
CLUSTERING EFFECTS IN THE EQUATION OF STATE ON NUCLEAR MATTER A Thesis by JARED LALMANSINGH Submitted to the Office of Graduate Studies of Texas A&M UniversityCommerce in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE May 2015 CLUSTERING EFFECTS IN THE EQUATION OF STATE ON NUCLEAR MATTER A Thesis by JARED LALMANSINGH Approved by: Advisor: William Newton Committee: Carlos Bertulani Kent Montgomery Head of Department: Matt. A. Wood Dean of the College: Brent Donham Dean of Graduate Studies: Arlene Horne iii ABSTRACT CLUSTERING EFFECTS IN THE EQUATION OF STATE ON NUCLEAR MATTER Jared Lalmansingh, MS Texas A&M UniversityCommerce, 2015 Advisor: William Newton, PhD In Nuclear AstroPhysics, clustering refers to the process by which atomic elements, known as nuclei, are formed from preexisting nucleons such as protons and neutrons. This process called nucleosynthesis, is thought to occur under the following scenarios: Big Bang nucleosynthesis, Stellar nucleosynthesis, explosive nucleosynthesis and nucleosynthesis derived from fragmentation processes in nuclear reactions. In the case of all except the latter, which deals with radioactive decay and fission (the dissociation of larger nuclei into smaller ones), nucleosynthesis is predominantly the result of fusion, which is the creation of larger nuclei from smaller ones. To date, our current understanding of nuclei formation has crucial limitations, which is due to the complexity of stellarnucleosynthesis and related processes. To overcome this gap in our understanding, of particular note and interest is explosive nucleosynthesis which involves the rprocess, rpprocess, iv sprocess and pprocess which are theorized to be the means by which elements more massive than iron are formed. Furthermore, the processes governing the different types of nucleosynthesis are additionally delineated by the relative rate at which nuclei formation occurs  which ranges from very rapidly in the order of seconds, to extremely long in the order of centuries, depending on the element and process in question. In attempting to reach an understanding of these processes, an additional caveat appears in that most of these processes are theorized to occur near the end of life of star  i.e. in a supernova event which is thought to supply both the high energy, temperatures and pressure with which to form these heavy nuclei. Consequently, neutron stars, which are highly dense stellar remnants of corecollapse supernova events, are potential candidates whose environments and existence are theorized to provide the ideal physical characteristics for the aforementioned processes hence explaining the formation of nuclei heavier than iron in neutron star mergers events. Although similar has work done on clustering in determining the physics, structure and evolution of neutron stars and their associated equations of state, it is not well known how such considerations derived from statistical physics affects heavy nuclei formation. Subsequently, one means of understanding these processes is to approach everything from a rigorous thermodynamical treatment. However, as nuclear v species can vary up in relative occupied volumes, we must account for the thermodynamic potential variance due to the occupied nuclear volume for interacting species in our treatment. And, as that variance could be potentially large, this fact violates the core tenet of the ideal gas law as we can no longer treat the particles as point particles but rather as particles with definite and noninfinitesimal volumes  which means that we can assert that the particles affect the occupied volume of the space in which they interact and occupy. Consequently, any treatment and inclusion of the ideal gas law and any derived thermodynamical potentials cannot be used or applied as the physics of those treatments are inapplicable in this situation. What this means is that for us to perform any useful calculation, we must account for the occupied particle species volumes in any thermodynamical potential  something that has to be derived and verified from scratch with many possible alternative methods. This idea of volume isolation or more accurately, exclusion, is called excludedvolume mechanism (excludedvolume can be thought to be the occupied volume minus the total volume occupied by all the particles), an approach that is fairly common in BioPhysics, but one that is not so common in nuclear astrophysics, with particular attention to how such an approach affects the formation of nuclei and the associated equations of state. To determine the effects, if any, that excluded volume may have on the equation of state, we begin by applying excludedvolume mechanism on classi vi cal thermodynamics and associated thermodynamical potentials by modifying the standard methodology of considering the nonrelativistic energetics of noninteracting particles from a purely statistical mechanical approach, known as a MaxwellBoltzmann statistics. Having considered excludedvolume statistically, we can derive thermodynamical potentials of interest such the reduced Gibb’s Free Energy, which can be used to formulate a rudimentary equation of state. But, as this approach doesn’t consider readily apparent physics that dominate neutron stars such as quantum mechanics, we have to extend this treatment even further by such an inclusion (via energy degeneracy and the inclusion of bound and occupied states of the particles), in addition to angular momentum and relativistic effects to create a more realistic physical model which can be applied and used in determining the associated equations of state and how nuclei formation is affected as a result. Summarily, as the thermodynamics determines the interparticle distance due to arguments based on particle energy, degeneracy, chemical potential, etc., it is our hope that such a treatment will not only prove to be definitive, but useful in constraining our understanding of these processes having utilized the relatively unexplored mechanism of excludedvolume, in its determination of the equation of state and how that affects nuclei formation in neutron stars. vii This work has been done in collaboration with the nuclear astrophysics group of Catania, Sicily, under the leadership of Prof. Spitaleri and with Dr. Stefan Typel from GSI, Darmstadt, Germany. viii ACKNOWLEDGEMENTS This work is dedicated to my mom, and my siblings for their unyielding support, care, patience and dedication to me and my work in pursuit of this goal. Were it not for them, this made this achievement would not have been possible. I would especially like to thank my thesis adviser, Dr. Carlos Bertulani for all his selfless help, care and unwavering support throughout my matriculation here at TAMUC, in addition to his introducing me to such an interesting problem. Thank you for everything that you’ve done  I couldn’t ask for a better adviser. Furthermore, I’d also like to thank the members of my thesis Committee: Dr. William Newton and Dr. Kent Montgomery for all their feedback and support in this endeavor, were it not for their suggestions and input, this document would have been incomplete. Of special note on that matter, I’d like to gratuitously thank Dr. Stefan Typel for his guidance through this process and greatly assisting me in the understanding of this material  it is immeasurably appreciated. And, on a separate note, I’m also especially grateful to the support and belief of Dr. Matt A. Wood, Dr. Mark Leising, and my undergraduate adviser Dr. Donald K. Walter  whose combined collected efforts have resulted in my reaching this point. I am greatly indebted to you all. ix Lastly, I wish to express my sincere gratitude, thanks and appreciation to everyone whose actions whether directly or indirectly has made this thesis a reality. I thank you all. x TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv CHAPTER 1. STATISTICS AND THERMODYNAMICS OF PARTICLES WITH FINITE SIZE . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1 Classical Excluded Volume Mechanism . . . . . . . . . . 2 Thermodynamic Potentials in Reduced Volume . . . . . . 11 Density Dependent Degeneracy Factors . . . . . . . . . . 18 2. EQUATION OF STATE WITH EXCLUDED VOLUME . . . 26 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 26 Classical Particles . . . . . . . . . . . . . . . . . . . . . . 27 Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . 29 Classical Gas with Excluded Volume . . . . . . . . 30 Series Expansion . . . . . . . . . . . . . . . . . . 32 Lowdensity Limit . . . . . . . . . . . . . . . . . . . . . . 32 Virial Expansion . . . . . . . . . . . . . . . . . . . 33 Classical Mechanics . . . . . . . . . . . . . . . . . 34 Quantum Mechanical Effects . . . . . . . . . . . . . . . . 35 xi CHAPTER 3. NUMERICAL STUDIES . . . . . . . . . . . . . . . . . . . . . 40 Single species with finite volume . . . . . . . . . . . . . . 40 Virial coefficients with excluded volume . . . . . . . . . . 41 Virial Equation of State . . . . . . . . . . . . . . . . . . 49 Mixture of Nucleons and Deuterons . . . . . . . . . . . . 55 Point Particles (Ri = 0 (i = n, p, d)) . . . . . . . 57 Finite Nonzero Radii Particles (Ri > 0 (i = n, p, d)) 61 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 66 4. APPLICATIONS TO BIG BANG NUCLEOSYNTHESIS . . . 67 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Appendix A. CALCULATION OF bnn . . . . . . . . . . . . . . . . . . 70 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 xii LIST OF TABLES TABLE 1. Particle masses and degeneracies. . . . . . . . . . . . . . . . . . 56 2. Particle Pairs Volumes . . . . . . . . . . . . . . . . . . . . . . . 61 xiii LIST OF FIGURES FIGURE 1. Neutronneutron scattering phaseshifts ( degrees) vs z (kRnn) within the limit [0; 20]. . . . . . . . . . . . . . . . . . . . . . . . 42 2. Neutronneutron phaseshift derivatives vs z = kRnn 8 z 2 [0; 20]. . . . . . . . . . . . . . . . . . . . . 43 3. Partial contributions for neutronneutron interaction vs energy (MeV) 8 E 2 [0MeV; 20MeV ]. . . . . . . . . . . . . . . . . . . 44 4. Quantum mechanical partial contributions for neutronneutron interaction vs temperature (K). . . . . . . . . . . . . . . . . . . . 46 5. Semilog plot of classical b(cl) nn vs temperature (K). . . . . . . . . 47 6. Semilog plot comparing classical and quantum mechanical partial contributions (b(qu) nn (T)) for NeutronNeutron Interaction vs Temperature (K). . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7. The second virial coefficient SFunction vs z = kRnn 8 z 2 [0; 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 8. The second virial coefficient SFunction tilde vs z = kRnn 8 z 2 [0; 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 9. Comparison of the second virial coefficient SFunction, and SFunction tilde vs z = kRnn 8 z 2 [0; 20]. . . . . . . . . . . . . 52 10. Semilog plot of the QM second virial coefficient tilde eb (qu) nn for neutronneutron interaction vs temperature (T). . . . . . . . . . 53 xiv 11. Semilog plot of the second virial coefficients b(cl) nn ; b(qu) nn ;eb (qu) nn for neutronneutron interaction vs temperature (K). . . . . . . . . . 54 12. nn Mass Fraction vs Baryon Density for point particles within T 2 [2; 10]MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 13. np Mass Fraction vs Baryon Density for point particles within T 2 [2; 10]MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 14. nd Mass Fraction vs Baryon Density for point particles within T 2 [2; 10]MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 15. nn Mass Fraction vs Baryon Density for volumeexcluded particles within T 2 [2; 10]MeV . . . . . . . . . . . . . . . . . . . . . . . 63 16. np Mass Fraction vs Baryon Density for volumeexcluded particles within T 2 [2; 10]MeV . . . . . . . . . . . . . . . . . . . . . . . 64 17. nd Mass Fraction vs Baryon Density for volumeexcluded particles within T 2 [2; 10]MeV . . . . . . . . . . . . . . . . . . . . . . . 65 1 Chapter 1 STATISTICS AND THERMODYNAMICS OF PARTICLES WITH FINITE SIZE Introduction If we consider a system composed of different particle species i with particle numbers Ni in a volume V at some temperature T, we can denote the particle densities as ni = Ni V ; (1.1) where we assume that every particle species i has a degeneracy factor gi and a chemical potential i for particles with rest mass mi. The energy of a particle is denoted by Ei and includes the rest mass. We will use canonical and grand canonical statistical ensembles and different particle statistics in the following. More specifically, according to Greiner et al. (1999), the canonical ensemble is suited for systems in a heat bath with a given T, V and N. On the other hand, the grand canonical  also called the macrocanonical ensemble  describes open systems where heat and particles are exchanged with the surroundings. 