweraofmin Eq. Since we are interested in the behavior at T !T0(ˇ!ˇ0), we denote all quantities that are constant in this limit by the symbol C (at each place where it occurs, C may have a different value and/or dimension ). Then the one-cluster partition function Z1;b.ˇ;V;b/ is given by Eq. 23.41), the only change is the dependence of the mass spectrum on the baryonic number b.
When n such clusters are present, but each with the same b, we find for the cluster function the usual expression (23.37). When groups with different scales are present, then we need to calculate the product of the different contributions. To obtain the partition function of an arbitrary number of clusters that have baryons together, we must in Eq. 23.48) clusters the sum over all possible numbers, since every such configuration is possible.
All values are allowed and stringfnj0g depends only on the fact that string members exist. Note that the existence of Z1.ˇ;V; /, the grand canonical partition function with one cluster, is not guaranteed. To proceed, we need to make an assumption about the b dependence of the mass of the cluster Mb.
While we can summarize the general formula (23.56), we will be interested here in the properties of bulk nuclear matter: that is, the case when a given number of nucleons. The bootstrap equation is then as before, ie, (23.9), but the input term describing only 'raw' pions and nucleons takes the form.
Properties of Nuclear Matter in the Bootstrap Model The Different Phases
We note that the behavior of the chemical potential for T ¤T0 is similar even when the pion term is turned off completely (dashed line in Fig. 23.2). However, it is possible to consider the analytic continuation of the grand canonical function in this domain - the inverse L-transform can then be used to find the canonical quantities. However, we have found other versions of the nuclear bootstrap model that allow a transition even to this region—but we will not discuss this possibility here.
This is partly due to the fact that Eq. Also, at 'D'0 the correct analytical behavior is obtained from Eq. 23.61) for G.'/and its first and second derivatives. Independent verification of our calculations was done using the Yellin extension (23.13) wherever possible. We will choose as the unit of V the 'elementary' volume of one baryon, VN D mNA, as represented in Eq. 23.22), together with the constant A (which is unrelated to the atomic and/or baryon number, denoted here by asb).
The first term is the only one remaining in the absence of pions and is shown as a dotted line in Figure 23-3. It is believed that the unexpected shape of the critical curve is due to the coexistence of pions and nucleons. We see that the onset of the pion component decreases the phase transition density, but at high temperatures the density increases again sharply.6.
In Figure 23.4 we show the baryon density in the gas phase: in Figure 23.4a as a function of the chemical potential with temperature as the parameter (isotherms), in Figure 23.4b as a function of temperature, with the chemical potential as the parameter. In Fig. 23.4c, d we have eliminated the chemical potential from Fig. 23.4a and replaced it with the pressure [see Eq. 6This reflects the behavior of the rapidly changing factor expŒ.m/=T; hadronic matter at the phase boundary is meson-dominated for T > m=2MeV.
Since the rest mass is included in the energy per nucleon volume, the lower part of the diagram remains empty. 23.66) and (23.62) are functions of and T, and we can numerically calculate any of these physical parameters in Eq. We record the almost linear behavior (in the gas phase) of the energy density: "C1CC2, with temperature dependent constantsC1;C2.
For higher temperatures, as we can see in Fig.23.6b, this is the lower limit of the thermal and interaction energyEbnr. The first term expresses the pion-nucleon interaction component and, as discussed in Section.23.4, is small at temperatures below 60 MeV.
Summary
In this regard, we recall that the volume of fireballs now increases with the mass of the fireball - so the average density must be bounded by T. In a stable model, we expect a finite energy density at T0, so the currently forbidden region beyondT0 will now become accessible. Looking ahead, we hope to extend our model by making the input more detailed, preserving the particle-antiparticle symmetry, and taking into account the special importance of alpha clusters.
It seems that a deep study of the 'liquid' phase will be rewarding since much of the structure of the liquid (perhaps even the existence of a new 'solid' phase) depends on how much nucleon structure we include in the terms of entry. An obvious first step in this direction is the possible introduction of effective masses ( Open Access This book is distributed under the terms of the Creative Commons Attribution Non-commercial License, which permits any non-commercial use, distribution, and reproduction in any medium, provided that the original author(s) and sources are acknowledged.