2 Classical Excluded Volume Mechanism If we also assume MaxwellBoltzmann statistics for noninteracting particles with nonrelativistic energy (in units of h = c = kB = 1); we have Ei(k) = k2 2mi + mi; (1.2) that depends on the particle momentum k. According to (Greiner et al., 1999, p. 187  189), MaxwellBoltzmann statistics assumes that for a given temperature, particles are distinguishable and that quantum mechanics plays a negligible role. These two assumptions while seemingly valid at first, create a number of problems depending on the particles and system. This indistinguishability is “corrected” by the inclusion of the Gibbs factor 1=N!  as noted in (Greiner et al., 1999, p. 133), which removes contradictions arising from the assumption that particles are distinguishable. And, depending on the particles and system in question, we will show that quantum statistics will have to be included where we will introduce Planck’s constant, h and other physical considerations. Given this information, it is more appropriate to consider the free energy of the system  which is energy that can be used to perform work (that includes the contributions of rest mass), in contrast to the total energy, E. More 3 specifically, if we begin with an ideal gas approximation with constant temperature and volume, the free energy in question is actually the Helmholtz energy. (Huang, 2001, p. 38) To begin our derivations, we use the definitions for the Helmholtz free energy F(T; V;N) = U TS; (1.3) dF = pdV SdT + X i idNi; (1.4) and the Gibbs free energy G(T; p;N) = U TS + pV; (1.5) dG = V dp SdT + X i idNi: (1.6) Physically, the meaning of the Helmholtz free energy is that of work done on the system plus heat loss from it, under constant pressure and entropy. The Gibbs free energy is the same, but for constant volume. In both cases, one adds the energy required to change the particle number. , the chemical potential, is the energy change by adding one particle to the system. And, to consider more realistic physical systems, if we apply the Gibbs correction factor (to “correct” contradictions arising from the initial assumption that for MaxwellBoltzmann statistics all particles are distinguishable), on an 4 ideal gas and consider phase space volume elements of size h3N to determine the mean number of states per energy interval, we can use the “absolute entropy” of an ideal gas, also known as the SackurTetrode equation, from which in (Greiner et al., 1999, p. 137  139), we can develop a more specialized form of the Helmholtz free energy to determine thermodynamic variables and potentials of interest S(U; V;N) = NkB " 5 2 + ln ( V N 4 mU 3N 3=2 )# : (1.7) Notably, we observe that h, Planck’s constant appears in the Sackur Tetrode equation  which comes from a “pseudoquantum mechanical” counting of the number of microstates where “the total energy is determined by the 3N quantum numbers of the occupied states” (Greiner et al., 1999, p. 135) U = E = h2 8mL2 X3N i=1 n2i (1.8) Then, solving for the internal energy, U U(S; V;N) = 3h2N5=3 4 mV 2=3 exp 2S 3NkB 5 3 ; (1.9) 5 the variation of the internal energy under constant pressure and temperature is given by (Greiner et al., 1999, p. 140) dU = TdS pdV + dN: (1.10) Hence, T = @U @S N;V = 2 3NkB U ! U = 3 2 NkBT; (1.11) p = @U @V S;N = 2 3V U ! U = pV = NkBT; (1.12) = @U @N S;V = U 5 3N 2S 3N2kB = kBTln ( N V h2 2 mkBT 3=2 ) : (1.13) Now, we can define a specific form the Helmholtz free energy using eqs. (1.7) and (1.11) F = U TS = NkBT " ln ( N V h2 2 mkBT 3=2 ) 1 # : (1.14) 6 The thermal wavelength  or the de Broglie wavelength for a gas in equilibrium at temperature, T as shown (Huang, 2010, p. 195)  is defined as (here we restore h and kB to explicitly show their dependence) i = s 2 h2 mikBT : (1.15) Then, setting h = kB = 1 (as done previously), we obtain i = r 2 miT : (1.16) Using eqs. (1.16) and (1.14) the free energy becomes F = U TS = NkBT ln N V i 3 1 : (1.17) The degeneracy increases the number of possible states in the system (similar to increasing the volume). Hence, we can write (note that the i subscript here means the particle in question  e.g. i = n) F(T; V;Ni) = X i Ni Tln Ni i 3 giV T + mi ; (1.18) Noting the presence of h in the thermal wavelength suggests that quantum mechanics is important to the calculation of the Helmholtz free energy. However, with this consideration we immediately violate one of the require 7 ments for MaxwellBoltzmann statistics (that quantum mechanical effects are considered to be minuscule such that they can be neglected altogether). This means that with the introduction of h we make the assertion that while MaxwellBoltzmann statistics are a good starting point to describe the physical constraints, we assume that quantum mechanical effects will have a part to play in the overall behavior and description of the system because of the nature of the particles in question; and, the scale of the overall interaction distances  as the average interparticle distance has to be compared to the deBroglie wavelength of the particles. Notably, when the thermal wavelength is much smaller than the inter particle distance, we recover the classical behavior. However, it should be noted that as this treatment is not entirely quantum mechanical derived, it is actually semiclassical. Additionally, we use  the deBroglie wavelength  in eq. (1.18), = h p2mEK ; (1.19) where EK is the average kinetic energy of the thermal particles. Indeed, using EK kBT, we recover eq. (1.15). 8 Furthermore, we note that the natural variables here are the temperature T, the volume V and the particle numbers Ni. Hence, we can then define the chemical potentials of the particles as i = @F @Ni T;V;Nj6=i : (1.20) Similarly, the entropy is given by eq. 3.37 of (Huang, 2001, p. 37) S = @F @T V;Ni : (1.21) And, in like fashion the pressure is obtained from P = @F @V T;Ni : (1.22) We begin determining the values of these potentials by using the relationship d dx x ln x = + ln x ; (1.23) hence, we can then find i = @F @Ni T;V;Nj6=i = T + T ln Ni i 3 giV T + mi = T ln Ni i 3 giV + mi: (1.24) 9 Similarly, for the entropy, S = @F @T V;Ni = X i Ni ln Ni i 3 giV 5 2 ; = X i Ni 5 2 ln Ni i 3 giV ; (1.25) where we have used the relationship d dT T ln T3=2 T = 5 2 + ln T3=2 : (1.26) Finally, for the pressure, we get P = @F @V T;Ni = X i Ni T V : (1.27) Using these solutions, we can then obtain the total internal energy (eq. 4.52) in Greiner et al. (1999). Using the relation between the internal energy and the free energy F = U TS X i iNi; (1.28) 10 we get U(S; V;Ni) = F(T; V;Ni) + TS = TS pV + G (1.29) = T X i Ni 5 2 ln Ni i 3 giV V X i Ni T V ! +T X i Ni ln Ni i 3 giV + X i Nimi; = T X i Ni 5 2 ln Ni i 3 giV T X i Ni +T X i Ni ln Ni i 3 giV + X i Nimi; = X i Ni 5 2 T T + T ln Ni i 3 giV T ln Ni i 3 giV + mi ; = X i Ni 3 2 T + mi ; (1.30) where we used mi= i T ln Ni i 3 giV : i = T ln Ni i 3 giV + mi: (1.31) Therefore, we can write the energy in natural variables as U(S; V;Ni) = X i Ni 3 2 T + i T ln Ni i 3 giV : (1.32) 11 And, using the grand canonical potential (a thermodynamic potential is a scalar function used to measure the state of a system), which describes the properties of an open system, i.e. a system with varying particle number, is defined as (T; V; i) = F(T; V;Ni) X i iNi = PV ; = V X i Ni T V ! ; = X i NiT = NT; (1.33) where the natural variables of the different thermodynamic potentials are given as arguments in parentheses. Sometimes it is also useful to introduce the free energy density f = F=V , the internal energy density u = U=V , the entropy density s = S=V and the grand canonical potential density ! = =V . Thermodynamic Potentials in Reduced Volume A finite volume Vi can be introduced for every particle species such that the available volume for the motion is not V any more but the reduced volume. 12 We will use the raised symbol to indicate quantities in the description with excluded volume mechanism and generalizations. eV = V X i NiVi: (1.34) Replacing V in eq. (1.18) by eV one obtains the classical excluded volume mechanism in the free energy e F(T; V;Ni) = X i Ni T ln Ni i 3 gi eV T + mi ; = X i Ni 2 6 4T ln 0 B@ Ni i 3 giV gi P j NjVj 1 CA T + mi 3 75 : (1.35) This approach is thermodynamically consistent since the natural variable of F is used in the formulation. Thus, using the developments in the section, we can then derive explicit expressions for the quantities i, S, P, E and in case of a gas with excluded volume. 13 The volume excluded chemical potential, e for each particle species i is e i = @F @Ni T;V;Nj6=i ; = T + T ln Ni i 3 gi eV T + mi + X k NkT gk P j ijVj gkV gk P j NjVj ; (1.36) = T ln Ni i 3 gi eV + mi + TVi X k Nk V P j NjVj : (1.37) Comparing this with the equation obtained for the chemical potential, i = T ln Ni i 3=giV +mi, we see that the chemical potential for a reduced volume is larger. Similarly, for the entropy eS = @F @T V;Ni ; = X i Ni ln Ni(2 =miT)3=2 gi eV 5 2 ; = X i Ni 5 2 ln Ni i 3 gi eV : (1.38) Comparing this with the original value obtained for the entropy S = X i Ni 5=2 ln Ni i 3=giV ; we see that the entropy is smaller for a reduced potential. 14 Finally, for the pressure, e P = @F @V T;Ni = T P i Ni V P j NjVj : (1.39) Comparing this against the value obtained previously for the pressure P = X i Ni T V ; we see that the pressure for reduced volume is higher. While seemingly unusual at first glance, this result is expected as we’re working in a smaller volume with the same particle numbers, and system constraints. More specifically, since the Temperature has remained the same and we have a smaller volume, the pressure has to increase. 15 Combining all of these solutions, we can hence derive the total internal energy for a reduced volume eU(eS; V;N) = e F(T; V;Ni) + T eS; = T eS epV + X i e iNi; = T X i Ni 5 2 ln Ni i 3 gi eV V T P i Ni V P j NjVj +T X i Ni ln Ni i 3 gi eV + X i Nimi +T X i NiVi X i Ni V P j NjVj ; = 5 2 T X i Ni V T P i Ni V P j NjVj + T X j NjVj X i Ni V P j NjVj + X i Nimi; = 5 2 T X i Ni T X i Ni 2 64 V V P j NjVj P j NjVj V P j NjVj 3 75 + X i Nimi; = X i Ni 3 2 T + mi : (1.40) 16 Noting that e i = T ln Ni i 3 gi eV + mi + TVi X k Nk V P j NjVj : (1.41) = T 2 64 ln Ni i 3 gi eV + mi T + Vi X k Nk V P j NjVj 3 75 ; we get mi = e i T 2 64 ln Ni i 3 gi eV + Vi X k Nk V P j NjVj 3 75: Hence, we can write the reduced volume energy as eU(eS; V;N) = X i Ni 2 64 3 2 T + e i T 0 B@ ln Ni i 3 gi eV + Vi X k Nk V P j NjVj 1 CA 3 75 : (1.42) Given this value, and comparing it against the value obtained previously for the internal energy, E = P i Ni (3T=2 + mi), we observe that the total internal energy is identical to the internal energy in the regularly case, the ideal gas. 17 Similarly, for the grand canonical potential for a reduced volume e (T; V; ) = e F(T; V;Ni) X i e iN; = e PV; = TV P i Ni V P j NjVj ; = T P i Ni 1 P j njVj : (1.43) Since e i = T ln Ni i 3 gi eV + mi + TVi X k Nk V P j NjVj ; we get T P i Ni 1 P j njVj = e i T ln Ni i 3 gi eV mi; or T P i Ni 1 P j njVj = T ln Ni i 3 gi eV + mi e i: (1.44) 18 Then we can write e as a function of natural variables e (T; V; e ) = e F(T; V;Ni) X i e iN = T P i Ni 1 P j njVj = T ln Ni i 3 gi eV + mi e i: (1.45) We can conclude that this potential is larger (in magnitude) for a given particle species within a reduced volume. Density Dependent Degeneracy Factors Using the definition for the reduced volume and degeneracy, we can define the effective degeneracy as a function of the volume, V as gi eV = gi V X i NiVi ! = gi 1 X i niVi ! V = egiV; (1.46) with effective degeneracy factors egi = gi 1 X i niVi ! ; (1.47) that depend on the particle densities ni. 19 We define the grand canonical potential as e (T; V; e ) = X i e i +W = T ln Ni i 3 gi eV + mi e i; (1.48) where the individual particle contributions can be written as e i = TegiV i Z d3k (2 )3 ln " 1 + iexp eUi e i T !# ; (1.49) with an additional contribution W. Here we introduce i as we wish to generalize the particle contributions with consideration of different statistics  i.e. FermiDirac and BoseEinstein statistics in which i = 1 for fermions and i = 1 for bosons as discussed in (Carroll, 2007, p. 43), a boson is an integerspin forcecarrying particle, as opposed to a matter particle (fermion); and, unlike fermions, bosons can be accommodated to the same state without limit  e.g. photons, gluons, gravitons, weak bosons, and the Higgs boson. Also we define the singleparticle energies as eUi(k) = k2 2mi + mi + Ui: (1.50) 20 Combining this with e i, we obtain eUi(k) e i = k2 2mi + mi + Ui 2 6 4T ln Ni i 3 gi eV + mi + TVi X k Nk V P j NjVj 3 75 ; = k2 2mi + Ui 2 6 4T ln Ni i 3 gi eV + TVi X k Nk V P j NjVj 3 75 : Since gi eV = gi (V P i NiVi) = gi (1 P i niVi) V = egiV , we obtain eUi(k) e i = k2 2mi + Ui T ln Ni i 3 gi eV TVi X i Ni gj eV ; = k2 2mi + Ui T " ln Ni i 3 gi eV Vi X i Ni gj eV # : That is eUi(k) e i T = k2 2miT + Ui T ln Ni i 3 gi eV + Vi X i Ni gj eV ; = k2 2miT + Ui T + ln gi eV Ni i 3 ! + Vi X i Ni gj eV : 21 Hence, we can calculate the particle number, Ni using @ @x ln a exp x + y b + 1 = a exp x+y b (a exp x+y b + 1) b = a b exp x+y b a exp x+y b + 1 ; = a b 1 a + exp (x+y) b : (1.51) Since e (T; V; e ) = T ln Ni i 3 gi eV + mi e i; then @e @e i T;V;e j!=i = 1: 22 Hence Ni != @e @e i T;V;e j!=i ; != + TegiV i Z d3k (2 )3 i T 1 i + exp ( eUie i) T @W @ e i ; = TegiV i i T Z d3k (2 )3 1 i + exp eUie i T @W @ e i ; = egiV Z d3k (2 )3 1 i + exp eUie i T @W @ e i ; = egiV Z d3k (2 )3 " i + exp eUi e i T !# 1 @W @ e i ; = egiV Z d3k (2 )3 " i + exp k2 2miT + Ui T + ln gi eV Ni i 3 ! + Vi X i Ni gj eV !# 1 @W @ e i : (1.52) Thus Ni != @e @e i T;V;e j!=i ; != egiV Z d3k (2 )3 2 6 4 i + exp 0 B@ k2 2miT + Ui T + Vi X i Ni gj eV + 2 4 1 ln gi eV Ni i 3 3 5 1 1 CA 3 75 1 @W @ e i ; 23 which contains the "rearrangement" potentials Ui, defined by the relation between Ui and W as shown in the above relationship, in addition to the following Ui = X j @egj @Ni e i egi : (1.53) Also noting that we can calculate @W=@ e i, @W @ e i = egiV Z d3k (2 )3 " i + exp eUi e i T !# 1 1: Now, giV = gi V X i NiVi ! = gi 1 X i niVi ! V = egiV; (1.54) egi = gi V V X i NiVi ! = gi 1 X i niVi ! ; (1.55) Since Ni = niV @egj @Ni = 1 V @egj @ni = gj P i Vi V : (1.56) 24 Using this in Ui Ui = X j @egj @Ni e i egi = 1 V @egj @ni e i egi ; = gj P i Vi V TV i Z d3k (2 )3 ln " 1 + iexp eUi e i T !#! ; = X i TgjVi i Z d3k (2 )3 ln " 1 + iexp eUi e i T !# : (1.57) and using Ni = niV , we can now calculate the work, W = X i NiUi; = X i TgiNiVi i Z d3k (2 )3 ln " 1 + iexp eUi e i T !# ; = X i TginiViV i Z d3k (2 )3 ln " 1 + iexp eUi e i T !# : (1.58) Combining these results, we get @egj @Ni = 1 V @egj @ni = gj P i Vi V ; (1.59) and Ni gj gj P i Vi V = P i NiVi V : (1.60) 25 Since, Ni = niV , one has P i NiVi V = P i niViV V = X i niVi: (1.61) Therefore, W = X i NiUi = X i TgjniV Vi i Z d3k (2 )3 ln " 1 + iexp eUi e i T !# : (1.62) We can combine all of these into a general equation W = X i X j niV egj @egj @Ni e j = X ij ij niV egj @egj @Ni e j : (1.63) Thus, we have developed a prescription to include the density dependence of the degeneracy factors on the excluded volumes of the clusters. 26 Chapter 2 EQUATION OF STATE WITH EXCLUDED VOLUME Introduction The effects of the finite size of particles on the equation of state of a gas can be described with the help of the excludedvolume mechanism. It can also be used in order to simulate the suppression of a particular particle species, e.g. nuclei (clusters) in a mixture with nucleons, at high densities. The excludedvolume mechanism can be formulated most simply by starting with the free energy F(T; V; fNig) of a classical ideal gas of particles i with singleparticle numbers Ni in a volume V at temperature T. Instead of the total volume V of the system it is assumed that for a particle i only a reduced volume Vi = V i < V; (2.1) is available since part of the whole volume is occupied already by other particles. The functions (ni) depend on the singleparticle densities ni = Ni=V . Applying a Legendre transformation (T; V; f ig) = F X i iNi = pV; (2.2) 27 with the chemical potentials (including rest masses mi of the particles) i = @F @Ni T;V; j6=i ; (2.3) where the grandcanonical potential depends on the natural variables T, V , and the set f ig. It is better suited for generalizations of the model than the free energy F. In the following, we use the usual system of units in nuclear physics with h = c = kB = 1. Thus these factors do not appear in the formulas explicitly but have to be considered in the conversion of quantities. Additionally, to more easily distinguish and compare the potentials derived from different physical constraints and arguments, we use labels of the form (id), (eff), (vi) , and (ex) as opposed to the use and absence of eX , (where X is the potential of interest), as in the previous chapter. Classical Particles Let us consider first classical particles with MaxwellBoltzmann statistics and nonrelativistic kinematics without explicit interactions. In this case the grandcanonical potential is given by = X i ( i NiUi) ; (2.4) 28 with singleparticle contributions i = Tg(e ) i V Z d3k (2 )3 exp Ei(k) i T ; (2.5) and rearrangement potentials Ui = X j @g(e ) j @Ni j g(e ) j ; (2.6) that contain the densitydependent effective degeneracy factors g(e ) i = gi i; (2.7) and the energies (including rest masses mi) Ei(k) = k2 2mi + mi + Ui: (2.8) of the particles. The usual degeneracy factors are denoted by gi. They are constants. The rearrangement potentials Ui are essential in order to obtain a thermodynamically consistent model. 29 Ideal Gas For the case of an ideal gas with pointlike particles, the functions i are just constants and equal to one such that the effective degeneracy factors g(e ) i are identical to the usual degeneracy factors gi of the particles and the rearrangement potentials Ui vanish. Then one finds explicitly as shown in (Typel, Röpkec, et al., 2010, p. 3) (id)(T; V; f (id) i g) = TV X i gi 3i exp (id) i mi T ! = p(id)V; (2.9) and Ni = @ (id) @ (id) i T;V; j6=i = V gi 3i exp (id) i mi T ! ; (2.10) with the thermal wavelengths i = r 2 miT ; (2.11) such that the pressure of an ideal gas is given by p(id) = T V X i Ni ; (2.12) i.e. the wellknown relation for a mixture of classical ideal gases. 30 Classical Gas with Excluded Volume Now we treat the case with explicit excludedvolume effects and assume that i(fnjg) = 1 X j vijnj ; (2.13) with constant volumes vij . Sometimes these functions are defined as i(fnjg) = 1 X j njvj ; (2.14) with volumes vj of particles j such that the functions i are identical for all i. However, this leads to problems when the comparison to the virial equation of state is made, see below. Their relation to the volumes of the individual particles will be specified later. One finds the singleparticle contributions (ex) i = TV g(e ) i 3i exp (ex) i mi Ui T ! ; (2.15) and the rearrangement potentials Ui = X j @ j @ni (ex) j V j = X j vij (ex) j V j : (2.16) 31 All of these combined lead to the total excluded grandcanonical potential (2.17) (ex)(T; V; f ig) = X i (ex) i + ni X j vij (ex) j j ! ; = X i X j ij j j (ex) i + ni X j vij (ex) j j ! ; = X j j j (ex) j + X i ni X j vij (ex) j j = X j (ex) j j ; = T X i gi 3i V exp (ex) i mi Ui T ! ; = p(ex)V; and the singleparticle densities ni = 1 V @ (ex) @ (ex) i = g(e ) i 3i exp (ex) i mi Ui T ! : (2.18) Thus we find the relation p(ex) = T X i ni i : (2.19) Since i < 1, the pressure p(ex) with excludedvolume effects is larger than that of an ideal gas p(id) with the same particle number densities ni. 32 Series Expansion The grandcanonical potential (ex) can be expanded in powers of fugacities as in (Typel, Röpkec, et al., 2010, p. 3) zi = exp (ex) i mi T ! ; (2.20) by using the results for the rearrangement potentials Ui, and using a similar approach as in eq. (2.17). We find (2.21) (ex)(T; V; f ig) = T X i gi 3i V zi exp Ui T ; = T X i gi 3i V zi " 1 Ui T + 1 2 Ui T 2 : : : # ; T X i gi 3i V zi 1 + 1 T X j vij (ex) j V j + : : : ! ; TV X i gi 3i zi 1 X j vij gj 3j zj + : : : ! ; = TV X i gi 3i zi X ij vij gigj 3i 3j zizj + : : : ! ; up to second order in the fugacities as also shown in Typel, Röpkec, et al. (2010). Lowdensity Limit So far we have considered the general formulation of the excludedvolume mechanism. However, the volumes vij are not fixed yet. In order to relate them 33 to the volumes of the individual particles, the exact lowdensity limit of the equation of state can be used to define them. Virial Expansion At finite temperatures, the lowdensity limit is given by the virial equation of state. It is an expansion of the grandcanonical potential (vi)(T; V; f (vi) i g) = TV X i bi 3i zi + X ij bij 3=2 i 3=2 j zizj + : : : ! ; (2.22) in powers of the fugacities zi = exp (vi) i mi T ! ; (2.23) with dimensionless virial coefficients bi, bij , . . . . The expansion is valid only for zi 1 or ni 3i 1. The first virial coefficients bi = gi are just the standard degeneracy factors. 34 Classical Mechanics The second virial coefficients in classical mechanics are given by b(cl) ij (T) = 1 + ij 2 gigj 3=2 i 3=2 j Z d3r exp Vij T 1 ; (2.24) depending on the potential Vij between the particles i and j. For incompressible spherical particles with radii Ri and volumes Vi = 4 R3 i =3 it is given by the hardsphere potential Vij(r) = 8>< >: 1 if r Rij ; 0 if r > Rij ; (2.25) with the sum of the radii Rij = Ri + Rj . Then we have b(cl) ij (T) = 1 + ij 2 gigj 3=2 i 3=2 j 4 Z Rij 0 dr r2 [1] = 1 + ij 2 gigj 3=2 i 3=2 j 4 3 R3 ij : (2.26) Comparing with the result for the series expansion of (ex) we can identify vij = 1 + ij 2 4 3 R3 ij ; (2.27) if we set (ex) i = (cl). In particular we find vii = 4 3 (2Ri)3 = 8Vi; (2.28) 35 because the centers of two equal particles cannot become closer than twice their radius. Quantum Mechanical Effects The classical expression for the second virial coefficient does not take into account quantum effects. Using the formalism described in both Uhlenbeck & Beth (1936) and Beth & Uhlenbeck (1937), the quantummechanical second virial coefficient (without additional quantumstatistical corrections for Fermions or Bosons) is given by b(qu) ij (T) = 1 + ij 2 3=2 i 3=2 j 3 ij Z dE exp E T Dij(E); (2.29) with ij = s 2 (mi + mj)T ; (2.30) and the kinetic energy of the relative motion E = k2 2 ij ; (2.31) that contains the reduced mass ij = mimj mi + mj : (2.32) 36 From these we can then determine the “level density” difference as shown in Typel, Röpkec, et al. (2010) Dij(E) = X k g(ij) k (E E(ij) k ) + X l g(ij) l d (ij) l dE ; (2.33) with g(ij) l = (2l + 1)gigj ; (2.34) which contains contributions from twoparticle bound states k and scattering states l in partialwave expansion (neglecting spindependent potentials). Notably, we don’t have bound states for the hardsphere potential but only scattering states. Hence, we need the scattering phase shifts (ij) l in all partial waves l as a function of the relative momentum k. Using that, we can then calculate the second virial coefficients b(qu) ij (T) = X l b(l) ij (T); (2.35) with b(l) ij (T) = 1 + ij 2 (2l + 1)gigj 3=2 i 3=2 j 3 ij Z 1 0 dk exp k2 2 ijT d (ij) l dk : (2.36) 37 To include quantum effects we need to solve the timeindependent Schrödinger equation Hij (r) = E (r); (2.37) with the Hamiltonian as Hij = p2 2 ij + Vij ; (2.38) to obtain the scattering wave function for the relative motion at energy E = k2=(2 ij) which can be written as ij(r) = 4 kr X lm ilu(ij) l (r)Ylm(^k)Y lm(^r); (2.39) with spherical harmonics Ylm that depend on the direction of the coordinate and momentum vectors r and k, respectively. The radial wave functions u(ij) l (r) have to fulfill the boundary conditions in each partial wave l. In case of the hardsphere potential, the radial wave functions for r Rij have the form u(ij) l (r) = 1 + S(ij) l 2 h Fl(kr) + K(ij) l Gl(kr) i ; (2.40) with the Smatrix elements are given in terms of the phase shifts by means of S(ij) l = exp 2i (ij) l ; (2.41) 38 and the Kmatrix elements are K(ij) l = 1 i S(ij) l 1 S(ij) l + 1 = tan (ij) l : (2.42) Of these, the regular and irregular scattering wave functions can be expressed with the help of spherical Bessel functions Fl(z) = zjl(z); (2.43) Gl(z) = zyl(z); (2.44) with the asymptotics Fl(z) ! sin z l 2 ; (2.45) Gl(z) ! cos z l 2 ; (2.46) for z ! 1. On the other hand, we have Fl(z) ! zl (2l + 1)! ! ; (2.47) Gl(z) ! (2l 1)! ! zl+1 ; (2.48) for z ! 0. For r Rij the radial wave function should vanish. This condition ul(Rij) = 0; (2.49) 39 leads to tan (ij) l = Fl(kRij) Gl(kRij) = jl(kRij) yl(kRij) : (2.50) Hence, we can calculate (ij) l for all k and l. For l = 0 we explicitly have tan (ij) 0 = sin(kRij) cos(kRij) ; (2.51) or (ij) 0 = kRij : (2.52) With help of the equations developed so far for the statistics and thermodynamics of particles with finite size, we can pave the road to understand the effects of excluded volume mechanism on the equation of state for a gas of nucleons and nuclei. Our analysis is not complete, but we show the effects in a few numerical exercises in the next chapter. 40 Chapter 3 NUMERICAL STUDIES In order to see the differences between the various equations of states and their approximations, numerical calculations have to be performed. We will study different systems in the following. Single species with finite volume As a first application we consider a single particle species i. In order to have realistic numbers we choose neutrons (i = n) as an example with rest mass mn = 939:565 MeV/c2 and degeneracy factor gn = 2. We set the radius to Rn = 0:5 fm, Rnn = 1:0 fm and the volume vnn = 1 2 4 3 R3 nn 2:094 fm3 : (3.1) Then the function n = 1 vnnnn ; (3.2) which appears in the effective degeneracy factor g(e ) n = gn n ; (3.3) with the neutron density nn becomes zero at the maximum density n(max) n = 1 vnn 0:4775 fm3 : (3.4) 41 Thus the equation of state with the excluded volume can be calculated only for neutron densities 0 < nn < n(max) n . Virial coefficients with excluded volume Recalling the virialized coefficients b(l) ij are functions of the partial waves, then as a first step in calculating the neutronneutron scattering phases shifts we calculate from eq. (2.50) tan (ij) l = Fl(kRij) Gl(kRij) = jl(kRij) yl(kRij) ; (ij) l = arctan jl(kRij) yl(kRij) (3.5) for the specific case where i = j = n is given by (nn) l = arctan jl(kRnn) yl(kRnn) (3.6) as a function of k for partial waves l = 0; 1; 2; : : : This is shown in Figure (1), for the interval [0 MeV/c; 200 MeV/c]. Then, using (3.6), we can also find the derivatives, d (nn) l =dk for the same interval [0 MeV/c; 200 MeV/c] as shown in Figure (2). All of this information can then be used to find the partial contributions b(l) nn(T) = (2l + 1)g2n 3 n 3 nn Z 1 0 dk exp k2 2 nnT d (ij) l dk (3.7) 42 0 2 4 6 8 10 12 14 16 18 20 kRnn 0 90 180 270 360 450 540 630 720 810 900 990 1080 1170 Degrees NeutronNeutron Scattering Phase Shifts vs z (kRnn) l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8 l = 9 Figure 1. Neutronneutron scattering phaseshifts ( degrees) vs z (kRnn) within the limit [0; 20]. 43 0 2 4 6 8 10 12 14 16 18 20 kRnn 0 5 10 15 20 25 30 35 40 45 50 55 60 Degrees NeutronNeutron Scattering Phase Shifts and Derivatives vs z (kRnn) l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8 l = 9 Figure 2. Neutronneutron phaseshift derivatives vs z = kRnn 8 z 2 [0; 20]. 44 0 2 4 6 8 10 12 14 16 18 20 E (MeV) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 b(l) nn Partial Contributions (b(l) nn) for NeutronNeutron Interaction vs E (MeV) l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8 l = 9 Figure 3. Partial contributions for neutronneutron interaction vs energy (MeV) 8 E 2 [0MeV; 20MeV ]. 45 to the second virial coefficient bnn(T) as a function of T inside the interval [0 MeV; 20 MeV] for l = 0; 1; 2; : : : shown in Figure (3). Using (3.7), we can then calculate the total second virial coefficient, which is given by as an infinite sum b(qu) nn (T) = 1X l=0 b(l) nn(T) : (3.8) However, while the total second virial coefficient (in the quantum mechanics case) can be analytically calculated as an infinite sum (for l > 0), the function converges relatively quickly and only 6 lvalues are required to achieve convergence (within 3.4 parts per million), as shown in Figure (4). Similarly, we can calculate the classical second virial coefficient as a function of the temperature, T which is given by the following equation (and as illustrated as shown in Figure (5)) b(cl) nn (T) = g2n 3 n 4 3 R3 nn (3.9) Finally, we can compare the classical second virial coefficient with the quantum mechanical version both as functions of T, and we obtain Figure (6) which definitely shows a disparity between both. This makes sense since a classical billiard ball treatment doesn’t take into consideration quantum effects at short distances. 46 108 109 1010 1011 T (K) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 b(qu) nn QM Partial Contributions for NeutronNeutron Interaction vs T (K) Pl=0 l=0 Pl=1 l=0 Pl=2 l=0 Pl=3 l=0 Pl=4 l=0 Pl=5 l=0 Pl=1 l=0 Figure 4. Quantum mechanical partial contributions for neutronneutron interaction vs temperature (K). 47 109 1010 1011 T (K) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 b(cl) nn Second Virial Coefficient (b(cl) nn ) vs T (K) b(cl) nn Figure 5. Semilog plot of classical b(cl) nn vs temperature (K). 48 108 109 1010 1011 T (K) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 bnn QM Partial Contributions for NeutronNeutron Interaction vs T (K) Pl=0 l=0 Pl=1 l=0 Pl=2 l=0 Pl=3 l=0 Pl=4 l=0 Pl=5 l=0 Pl=1 l=0 b(cl) nn Figure 6. Semilog plot comparing classical and quantum mechanical partial contributions (b(qu) nn (T)) for NeutronNeutron Interaction vs Temperature (K). 49 To examine this even further, we then calculate the function, called the SFunction S(z) = (3.10) 1 z2 X l=0 2l + 1 [jl(z)2] + [jl(z)2] : Which, for the interval, z 2 [0; 20] is shown in Figure (7). Then, defining an approximation to this function (i.e. a fitting function) as Se(z) = 1 (3.11) 3 2 z 2 3 z2: Next, after comparing both of these together in Figure (9) we observe that eS is a good (and fast) approximation to the general Sfunction which involves spherical Bessel functions. Finally, we can compare all three second virial coefficients b(cl) nn (T), b(qu) nn (T), eb (qu) nn (T) as shown in Figure (10) and Figure (11) which illustrate that the b(qu) nn (T),eb (qu) nn (T) coefficients deviate from the classical case. Virial Equation of State With the known second virial coefficients b(qu) nn (T) and b(cl) nn (T) we can calculate the pressure p(vi;qu) = (vi;qu)(T; V; (vi;qu) n ) V = T 3 n gnz(vi;qm) n + b(qu) nn z(vi;qm) n 2 (3.12) and p(vi;cl) = (vi;cl)(T; V; (vi;cl) n ) V = T 3 n gnz(vi;cl) n + b(cl) nn z(vi;cl) n 2 (3.13) 50 0 2 4 6 8 10 12 14 16 18 20 z 0 50 100 150 200 250 300 S(z) SFunction vs z S(z) Figure 7. The second virial coefficient SFunction vs z = kRnn 8 z 2 [0; 20]. 51 0 2 4 6 8 10 12 14 16 18 20 z 0 50 100 150 200 250 300 eS(z) SFunction Tilde vs z eS(z) Figure 8. The second virial coefficient SFunction tilde vs z = kRnn 8 z 2 [0; 20]. 52 0 2 4 6 8 10 12 14 16 18 20 z 0 50 100 150 200 250 300 S(z)keS(z) SFunction and SFunction Tilde vs z S(z) eS(z) Figure 9. Comparison of the second virial coefficient SFunction, and SFunction tilde vs z = kRnn 8 z 2 [0; 20]. 53 108 109 1010 1011 T (K) 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 b(qu) nn QM Second Virial Coefficient (eb (qu) ij ) for NeutronNeutron Interaction vs T (K) eb (qu) ij (T) Figure 10. Semilog plot of the QM second virial coefficient tilde eb (qu) nn for neutronneutron interaction vs temperature (T). 54 1010 1011 1012 1013 1014 T (K) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 b(l) nn Second Virial Coefficients b(cl) nn , b(qu) nn ,eb (qu) nn for NeutronNeutron Interaction vs T (K) Pl=1 l=0 b(cl) nn eb (qu) nn (T) Figure 11. Semilog plot of the second virial coefficients b(cl) nn ; b(qu) nn ;eb (qu) nn for neutronneutron interaction vs temperature (K). 55 in the two approximations for the virial equation of state. In order to obtain the fugacities z(vi;qm) n = exp (vi;qm) n mn T ! (3.14) and z(vi;cl) n = exp (vi;cl) n mn T ! (3.15) we consider the neutron densities nn = @p(vi;qu) @ n T = 1 3 n gnz(vi;qm) n + 2b(qu) nn z(vi;qm) n 2 (3.16) and nn = @p(vi;cl) @ n T = 1 3 n gnz(vi;cl) n + 2b(cl) nn z(vi;cl) n 2 : (3.17) Using these two (quadratic) equations, we can find the fugacities for given temperature T and density nn. In a second step, the pressure can be calculated. Mixture of Nucleons and Deuterons To observe the effect of volumeexclusion, we consider a classical mixture of neutrons, protons and deuterons in chemical and thermal equilibrium. From these considerations we can calculate the particle number densities from ni = gi 3i exp (id) i mi T ! (3.18) 56 where we use the following constraints Table 1 Particle masses and degeneracies. Particle Mass (MeV=c2) Degeneracy p mp = 939.565 gp = 2 n mn = 938.272 gn = 2 d md = 1875.612 gd = 3 and where the mass of Deuteron is calculated as: md = mn + mp Bd = 939:565MeV=c2 + 938:272MeV=c2 2:2225MeV=c2 = 1875:612MeV=c2, (Bd is the binding energy of the Deuteron nucleus (2:2225MeV=c2)). As there are two independent densities, we can call this the total neutron number density as shown in (Typel, Röpke, et al., 2010, p. 3) n(tot) n = nn + nd: (3.19) Similarly, for the proton number density (Typel, Röpke, et al., 2010, p. 3) n(tot) p = np + nd: (3.20) However, it is more convenient to introduce the total baryon number density nB, which is a sum of the two nB = n(tot) n + n(tot) p = nn + np + 2nd: (3.21) 57 Next, we also consider the isospin asymmetry which is defined as = n(tot) n n(tot) p nB = nn np nB (3.22) However, while the total neutron and proton number densities are independent, the individual number densities of the neutrons, protons and deuterons are not  as they are constrained by the chemical equilibrium such that n + p = d: (3.23) Point Particles (Ri = 0 (i = n, p, d)) To further understand this, we first consider a mixture of point particles of neutrons, protons and deuterons (i = n; p; d) such that Rn = Rp = Rd = 0 fm, and with = 0 as shown in Figures (12), (13), and (14) for neutrons, protons and deuterons respectively. From Figures (12), (13), and (14) we can assert that the deuteron mass fraction decreases with rising temperature, and increases slowly with increasing temperature and baryon density. The opposite effect is observed with the symmetric mixture of protons and neutrons, for increasing temperature and baryon density. 58 0.00 0.05 0.10 0.15 0.20 nb(fm−3) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Xn = nn nB Neutron Mass Fractions vs Baryon Density for T 2 [2,10] MeV T = 2 MeV T = 4 MeV T = 6 MeV T = 8 MeV T = 10 MeV Figure 12. nn Mass Fraction vs Baryon Density for point particles within T 2 [2; 10]MeV . 59 0.00 0.05 0.10 0.15 0.20 nb(fm−3) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Xp = np nB Proton Mass Fractions vs Baryon Density for T 2 [2,10] MeV T = 2 MeV T = 4 MeV T = 6 MeV T = 8 MeV T = 10 MeV Figure 13. np Mass Fraction vs Baryon Density for point particles within T 2 [2; 10]MeV . 60 0.00 0.05 0.10 0.15 0.20 nb(fm−3) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Xd = 2nd nB Deuteron Mass Fractions vs Baryon Density for T 2 [2,10] MeV T = 2 MeV T = 4 MeV T = 6 MeV T = 8 MeV T = 10 MeV Figure 14. nd Mass Fraction vs Baryon Density for point particles within T 2 [2; 10]MeV . 61 Finite Nonzero Radii Particles (Ri > 0 (i = n, p, d)) In this case, we assume that deuteron has a finite radius Rd = 1:2fm Table 2 Particle Pairs Volumes Particle Pair Volume (fm3) vnn = vpp 2.094 vpn = vnp 2.094 vdn = vnd 7.238 vpd = vdp 7.238 vdd 57.906 where vdn = vnd = vdp = vpd = 4 3 R3d = 7.238 fm3; (3.24) and vdd 4 3 (2Rd)3 = 57.906 fm3: (3.25) Using these terms we can introduce the rearrangement potentials into the number densities Ui = X j vij (ex) j V j ; (3.26) 62 hence Ui = X j vij (ex) j V j (ex) i = TV g(eff) i i 3i exp (ex) i mi Ui T ! : (3.27) Combining these last two equations, we express the potentials more simply as Ui = T X j vij gj 3i exp (ex) i mi Ui T ! = T X j vij nj V j : (3.28) And, using this, we can finally include volumeexclusion in our calculation of the particle densities ni = g(eff) i 3i exp (ex) i mi Ui T ! (3.29) = gi i 3i exp (ex) i mi Ui T ! ; (3.30) the results of which are shown in figures (15), (16), and (17). Comparing these to Figures (12), (13), and (14) for the point particles, we have a vastly different scenario. Having considered volume exclusion, Figures (15), (16), and (17) suggest that we get a decrease in deuteron mass fraction as the baryon density increases. And, as temperature increases deuteron mass fraction decreases. As in the previous case for point particles, the trend of neutrons and protons against the deuterons is reversed. But this time, the proton and neutron 63 0.00 0.05 0.10 0.15 0.20 nb(fm−3) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Xn = nn nB ExcludedVolume Neutron Mass Fractions vs Baryon Density for T 2 [2,10] MeV T = 2 MeV T = 4 MeV T = 6 MeV T = 8 MeV T = 10 MeV Figure 15. nn Mass Fraction vs Baryon Density for volumeexcluded particles within T 2 [2; 10]MeV . 64 0.00 0.05 0.10 0.15 0.20 nb(fm−3) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Xp = np nB ExcludedVolume Proton Mass Fractions vs Baryon Density for T 2 [2,10] MeV T = 2 MeV T = 4 MeV T = 6 MeV T = 8 MeV T = 10 MeV Figure 16. np Mass Fraction vs Baryon Density for volumeexcluded particles within T 2 [2; 10]MeV . 65 0.00 0.05 0.10 0.15 0.20 nb(fm−3) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Xd = 2nd nB ExcludedVolume Deuteron Mass Fractions vs Baryon Density for T 2 [2,10] MeV T = 2 MeV T = 4 MeV T = 6 MeV T = 8 MeV T = 10 MeV Figure 17. nd Mass Fraction vs Baryon Density for volumeexcluded particles within T 2 [2; 10]MeV . 66 mass fractions increase with baryon density, and also for increasing temperature. Future Work Hence, while the results presented in this chapter are for the special case of a symmetric mixture of nucleons (i.e. = 0) for three particle species, neutrons, protons and deuterons, this methodology should be applied and compared against actual data for different nuclei species to confirm its validity. Furthermore, more realistic consideration involving particle number conservation, nonconserved chemical potential, and energy loss or gain, and inclusion of relativistic physical situations should also be considered. However, while such an approach is beyond the scope of this study, the arguments and approach made herein can be extended and applied in a more general sense beyond a finite mixture of gases of simple nucleons  as we have shown here. 67 Chapter 4 APPLICATIONS TO BIG BANG NUCLEOSYNTHESIS We also note that this methodology  with the inclusion of relativity  can be applied to understanding Big Bang Nucleosynthesis. Additional but unrelated work to the one shown herein, was done with a collaboration with Prof. Spitaleri and his nuclear astrophysics group in Catania, Sicily as shown in Pizzone et al. (2014), the abstract of which is cited below. “Nuclear reaction rates are among the most important input for understanding the primordial nucleosynthesis and therefore for a quantitative description of the early Universe. An uptodate compilation of direct cross sections of 2H(d,p) 3H, 2H(d,n) 3He, 7Li(p, ) 4He and 3He(d,p) 4He reactions is given. These are among the most uncertain cross sections used and input for Big Bang nucleosynthesis calculations. Their measurements through the Trojan Horse Method (THM) are also reviewed and compared with direct data. The reaction rates and the corresponding recommended errors in this work were used as input for primordial nucleosynthesis calculations to evaluate their impact on the 2H, 3;4He and 7Li primordial abundances, which are then compared with observations.” 68 APPENDICES 69 APPENDIX A CALCULATION OF bnn 70 CALCULATION OF bnn From eq. (2.22) p(vi;cl) = (vi;cl)(T; V; (vi;cl) n ) V = T 3 n gnz(vi;cl) n + b(cl) nn z(vi;cl) n 2 ; (4.1) and p(vi;qu) = (vi;qu)(T; V; (vi;qu) n ) V = T 3 n gnz(vi;qu) n + b(qu) nn z(vi;qu) n 2 ; (4.2) we have nn = @p(vi;cl) @ n = 1 3 n gnz(vi;cl) n + 2b(cl) nn z(vi;cl) n 2 ; (4.3) and nn = @p(vi;qu) @ n = 1 3 n gnz(vi;qu) n + 2b(qu) nn z(vi;qu) n 2 : (4.4) Simplifying this, we arrive at the relations nn = 1 3 n gnz(vi) n + 2bnn z(vi) n 2 ; (4.5) nn 3 n = gnz(vi) n + 2bnn z(vi) n 2 ; (4.6) 71 leading to z(vi) n 2 + z(vi) n gn 2bnn nn 3 n 2bnn = 0; (4.7) which has the solution: z(vi) n = 1 4bnn p g2n + 8bnnnn 3 n gn : (4.8) Hence, inserting into eqs. (4.1) and (4.2) respectively p(vi) = T 3 n gnz(vi) n + bnn z(vi) n 2 ; = T 3 n gnz(vi) n + bnn nn 3 n 2bnn z(vi) n gn 2bnn ; = T 3 n gnz(vi) n + 1 2 nn 3 n gnz(vi) n ; = T 2 3 n h gnz(vi) n + nn 3 n i ; = T 2 3 n h nn 3 n + gn p g2n + 8bnnnn 3 n gn i ; = T 2 3 n h nn 3 n g2n p g4n + 8g2n bnnnn 3 n i : (4.9) 72 To produce real values for p(vi) we must have the condition (constraint to bnn) g4n 8g2n bnnnn 3 n; (4.10) or g2n 8 3 nnn bnn: (4.11) For the classical case for neutrons, this is: b(cl) nn (T) = g2n 3 n 4 3 R3 nn; (4.12) or g2n 8 3 nnn + g2n 3 n 4 3 R3 nn: (4.13) Thus, the limiting density is g2n 8 3 nnn g2n 3 n 4 3 R3 nn; (4.14) 73 or 1 8nn 4 3 R3 nn: (4.15) In terms of the neutron number density (and recalling that Rnn 1fm) 3 32 1 R3 nn nn 0:029842 fm3; (4.16) the excluded volume is vnn = 1 2 4 3 R3 nn; (4.17) leading to 3 32 1 R3 nn ! 1 16vnn nn 0:029842 fm3: (4.18) Noticeably, this is temperature independent and is only constrained to the the volume of the sphere in question  i.e. volume dependent  as it should be. 74 For the quantum mechanical case b(vi;qm) (4.19) nn (T) = lX=1 l=0 b(l) nn(T); thus, using b(l) (4.20) nn(T) = (2l + 1)g2n 3 n 3 nn Z 1 0 exp k2 2 nnT d (ij) l dk ; we obtain b(vi;qm) nn (T) = lX=1 l=0 b(l) nn(T); (4.21) where g2n 8 3 nnn lX=1 l=0 b(l) nn(T): (4.22) Defining the integration result as f(T), we have b(vi;qm) nn (T) = lX=1 l=0 (2l + 1)g2n 3 n 3 nn f(T) = (2l + 1)g2n 3 n 3 nn lX=1 l=0 f(T); (4.23) since g2n 8 3 nb(vi;qm) nn (T) nn; (4.24) 75 we get g2n 8 3 n g2n 3 n 3 nn nn X (2l + 1)f(T); (4.25) or g2n 3 nn 8 3 ng2n 3 n nn X (2l + 1)f(T); (4.26) that is 3 nn 8 6 n P (2l + 1)f(T) nn: (4.27) From an examination of the terms, we observe that P (2l + 1)f(T) requires that the overall integrand result be unitless, so that it can be thought of as a single function dependent on T. Hence at a given temperature and nucleon density, we can write 3 nn 8 6 nnn X (2l + 1)f(T): (4.28) 76 REFERENCES Beth, E., & Uhlenbeck, G. E. (1937). The quantum theory of the nonideal gas. ii. behaviour at low temperatures. Physica, 4 (10), 915–924. Carroll, S. (2007). Dark matter, dark energy: The dark side of the universe (guidebook part 2). The Teaching Company. Greiner, W., Neise, L., & Stöcker, H. (1999). Thermodynamics and statistical mechanics. Springer. Huang, K. (2001). Introduction to statistical physics. Boca Raton: CRC Press. Huang, K. (2010). Introduction to statistical physics (2nd edition) (second ed.). Boca Raton: CRC Press. Pizzone, R., Spartá, R., Bertulani, C., Spitaleri, C., La Cognata, M., Lalmansingh, J., . . . Tumino, A. (2014). Big bang nucleosynthesis revisited via trojan horse method measurements. The Astrophysical Journal, 786 (2), 112. Typel, S., Röpke, G., Klähn, T., Blaschke, D., & Wolter, H. (2010). Composition and thermodynamics of nuclear matter with light clusters. Physical Review C, 81 (1), 015803. Typel, S., Röpkec, G., Klähnd, T., Blaschked, D., Woltere, H., & Voskresenskayab, M. (2010). Clusters in dense matter and the equation of state. Nuclei in the Cosmos, 1 , 39. 77 Uhlenbeck, G. E., & Beth, E. (1936). The quantum theory of the nonideal gas i. deviations from the classical theory. Physica, 3 (8), 729–745. 78 VITA Jared Lalmansingh received his Bachelor of Science in Physics from South Carolina State University in 2011. He next pursued his Master of Science in Physics at Texas A&M UniversityCommerce, where he is expected to graduate in 2015. Permanent address: Department of Physics and Astronomy, Texas A&M UniversityCommerce Commerce, Texas 75429 Email: jlalmansingh@leomail.tamuc.edu
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Title  Clustering Effects in the Equation of State on Nuclear Matter 
Author  Lalmansingh, Jared 
Subject  Nuclear physics; Theoretical physics; Physics 
Abstract  CLUSTERING EFFECTS IN THE EQUATION OF STATE ON NUCLEAR MATTER A Thesis by JARED LALMANSINGH Submitted to the Office of Graduate Studies of Texas A&M UniversityCommerce in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE May 2015 CLUSTERING EFFECTS IN THE EQUATION OF STATE ON NUCLEAR MATTER A Thesis by JARED LALMANSINGH Approved by: Advisor: William Newton Committee: Carlos Bertulani Kent Montgomery Head of Department: Matt. A. Wood Dean of the College: Brent Donham Dean of Graduate Studies: Arlene Horne iii ABSTRACT CLUSTERING EFFECTS IN THE EQUATION OF STATE ON NUCLEAR MATTER Jared Lalmansingh, MS Texas A&M UniversityCommerce, 2015 Advisor: William Newton, PhD In Nuclear AstroPhysics, clustering refers to the process by which atomic elements, known as nuclei, are formed from preexisting nucleons such as protons and neutrons. This process called nucleosynthesis, is thought to occur under the following scenarios: Big Bang nucleosynthesis, Stellar nucleosynthesis, explosive nucleosynthesis and nucleosynthesis derived from fragmentation processes in nuclear reactions. In the case of all except the latter, which deals with radioactive decay and fission (the dissociation of larger nuclei into smaller ones), nucleosynthesis is predominantly the result of fusion, which is the creation of larger nuclei from smaller ones. To date, our current understanding of nuclei formation has crucial limitations, which is due to the complexity of stellarnucleosynthesis and related processes. To overcome this gap in our understanding, of particular note and interest is explosive nucleosynthesis which involves the rprocess, rpprocess, iv sprocess and pprocess which are theorized to be the means by which elements more massive than iron are formed. Furthermore, the processes governing the different types of nucleosynthesis are additionally delineated by the relative rate at which nuclei formation occurs  which ranges from very rapidly in the order of seconds, to extremely long in the order of centuries, depending on the element and process in question. In attempting to reach an understanding of these processes, an additional caveat appears in that most of these processes are theorized to occur near the end of life of star  i.e. in a supernova event which is thought to supply both the high energy, temperatures and pressure with which to form these heavy nuclei. Consequently, neutron stars, which are highly dense stellar remnants of corecollapse supernova events, are potential candidates whose environments and existence are theorized to provide the ideal physical characteristics for the aforementioned processes hence explaining the formation of nuclei heavier than iron in neutron star mergers events. Although similar has work done on clustering in determining the physics, structure and evolution of neutron stars and their associated equations of state, it is not well known how such considerations derived from statistical physics affects heavy nuclei formation. Subsequently, one means of understanding these processes is to approach everything from a rigorous thermodynamical treatment. However, as nuclear v species can vary up in relative occupied volumes, we must account for the thermodynamic potential variance due to the occupied nuclear volume for interacting species in our treatment. And, as that variance could be potentially large, this fact violates the core tenet of the ideal gas law as we can no longer treat the particles as point particles but rather as particles with definite and noninfinitesimal volumes  which means that we can assert that the particles affect the occupied volume of the space in which they interact and occupy. Consequently, any treatment and inclusion of the ideal gas law and any derived thermodynamical potentials cannot be used or applied as the physics of those treatments are inapplicable in this situation. What this means is that for us to perform any useful calculation, we must account for the occupied particle species volumes in any thermodynamical potential  something that has to be derived and verified from scratch with many possible alternative methods. This idea of volume isolation or more accurately, exclusion, is called excludedvolume mechanism (excludedvolume can be thought to be the occupied volume minus the total volume occupied by all the particles), an approach that is fairly common in BioPhysics, but one that is not so common in nuclear astrophysics, with particular attention to how such an approach affects the formation of nuclei and the associated equations of state. To determine the effects, if any, that excluded volume may have on the equation of state, we begin by applying excludedvolume mechanism on classi vi cal thermodynamics and associated thermodynamical potentials by modifying the standard methodology of considering the nonrelativistic energetics of noninteracting particles from a purely statistical mechanical approach, known as a MaxwellBoltzmann statistics. Having considered excludedvolume statistically, we can derive thermodynamical potentials of interest such the reduced Gibb’s Free Energy, which can be used to formulate a rudimentary equation of state. But, as this approach doesn’t consider readily apparent physics that dominate neutron stars such as quantum mechanics, we have to extend this treatment even further by such an inclusion (via energy degeneracy and the inclusion of bound and occupied states of the particles), in addition to angular momentum and relativistic effects to create a more realistic physical model which can be applied and used in determining the associated equations of state and how nuclei formation is affected as a result. Summarily, as the thermodynamics determines the interparticle distance due to arguments based on particle energy, degeneracy, chemical potential, etc., it is our hope that such a treatment will not only prove to be definitive, but useful in constraining our understanding of these processes having utilized the relatively unexplored mechanism of excludedvolume, in its determination of the equation of state and how that affects nuclei formation in neutron stars. vii This work has been done in collaboration with the nuclear astrophysics group of Catania, Sicily, under the leadership of Prof. Spitaleri and with Dr. Stefan Typel from GSI, Darmstadt, Germany. viii ACKNOWLEDGEMENTS This work is dedicated to my mom, and my siblings for their unyielding support, care, patience and dedication to me and my work in pursuit of this goal. Were it not for them, this made this achievement would not have been possible. I would especially like to thank my thesis adviser, Dr. Carlos Bertulani for all his selfless help, care and unwavering support throughout my matriculation here at TAMUC, in addition to his introducing me to such an interesting problem. Thank you for everything that you’ve done  I couldn’t ask for a better adviser. Furthermore, I’d also like to thank the members of my thesis Committee: Dr. William Newton and Dr. Kent Montgomery for all their feedback and support in this endeavor, were it not for their suggestions and input, this document would have been incomplete. Of special note on that matter, I’d like to gratuitously thank Dr. Stefan Typel for his guidance through this process and greatly assisting me in the understanding of this material  it is immeasurably appreciated. And, on a separate note, I’m also especially grateful to the support and belief of Dr. Matt A. Wood, Dr. Mark Leising, and my undergraduate adviser Dr. Donald K. Walter  whose combined collected efforts have resulted in my reaching this point. I am greatly indebted to you all. ix Lastly, I wish to express my sincere gratitude, thanks and appreciation to everyone whose actions whether directly or indirectly has made this thesis a reality. I thank you all. x TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv CHAPTER 1. STATISTICS AND THERMODYNAMICS OF PARTICLES WITH FINITE SIZE . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1 Classical Excluded Volume Mechanism . . . . . . . . . . 2 Thermodynamic Potentials in Reduced Volume . . . . . . 11 Density Dependent Degeneracy Factors . . . . . . . . . . 18 2. EQUATION OF STATE WITH EXCLUDED VOLUME . . . 26 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 26 Classical Particles . . . . . . . . . . . . . . . . . . . . . . 27 Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . 29 Classical Gas with Excluded Volume . . . . . . . . 30 Series Expansion . . . . . . . . . . . . . . . . . . 32 Lowdensity Limit . . . . . . . . . . . . . . . . . . . . . . 32 Virial Expansion . . . . . . . . . . . . . . . . . . . 33 Classical Mechanics . . . . . . . . . . . . . . . . . 34 Quantum Mechanical Effects . . . . . . . . . . . . . . . . 35 xi CHAPTER 3. NUMERICAL STUDIES . . . . . . . . . . . . . . . . . . . . . 40 Single species with finite volume . . . . . . . . . . . . . . 40 Virial coefficients with excluded volume . . . . . . . . . . 41 Virial Equation of State . . . . . . . . . . . . . . . . . . 49 Mixture of Nucleons and Deuterons . . . . . . . . . . . . 55 Point Particles (Ri = 0 (i = n, p, d)) . . . . . . . 57 Finite Nonzero Radii Particles (Ri > 0 (i = n, p, d)) 61 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 66 4. APPLICATIONS TO BIG BANG NUCLEOSYNTHESIS . . . 67 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Appendix A. CALCULATION OF bnn . . . . . . . . . . . . . . . . . . 70 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 xii LIST OF TABLES TABLE 1. Particle masses and degeneracies. . . . . . . . . . . . . . . . . . 56 2. Particle Pairs Volumes . . . . . . . . . . . . . . . . . . . . . . . 61 xiii LIST OF FIGURES FIGURE 1. Neutronneutron scattering phaseshifts ( degrees) vs z (kRnn) within the limit [0; 20]. . . . . . . . . . . . . . . . . . . . . . . . 42 2. Neutronneutron phaseshift derivatives vs z = kRnn 8 z 2 [0; 20]. . . . . . . . . . . . . . . . . . . . . 43 3. Partial contributions for neutronneutron interaction vs energy (MeV) 8 E 2 [0MeV; 20MeV ]. . . . . . . . . . . . . . . . . . . 44 4. Quantum mechanical partial contributions for neutronneutron interaction vs temperature (K). . . . . . . . . . . . . . . . . . . . 46 5. Semilog plot of classical b(cl) nn vs temperature (K). . . . . . . . . 47 6. Semilog plot comparing classical and quantum mechanical partial contributions (b(qu) nn (T)) for NeutronNeutron Interaction vs Temperature (K). . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7. The second virial coefficient SFunction vs z = kRnn 8 z 2 [0; 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 8. The second virial coefficient SFunction tilde vs z = kRnn 8 z 2 [0; 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 9. Comparison of the second virial coefficient SFunction, and SFunction tilde vs z = kRnn 8 z 2 [0; 20]. . . . . . . . . . . . . 52 10. Semilog plot of the QM second virial coefficient tilde eb (qu) nn for neutronneutron interaction vs temperature (T). . . . . . . . . . 53 xiv 11. Semilog plot of the second virial coefficients b(cl) nn ; b(qu) nn ;eb (qu) nn for neutronneutron interaction vs temperature (K). . . . . . . . . . 54 12. nn Mass Fraction vs Baryon Density for point particles within T 2 [2; 10]MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 13. np Mass Fraction vs Baryon Density for point particles within T 2 [2; 10]MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 14. nd Mass Fraction vs Baryon Density for point particles within T 2 [2; 10]MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 15. nn Mass Fraction vs Baryon Density for volumeexcluded particles within T 2 [2; 10]MeV . . . . . . . . . . . . . . . . . . . . . . . 63 16. np Mass Fraction vs Baryon Density for volumeexcluded particles within T 2 [2; 10]MeV . . . . . . . . . . . . . . . . . . . . . . . 64 17. nd Mass Fraction vs Baryon Density for volumeexcluded particles within T 2 [2; 10]MeV . . . . . . . . . . . . . . . . . . . . . . . 65 1 Chapter 1 STATISTICS AND THERMODYNAMICS OF PARTICLES WITH FINITE SIZE Introduction If we consider a system composed of different particle species i with particle numbers Ni in a volume V at some temperature T, we can denote the particle densities as ni = Ni V ; (1.1) where we assume that every particle species i has a degeneracy factor gi and a chemical potential i for particles with rest mass mi. The energy of a particle is denoted by Ei and includes the rest mass. We will use canonical and grand canonical statistical ensembles and different particle statistics in the following. More specifically, according to Greiner et al. (1999), the canonical ensemble is suited for systems in a heat bath with a given T, V and N. On the other hand, the grand canonical  also called the macrocanonical ensemble  describes open systems where heat and particles are exchanged with the surroundings. 2 Classical Excluded Volume Mechanism If we also assume MaxwellBoltzmann statistics for noninteracting particles with nonrelativistic energy (in units of h = c = kB = 1); we have Ei(k) = k2 2mi + mi; (1.2) that depends on the particle momentum k. According to (Greiner et al., 1999, p. 187  189), MaxwellBoltzmann statistics assumes that for a given temperature, particles are distinguishable and that quantum mechanics plays a negligible role. These two assumptions while seemingly valid at first, create a number of problems depending on the particles and system. This indistinguishability is “corrected” by the inclusion of the Gibbs factor 1=N!  as noted in (Greiner et al., 1999, p. 133), which removes contradictions arising from the assumption that particles are distinguishable. And, depending on the particles and system in question, we will show that quantum statistics will have to be included where we will introduce Planck’s constant, h and other physical considerations. Given this information, it is more appropriate to consider the free energy of the system  which is energy that can be used to perform work (that includes the contributions of rest mass), in contrast to the total energy, E. More 3 specifically, if we begin with an ideal gas approximation with constant temperature and volume, the free energy in question is actually the Helmholtz energy. (Huang, 2001, p. 38) To begin our derivations, we use the definitions for the Helmholtz free energy F(T; V;N) = U TS; (1.3) dF = pdV SdT + X i idNi; (1.4) and the Gibbs free energy G(T; p;N) = U TS + pV; (1.5) dG = V dp SdT + X i idNi: (1.6) Physically, the meaning of the Helmholtz free energy is that of work done on the system plus heat loss from it, under constant pressure and entropy. The Gibbs free energy is the same, but for constant volume. In both cases, one adds the energy required to change the particle number. , the chemical potential, is the energy change by adding one particle to the system. And, to consider more realistic physical systems, if we apply the Gibbs correction factor (to “correct” contradictions arising from the initial assumption that for MaxwellBoltzmann statistics all particles are distinguishable), on an 4 ideal gas and consider phase space volume elements of size h3N to determine the mean number of states per energy interval, we can use the “absolute entropy” of an ideal gas, also known as the SackurTetrode equation, from which in (Greiner et al., 1999, p. 137  139), we can develop a more specialized form of the Helmholtz free energy to determine thermodynamic variables and potentials of interest S(U; V;N) = NkB " 5 2 + ln ( V N 4 mU 3N 3=2 )# : (1.7) Notably, we observe that h, Planck’s constant appears in the Sackur Tetrode equation  which comes from a “pseudoquantum mechanical” counting of the number of microstates where “the total energy is determined by the 3N quantum numbers of the occupied states” (Greiner et al., 1999, p. 135) U = E = h2 8mL2 X3N i=1 n2i (1.8) Then, solving for the internal energy, U U(S; V;N) = 3h2N5=3 4 mV 2=3 exp 2S 3NkB 5 3 ; (1.9) 5 the variation of the internal energy under constant pressure and temperature is given by (Greiner et al., 1999, p. 140) dU = TdS pdV + dN: (1.10) Hence, T = @U @S N;V = 2 3NkB U ! U = 3 2 NkBT; (1.11) p = @U @V S;N = 2 3V U ! U = pV = NkBT; (1.12) = @U @N S;V = U 5 3N 2S 3N2kB = kBTln ( N V h2 2 mkBT 3=2 ) : (1.13) Now, we can define a specific form the Helmholtz free energy using eqs. (1.7) and (1.11) F = U TS = NkBT " ln ( N V h2 2 mkBT 3=2 ) 1 # : (1.14) 6 The thermal wavelength  or the de Broglie wavelength for a gas in equilibrium at temperature, T as shown (Huang, 2010, p. 195)  is defined as (here we restore h and kB to explicitly show their dependence) i = s 2 h2 mikBT : (1.15) Then, setting h = kB = 1 (as done previously), we obtain i = r 2 miT : (1.16) Using eqs. (1.16) and (1.14) the free energy becomes F = U TS = NkBT ln N V i 3 1 : (1.17) The degeneracy increases the number of possible states in the system (similar to increasing the volume). Hence, we can write (note that the i subscript here means the particle in question  e.g. i = n) F(T; V;Ni) = X i Ni Tln Ni i 3 giV T + mi ; (1.18) Noting the presence of h in the thermal wavelength suggests that quantum mechanics is important to the calculation of the Helmholtz free energy. However, with this consideration we immediately violate one of the require 7 ments for MaxwellBoltzmann statistics (that quantum mechanical effects are considered to be minuscule such that they can be neglected altogether). This means that with the introduction of h we make the assertion that while MaxwellBoltzmann statistics are a good starting point to describe the physical constraints, we assume that quantum mechanical effects will have a part to play in the overall behavior and description of the system because of the nature of the particles in question; and, the scale of the overall interaction distances  as the average interparticle distance has to be compared to the deBroglie wavelength of the particles. Notably, when the thermal wavelength is much smaller than the inter particle distance, we recover the classical behavior. However, it should be noted that as this treatment is not entirely quantum mechanical derived, it is actually semiclassical. Additionally, we use  the deBroglie wavelength  in eq. (1.18), = h p2mEK ; (1.19) where EK is the average kinetic energy of the thermal particles. Indeed, using EK kBT, we recover eq. (1.15). 8 Furthermore, we note that the natural variables here are the temperature T, the volume V and the particle numbers Ni. Hence, we can then define the chemical potentials of the particles as i = @F @Ni T;V;Nj6=i : (1.20) Similarly, the entropy is given by eq. 3.37 of (Huang, 2001, p. 37) S = @F @T V;Ni : (1.21) And, in like fashion the pressure is obtained from P = @F @V T;Ni : (1.22) We begin determining the values of these potentials by using the relationship d dx x ln x = + ln x ; (1.23) hence, we can then find i = @F @Ni T;V;Nj6=i = T + T ln Ni i 3 giV T + mi = T ln Ni i 3 giV + mi: (1.24) 9 Similarly, for the entropy, S = @F @T V;Ni = X i Ni ln Ni i 3 giV 5 2 ; = X i Ni 5 2 ln Ni i 3 giV ; (1.25) where we have used the relationship d dT T ln T3=2 T = 5 2 + ln T3=2 : (1.26) Finally, for the pressure, we get P = @F @V T;Ni = X i Ni T V : (1.27) Using these solutions, we can then obtain the total internal energy (eq. 4.52) in Greiner et al. (1999). Using the relation between the internal energy and the free energy F = U TS X i iNi; (1.28) 10 we get U(S; V;Ni) = F(T; V;Ni) + TS = TS pV + G (1.29) = T X i Ni 5 2 ln Ni i 3 giV V X i Ni T V ! +T X i Ni ln Ni i 3 giV + X i Nimi; = T X i Ni 5 2 ln Ni i 3 giV T X i Ni +T X i Ni ln Ni i 3 giV + X i Nimi; = X i Ni 5 2 T T + T ln Ni i 3 giV T ln Ni i 3 giV + mi ; = X i Ni 3 2 T + mi ; (1.30) where we used mi= i T ln Ni i 3 giV : i = T ln Ni i 3 giV + mi: (1.31) Therefore, we can write the energy in natural variables as U(S; V;Ni) = X i Ni 3 2 T + i T ln Ni i 3 giV : (1.32) 11 And, using the grand canonical potential (a thermodynamic potential is a scalar function used to measure the state of a system), which describes the properties of an open system, i.e. a system with varying particle number, is defined as (T; V; i) = F(T; V;Ni) X i iNi = PV ; = V X i Ni T V ! ; = X i NiT = NT; (1.33) where the natural variables of the different thermodynamic potentials are given as arguments in parentheses. Sometimes it is also useful to introduce the free energy density f = F=V , the internal energy density u = U=V , the entropy density s = S=V and the grand canonical potential density ! = =V . Thermodynamic Potentials in Reduced Volume A finite volume Vi can be introduced for every particle species such that the available volume for the motion is not V any more but the reduced volume. 12 We will use the raised symbol to indicate quantities in the description with excluded volume mechanism and generalizations. eV = V X i NiVi: (1.34) Replacing V in eq. (1.18) by eV one obtains the classical excluded volume mechanism in the free energy e F(T; V;Ni) = X i Ni T ln Ni i 3 gi eV T + mi ; = X i Ni 2 6 4T ln 0 B@ Ni i 3 giV gi P j NjVj 1 CA T + mi 3 75 : (1.35) This approach is thermodynamically consistent since the natural variable of F is used in the formulation. Thus, using the developments in the section, we can then derive explicit expressions for the quantities i, S, P, E and in case of a gas with excluded volume. 13 The volume excluded chemical potential, e for each particle species i is e i = @F @Ni T;V;Nj6=i ; = T + T ln Ni i 3 gi eV T + mi + X k NkT gk P j ijVj gkV gk P j NjVj ; (1.36) = T ln Ni i 3 gi eV + mi + TVi X k Nk V P j NjVj : (1.37) Comparing this with the equation obtained for the chemical potential, i = T ln Ni i 3=giV +mi, we see that the chemical potential for a reduced volume is larger. Similarly, for the entropy eS = @F @T V;Ni ; = X i Ni ln Ni(2 =miT)3=2 gi eV 5 2 ; = X i Ni 5 2 ln Ni i 3 gi eV : (1.38) Comparing this with the original value obtained for the entropy S = X i Ni 5=2 ln Ni i 3=giV ; we see that the entropy is smaller for a reduced potential. 14 Finally, for the pressure, e P = @F @V T;Ni = T P i Ni V P j NjVj : (1.39) Comparing this against the value obtained previously for the pressure P = X i Ni T V ; we see that the pressure for reduced volume is higher. While seemingly unusual at first glance, this result is expected as we’re working in a smaller volume with the same particle numbers, and system constraints. More specifically, since the Temperature has remained the same and we have a smaller volume, the pressure has to increase. 15 Combining all of these solutions, we can hence derive the total internal energy for a reduced volume eU(eS; V;N) = e F(T; V;Ni) + T eS; = T eS epV + X i e iNi; = T X i Ni 5 2 ln Ni i 3 gi eV V T P i Ni V P j NjVj +T X i Ni ln Ni i 3 gi eV + X i Nimi +T X i NiVi X i Ni V P j NjVj ; = 5 2 T X i Ni V T P i Ni V P j NjVj + T X j NjVj X i Ni V P j NjVj + X i Nimi; = 5 2 T X i Ni T X i Ni 2 64 V V P j NjVj P j NjVj V P j NjVj 3 75 + X i Nimi; = X i Ni 3 2 T + mi : (1.40) 16 Noting that e i = T ln Ni i 3 gi eV + mi + TVi X k Nk V P j NjVj : (1.41) = T 2 64 ln Ni i 3 gi eV + mi T + Vi X k Nk V P j NjVj 3 75 ; we get mi = e i T 2 64 ln Ni i 3 gi eV + Vi X k Nk V P j NjVj 3 75: Hence, we can write the reduced volume energy as eU(eS; V;N) = X i Ni 2 64 3 2 T + e i T 0 B@ ln Ni i 3 gi eV + Vi X k Nk V P j NjVj 1 CA 3 75 : (1.42) Given this value, and comparing it against the value obtained previously for the internal energy, E = P i Ni (3T=2 + mi), we observe that the total internal energy is identical to the internal energy in the regularly case, the ideal gas. 17 Similarly, for the grand canonical potential for a reduced volume e (T; V; ) = e F(T; V;Ni) X i e iN; = e PV; = TV P i Ni V P j NjVj ; = T P i Ni 1 P j njVj : (1.43) Since e i = T ln Ni i 3 gi eV + mi + TVi X k Nk V P j NjVj ; we get T P i Ni 1 P j njVj = e i T ln Ni i 3 gi eV mi; or T P i Ni 1 P j njVj = T ln Ni i 3 gi eV + mi e i: (1.44) 18 Then we can write e as a function of natural variables e (T; V; e ) = e F(T; V;Ni) X i e iN = T P i Ni 1 P j njVj = T ln Ni i 3 gi eV + mi e i: (1.45) We can conclude that this potential is larger (in magnitude) for a given particle species within a reduced volume. Density Dependent Degeneracy Factors Using the definition for the reduced volume and degeneracy, we can define the effective degeneracy as a function of the volume, V as gi eV = gi V X i NiVi ! = gi 1 X i niVi ! V = egiV; (1.46) with effective degeneracy factors egi = gi 1 X i niVi ! ; (1.47) that depend on the particle densities ni. 19 We define the grand canonical potential as e (T; V; e ) = X i e i +W = T ln Ni i 3 gi eV + mi e i; (1.48) where the individual particle contributions can be written as e i = TegiV i Z d3k (2 )3 ln " 1 + iexp eUi e i T !# ; (1.49) with an additional contribution W. Here we introduce i as we wish to generalize the particle contributions with consideration of different statistics  i.e. FermiDirac and BoseEinstein statistics in which i = 1 for fermions and i = 1 for bosons as discussed in (Carroll, 2007, p. 43), a boson is an integerspin forcecarrying particle, as opposed to a matter particle (fermion); and, unlike fermions, bosons can be accommodated to the same state without limit  e.g. photons, gluons, gravitons, weak bosons, and the Higgs boson. Also we define the singleparticle energies as eUi(k) = k2 2mi + mi + Ui: (1.50) 20 Combining this with e i, we obtain eUi(k) e i = k2 2mi + mi + Ui 2 6 4T ln Ni i 3 gi eV + mi + TVi X k Nk V P j NjVj 3 75 ; = k2 2mi + Ui 2 6 4T ln Ni i 3 gi eV + TVi X k Nk V P j NjVj 3 75 : Since gi eV = gi (V P i NiVi) = gi (1 P i niVi) V = egiV , we obtain eUi(k) e i = k2 2mi + Ui T ln Ni i 3 gi eV TVi X i Ni gj eV ; = k2 2mi + Ui T " ln Ni i 3 gi eV Vi X i Ni gj eV # : That is eUi(k) e i T = k2 2miT + Ui T ln Ni i 3 gi eV + Vi X i Ni gj eV ; = k2 2miT + Ui T + ln gi eV Ni i 3 ! + Vi X i Ni gj eV : 21 Hence, we can calculate the particle number, Ni using @ @x ln a exp x + y b + 1 = a exp x+y b (a exp x+y b + 1) b = a b exp x+y b a exp x+y b + 1 ; = a b 1 a + exp (x+y) b : (1.51) Since e (T; V; e ) = T ln Ni i 3 gi eV + mi e i; then @e @e i T;V;e j!=i = 1: 22 Hence Ni != @e @e i T;V;e j!=i ; != + TegiV i Z d3k (2 )3 i T 1 i + exp ( eUie i) T @W @ e i ; = TegiV i i T Z d3k (2 )3 1 i + exp eUie i T @W @ e i ; = egiV Z d3k (2 )3 1 i + exp eUie i T @W @ e i ; = egiV Z d3k (2 )3 " i + exp eUi e i T !# 1 @W @ e i ; = egiV Z d3k (2 )3 " i + exp k2 2miT + Ui T + ln gi eV Ni i 3 ! + Vi X i Ni gj eV !# 1 @W @ e i : (1.52) Thus Ni != @e @e i T;V;e j!=i ; != egiV Z d3k (2 )3 2 6 4 i + exp 0 B@ k2 2miT + Ui T + Vi X i Ni gj eV + 2 4 1 ln gi eV Ni i 3 3 5 1 1 CA 3 75 1 @W @ e i ; 23 which contains the "rearrangement" potentials Ui, defined by the relation between Ui and W as shown in the above relationship, in addition to the following Ui = X j @egj @Ni e i egi : (1.53) Also noting that we can calculate @W=@ e i, @W @ e i = egiV Z d3k (2 )3 " i + exp eUi e i T !# 1 1: Now, giV = gi V X i NiVi ! = gi 1 X i niVi ! V = egiV; (1.54) egi = gi V V X i NiVi ! = gi 1 X i niVi ! ; (1.55) Since Ni = niV @egj @Ni = 1 V @egj @ni = gj P i Vi V : (1.56) 24 Using this in Ui Ui = X j @egj @Ni e i egi = 1 V @egj @ni e i egi ; = gj P i Vi V TV i Z d3k (2 )3 ln " 1 + iexp eUi e i T !#! ; = X i TgjVi i Z d3k (2 )3 ln " 1 + iexp eUi e i T !# : (1.57) and using Ni = niV , we can now calculate the work, W = X i NiUi; = X i TgiNiVi i Z d3k (2 )3 ln " 1 + iexp eUi e i T !# ; = X i TginiViV i Z d3k (2 )3 ln " 1 + iexp eUi e i T !# : (1.58) Combining these results, we get @egj @Ni = 1 V @egj @ni = gj P i Vi V ; (1.59) and Ni gj gj P i Vi V = P i NiVi V : (1.60) 25 Since, Ni = niV , one has P i NiVi V = P i niViV V = X i niVi: (1.61) Therefore, W = X i NiUi = X i TgjniV Vi i Z d3k (2 )3 ln " 1 + iexp eUi e i T !# : (1.62) We can combine all of these into a general equation W = X i X j niV egj @egj @Ni e j = X ij ij niV egj @egj @Ni e j : (1.63) Thus, we have developed a prescription to include the density dependence of the degeneracy factors on the excluded volumes of the clusters. 26 Chapter 2 EQUATION OF STATE WITH EXCLUDED VOLUME Introduction The effects of the finite size of particles on the equation of state of a gas can be described with the help of the excludedvolume mechanism. It can also be used in order to simulate the suppression of a particular particle species, e.g. nuclei (clusters) in a mixture with nucleons, at high densities. The excludedvolume mechanism can be formulated most simply by starting with the free energy F(T; V; fNig) of a classical ideal gas of particles i with singleparticle numbers Ni in a volume V at temperature T. Instead of the total volume V of the system it is assumed that for a particle i only a reduced volume Vi = V i < V; (2.1) is available since part of the whole volume is occupied already by other particles. The functions (ni) depend on the singleparticle densities ni = Ni=V . Applying a Legendre transformation (T; V; f ig) = F X i iNi = pV; (2.2) 27 with the chemical potentials (including rest masses mi of the particles) i = @F @Ni T;V; j6=i ; (2.3) where the grandcanonical potential depends on the natural variables T, V , and the set f ig. It is better suited for generalizations of the model than the free energy F. In the following, we use the usual system of units in nuclear physics with h = c = kB = 1. Thus these factors do not appear in the formulas explicitly but have to be considered in the conversion of quantities. Additionally, to more easily distinguish and compare the potentials derived from different physical constraints and arguments, we use labels of the form (id), (eff), (vi) , and (ex) as opposed to the use and absence of eX , (where X is the potential of interest), as in the previous chapter. Classical Particles Let us consider first classical particles with MaxwellBoltzmann statistics and nonrelativistic kinematics without explicit interactions. In this case the grandcanonical potential is given by = X i ( i NiUi) ; (2.4) 28 with singleparticle contributions i = Tg(e ) i V Z d3k (2 )3 exp Ei(k) i T ; (2.5) and rearrangement potentials Ui = X j @g(e ) j @Ni j g(e ) j ; (2.6) that contain the densitydependent effective degeneracy factors g(e ) i = gi i; (2.7) and the energies (including rest masses mi) Ei(k) = k2 2mi + mi + Ui: (2.8) of the particles. The usual degeneracy factors are denoted by gi. They are constants. The rearrangement potentials Ui are essential in order to obtain a thermodynamically consistent model. 29 Ideal Gas For the case of an ideal gas with pointlike particles, the functions i are just constants and equal to one such that the effective degeneracy factors g(e ) i are identical to the usual degeneracy factors gi of the particles and the rearrangement potentials Ui vanish. Then one finds explicitly as shown in (Typel, Röpkec, et al., 2010, p. 3) (id)(T; V; f (id) i g) = TV X i gi 3i exp (id) i mi T ! = p(id)V; (2.9) and Ni = @ (id) @ (id) i T;V; j6=i = V gi 3i exp (id) i mi T ! ; (2.10) with the thermal wavelengths i = r 2 miT ; (2.11) such that the pressure of an ideal gas is given by p(id) = T V X i Ni ; (2.12) i.e. the wellknown relation for a mixture of classical ideal gases. 30 Classical Gas with Excluded Volume Now we treat the case with explicit excludedvolume effects and assume that i(fnjg) = 1 X j vijnj ; (2.13) with constant volumes vij . Sometimes these functions are defined as i(fnjg) = 1 X j njvj ; (2.14) with volumes vj of particles j such that the functions i are identical for all i. However, this leads to problems when the comparison to the virial equation of state is made, see below. Their relation to the volumes of the individual particles will be specified later. One finds the singleparticle contributions (ex) i = TV g(e ) i 3i exp (ex) i mi Ui T ! ; (2.15) and the rearrangement potentials Ui = X j @ j @ni (ex) j V j = X j vij (ex) j V j : (2.16) 31 All of these combined lead to the total excluded grandcanonical potential (2.17) (ex)(T; V; f ig) = X i (ex) i + ni X j vij (ex) j j ! ; = X i X j ij j j (ex) i + ni X j vij (ex) j j ! ; = X j j j (ex) j + X i ni X j vij (ex) j j = X j (ex) j j ; = T X i gi 3i V exp (ex) i mi Ui T ! ; = p(ex)V; and the singleparticle densities ni = 1 V @ (ex) @ (ex) i = g(e ) i 3i exp (ex) i mi Ui T ! : (2.18) Thus we find the relation p(ex) = T X i ni i : (2.19) Since i < 1, the pressure p(ex) with excludedvolume effects is larger than that of an ideal gas p(id) with the same particle number densities ni. 32 Series Expansion The grandcanonical potential (ex) can be expanded in powers of fugacities as in (Typel, Röpkec, et al., 2010, p. 3) zi = exp (ex) i mi T ! ; (2.20) by using the results for the rearrangement potentials Ui, and using a similar approach as in eq. (2.17). We find (2.21) (ex)(T; V; f ig) = T X i gi 3i V zi exp Ui T ; = T X i gi 3i V zi " 1 Ui T + 1 2 Ui T 2 : : : # ; T X i gi 3i V zi 1 + 1 T X j vij (ex) j V j + : : : ! ; TV X i gi 3i zi 1 X j vij gj 3j zj + : : : ! ; = TV X i gi 3i zi X ij vij gigj 3i 3j zizj + : : : ! ; up to second order in the fugacities as also shown in Typel, Röpkec, et al. (2010). Lowdensity Limit So far we have considered the general formulation of the excludedvolume mechanism. However, the volumes vij are not fixed yet. In order to relate them 33 to the volumes of the individual particles, the exact lowdensity limit of the equation of state can be used to define them. Virial Expansion At finite temperatures, the lowdensity limit is given by the virial equation of state. It is an expansion of the grandcanonical potential (vi)(T; V; f (vi) i g) = TV X i bi 3i zi + X ij bij 3=2 i 3=2 j zizj + : : : ! ; (2.22) in powers of the fugacities zi = exp (vi) i mi T ! ; (2.23) with dimensionless virial coefficients bi, bij , . . . . The expansion is valid only for zi 1 or ni 3i 1. The first virial coefficients bi = gi are just the standard degeneracy factors. 34 Classical Mechanics The second virial coefficients in classical mechanics are given by b(cl) ij (T) = 1 + ij 2 gigj 3=2 i 3=2 j Z d3r exp Vij T 1 ; (2.24) depending on the potential Vij between the particles i and j. For incompressible spherical particles with radii Ri and volumes Vi = 4 R3 i =3 it is given by the hardsphere potential Vij(r) = 8>< >: 1 if r Rij ; 0 if r > Rij ; (2.25) with the sum of the radii Rij = Ri + Rj . Then we have b(cl) ij (T) = 1 + ij 2 gigj 3=2 i 3=2 j 4 Z Rij 0 dr r2 [1] = 1 + ij 2 gigj 3=2 i 3=2 j 4 3 R3 ij : (2.26) Comparing with the result for the series expansion of (ex) we can identify vij = 1 + ij 2 4 3 R3 ij ; (2.27) if we set (ex) i = (cl). In particular we find vii = 4 3 (2Ri)3 = 8Vi; (2.28) 35 because the centers of two equal particles cannot become closer than twice their radius. Quantum Mechanical Effects The classical expression for the second virial coefficient does not take into account quantum effects. Using the formalism described in both Uhlenbeck & Beth (1936) and Beth & Uhlenbeck (1937), the quantummechanical second virial coefficient (without additional quantumstatistical corrections for Fermions or Bosons) is given by b(qu) ij (T) = 1 + ij 2 3=2 i 3=2 j 3 ij Z dE exp E T Dij(E); (2.29) with ij = s 2 (mi + mj)T ; (2.30) and the kinetic energy of the relative motion E = k2 2 ij ; (2.31) that contains the reduced mass ij = mimj mi + mj : (2.32) 36 From these we can then determine the “level density” difference as shown in Typel, Röpkec, et al. (2010) Dij(E) = X k g(ij) k (E E(ij) k ) + X l g(ij) l d (ij) l dE ; (2.33) with g(ij) l = (2l + 1)gigj ; (2.34) which contains contributions from twoparticle bound states k and scattering states l in partialwave expansion (neglecting spindependent potentials). Notably, we don’t have bound states for the hardsphere potential but only scattering states. Hence, we need the scattering phase shifts (ij) l in all partial waves l as a function of the relative momentum k. Using that, we can then calculate the second virial coefficients b(qu) ij (T) = X l b(l) ij (T); (2.35) with b(l) ij (T) = 1 + ij 2 (2l + 1)gigj 3=2 i 3=2 j 3 ij Z 1 0 dk exp k2 2 ijT d (ij) l dk : (2.36) 37 To include quantum effects we need to solve the timeindependent Schrödinger equation Hij (r) = E (r); (2.37) with the Hamiltonian as Hij = p2 2 ij + Vij ; (2.38) to obtain the scattering wave function for the relative motion at energy E = k2=(2 ij) which can be written as ij(r) = 4 kr X lm ilu(ij) l (r)Ylm(^k)Y lm(^r); (2.39) with spherical harmonics Ylm that depend on the direction of the coordinate and momentum vectors r and k, respectively. The radial wave functions u(ij) l (r) have to fulfill the boundary conditions in each partial wave l. In case of the hardsphere potential, the radial wave functions for r Rij have the form u(ij) l (r) = 1 + S(ij) l 2 h Fl(kr) + K(ij) l Gl(kr) i ; (2.40) with the Smatrix elements are given in terms of the phase shifts by means of S(ij) l = exp 2i (ij) l ; (2.41) 38 and the Kmatrix elements are K(ij) l = 1 i S(ij) l 1 S(ij) l + 1 = tan (ij) l : (2.42) Of these, the regular and irregular scattering wave functions can be expressed with the help of spherical Bessel functions Fl(z) = zjl(z); (2.43) Gl(z) = zyl(z); (2.44) with the asymptotics Fl(z) ! sin z l 2 ; (2.45) Gl(z) ! cos z l 2 ; (2.46) for z ! 1. On the other hand, we have Fl(z) ! zl (2l + 1)! ! ; (2.47) Gl(z) ! (2l 1)! ! zl+1 ; (2.48) for z ! 0. For r Rij the radial wave function should vanish. This condition ul(Rij) = 0; (2.49) 39 leads to tan (ij) l = Fl(kRij) Gl(kRij) = jl(kRij) yl(kRij) : (2.50) Hence, we can calculate (ij) l for all k and l. For l = 0 we explicitly have tan (ij) 0 = sin(kRij) cos(kRij) ; (2.51) or (ij) 0 = kRij : (2.52) With help of the equations developed so far for the statistics and thermodynamics of particles with finite size, we can pave the road to understand the effects of excluded volume mechanism on the equation of state for a gas of nucleons and nuclei. Our analysis is not complete, but we show the effects in a few numerical exercises in the next chapter. 40 Chapter 3 NUMERICAL STUDIES In order to see the differences between the various equations of states and their approximations, numerical calculations have to be performed. We will study different systems in the following. Single species with finite volume As a first application we consider a single particle species i. In order to have realistic numbers we choose neutrons (i = n) as an example with rest mass mn = 939:565 MeV/c2 and degeneracy factor gn = 2. We set the radius to Rn = 0:5 fm, Rnn = 1:0 fm and the volume vnn = 1 2 4 3 R3 nn 2:094 fm3 : (3.1) Then the function n = 1 vnnnn ; (3.2) which appears in the effective degeneracy factor g(e ) n = gn n ; (3.3) with the neutron density nn becomes zero at the maximum density n(max) n = 1 vnn 0:4775 fm3 : (3.4) 41 Thus the equation of state with the excluded volume can be calculated only for neutron densities 0 < nn < n(max) n . Virial coefficients with excluded volume Recalling the virialized coefficients b(l) ij are functions of the partial waves, then as a first step in calculating the neutronneutron scattering phases shifts we calculate from eq. (2.50) tan (ij) l = Fl(kRij) Gl(kRij) = jl(kRij) yl(kRij) ; (ij) l = arctan jl(kRij) yl(kRij) (3.5) for the specific case where i = j = n is given by (nn) l = arctan jl(kRnn) yl(kRnn) (3.6) as a function of k for partial waves l = 0; 1; 2; : : : This is shown in Figure (1), for the interval [0 MeV/c; 200 MeV/c]. Then, using (3.6), we can also find the derivatives, d (nn) l =dk for the same interval [0 MeV/c; 200 MeV/c] as shown in Figure (2). All of this information can then be used to find the partial contributions b(l) nn(T) = (2l + 1)g2n 3 n 3 nn Z 1 0 dk exp k2 2 nnT d (ij) l dk (3.7) 42 0 2 4 6 8 10 12 14 16 18 20 kRnn 0 90 180 270 360 450 540 630 720 810 900 990 1080 1170 Degrees NeutronNeutron Scattering Phase Shifts vs z (kRnn) l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8 l = 9 Figure 1. Neutronneutron scattering phaseshifts ( degrees) vs z (kRnn) within the limit [0; 20]. 43 0 2 4 6 8 10 12 14 16 18 20 kRnn 0 5 10 15 20 25 30 35 40 45 50 55 60 Degrees NeutronNeutron Scattering Phase Shifts and Derivatives vs z (kRnn) l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8 l = 9 Figure 2. Neutronneutron phaseshift derivatives vs z = kRnn 8 z 2 [0; 20]. 44 0 2 4 6 8 10 12 14 16 18 20 E (MeV) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 b(l) nn Partial Contributions (b(l) nn) for NeutronNeutron Interaction vs E (MeV) l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8 l = 9 Figure 3. Partial contributions for neutronneutron interaction vs energy (MeV) 8 E 2 [0MeV; 20MeV ]. 45 to the second virial coefficient bnn(T) as a function of T inside the interval [0 MeV; 20 MeV] for l = 0; 1; 2; : : : shown in Figure (3). Using (3.7), we can then calculate the total second virial coefficient, which is given by as an infinite sum b(qu) nn (T) = 1X l=0 b(l) nn(T) : (3.8) However, while the total second virial coefficient (in the quantum mechanics case) can be analytically calculated as an infinite sum (for l > 0), the function converges relatively quickly and only 6 lvalues are required to achieve convergence (within 3.4 parts per million), as shown in Figure (4). Similarly, we can calculate the classical second virial coefficient as a function of the temperature, T which is given by the following equation (and as illustrated as shown in Figure (5)) b(cl) nn (T) = g2n 3 n 4 3 R3 nn (3.9) Finally, we can compare the classical second virial coefficient with the quantum mechanical version both as functions of T, and we obtain Figure (6) which definitely shows a disparity between both. This makes sense since a classical billiard ball treatment doesn’t take into consideration quantum effects at short distances. 46 108 109 1010 1011 T (K) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 b(qu) nn QM Partial Contributions for NeutronNeutron Interaction vs T (K) Pl=0 l=0 Pl=1 l=0 Pl=2 l=0 Pl=3 l=0 Pl=4 l=0 Pl=5 l=0 Pl=1 l=0 Figure 4. Quantum mechanical partial contributions for neutronneutron interaction vs temperature (K). 47 109 1010 1011 T (K) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 b(cl) nn Second Virial Coefficient (b(cl) nn ) vs T (K) b(cl) nn Figure 5. Semilog plot of classical b(cl) nn vs temperature (K). 48 108 109 1010 1011 T (K) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 bnn QM Partial Contributions for NeutronNeutron Interaction vs T (K) Pl=0 l=0 Pl=1 l=0 Pl=2 l=0 Pl=3 l=0 Pl=4 l=0 Pl=5 l=0 Pl=1 l=0 b(cl) nn Figure 6. Semilog plot comparing classical and quantum mechanical partial contributions (b(qu) nn (T)) for NeutronNeutron Interaction vs Temperature (K). 49 To examine this even further, we then calculate the function, called the SFunction S(z) = (3.10) 1 z2 X l=0 2l + 1 [jl(z)2] + [jl(z)2] : Which, for the interval, z 2 [0; 20] is shown in Figure (7). Then, defining an approximation to this function (i.e. a fitting function) as Se(z) = 1 (3.11) 3 2 z 2 3 z2: Next, after comparing both of these together in Figure (9) we observe that eS is a good (and fast) approximation to the general Sfunction which involves spherical Bessel functions. Finally, we can compare all three second virial coefficients b(cl) nn (T), b(qu) nn (T), eb (qu) nn (T) as shown in Figure (10) and Figure (11) which illustrate that the b(qu) nn (T),eb (qu) nn (T) coefficients deviate from the classical case. Virial Equation of State With the known second virial coefficients b(qu) nn (T) and b(cl) nn (T) we can calculate the pressure p(vi;qu) = (vi;qu)(T; V; (vi;qu) n ) V = T 3 n gnz(vi;qm) n + b(qu) nn z(vi;qm) n 2 (3.12) and p(vi;cl) = (vi;cl)(T; V; (vi;cl) n ) V = T 3 n gnz(vi;cl) n + b(cl) nn z(vi;cl) n 2 (3.13) 50 0 2 4 6 8 10 12 14 16 18 20 z 0 50 100 150 200 250 300 S(z) SFunction vs z S(z) Figure 7. The second virial coefficient SFunction vs z = kRnn 8 z 2 [0; 20]. 51 0 2 4 6 8 10 12 14 16 18 20 z 0 50 100 150 200 250 300 eS(z) SFunction Tilde vs z eS(z) Figure 8. The second virial coefficient SFunction tilde vs z = kRnn 8 z 2 [0; 20]. 52 0 2 4 6 8 10 12 14 16 18 20 z 0 50 100 150 200 250 300 S(z)keS(z) SFunction and SFunction Tilde vs z S(z) eS(z) Figure 9. Comparison of the second virial coefficient SFunction, and SFunction tilde vs z = kRnn 8 z 2 [0; 20]. 53 108 109 1010 1011 T (K) 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 b(qu) nn QM Second Virial Coefficient (eb (qu) ij ) for NeutronNeutron Interaction vs T (K) eb (qu) ij (T) Figure 10. Semilog plot of the QM second virial coefficient tilde eb (qu) nn for neutronneutron interaction vs temperature (T). 54 1010 1011 1012 1013 1014 T (K) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 b(l) nn Second Virial Coefficients b(cl) nn , b(qu) nn ,eb (qu) nn for NeutronNeutron Interaction vs T (K) Pl=1 l=0 b(cl) nn eb (qu) nn (T) Figure 11. Semilog plot of the second virial coefficients b(cl) nn ; b(qu) nn ;eb (qu) nn for neutronneutron interaction vs temperature (K). 55 in the two approximations for the virial equation of state. In order to obtain the fugacities z(vi;qm) n = exp (vi;qm) n mn T ! (3.14) and z(vi;cl) n = exp (vi;cl) n mn T ! (3.15) we consider the neutron densities nn = @p(vi;qu) @ n T = 1 3 n gnz(vi;qm) n + 2b(qu) nn z(vi;qm) n 2 (3.16) and nn = @p(vi;cl) @ n T = 1 3 n gnz(vi;cl) n + 2b(cl) nn z(vi;cl) n 2 : (3.17) Using these two (quadratic) equations, we can find the fugacities for given temperature T and density nn. In a second step, the pressure can be calculated. Mixture of Nucleons and Deuterons To observe the effect of volumeexclusion, we consider a classical mixture of neutrons, protons and deuterons in chemical and thermal equilibrium. From these considerations we can calculate the particle number densities from ni = gi 3i exp (id) i mi T ! (3.18) 56 where we use the following constraints Table 1 Particle masses and degeneracies. Particle Mass (MeV=c2) Degeneracy p mp = 939.565 gp = 2 n mn = 938.272 gn = 2 d md = 1875.612 gd = 3 and where the mass of Deuteron is calculated as: md = mn + mp Bd = 939:565MeV=c2 + 938:272MeV=c2 2:2225MeV=c2 = 1875:612MeV=c2, (Bd is the binding energy of the Deuteron nucleus (2:2225MeV=c2)). As there are two independent densities, we can call this the total neutron number density as shown in (Typel, Röpke, et al., 2010, p. 3) n(tot) n = nn + nd: (3.19) Similarly, for the proton number density (Typel, Röpke, et al., 2010, p. 3) n(tot) p = np + nd: (3.20) However, it is more convenient to introduce the total baryon number density nB, which is a sum of the two nB = n(tot) n + n(tot) p = nn + np + 2nd: (3.21) 57 Next, we also consider the isospin asymmetry which is defined as = n(tot) n n(tot) p nB = nn np nB (3.22) However, while the total neutron and proton number densities are independent, the individual number densities of the neutrons, protons and deuterons are not  as they are constrained by the chemical equilibrium such that n + p = d: (3.23) Point Particles (Ri = 0 (i = n, p, d)) To further understand this, we first consider a mixture of point particles of neutrons, protons and deuterons (i = n; p; d) such that Rn = Rp = Rd = 0 fm, and with = 0 as shown in Figures (12), (13), and (14) for neutrons, protons and deuterons respectively. From Figures (12), (13), and (14) we can assert that the deuteron mass fraction decreases with rising temperature, and increases slowly with increasing temperature and baryon density. The opposite effect is observed with the symmetric mixture of protons and neutrons, for increasing temperature and baryon density. 58 0.00 0.05 0.10 0.15 0.20 nb(fm−3) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Xn = nn nB Neutron Mass Fractions vs Baryon Density for T 2 [2,10] MeV T = 2 MeV T = 4 MeV T = 6 MeV T = 8 MeV T = 10 MeV Figure 12. nn Mass Fraction vs Baryon Density for point particles within T 2 [2; 10]MeV . 59 0.00 0.05 0.10 0.15 0.20 nb(fm−3) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Xp = np nB Proton Mass Fractions vs Baryon Density for T 2 [2,10] MeV T = 2 MeV T = 4 MeV T = 6 MeV T = 8 MeV T = 10 MeV Figure 13. np Mass Fraction vs Baryon Density for point particles within T 2 [2; 10]MeV . 60 0.00 0.05 0.10 0.15 0.20 nb(fm−3) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Xd = 2nd nB Deuteron Mass Fractions vs Baryon Density for T 2 [2,10] MeV T = 2 MeV T = 4 MeV T = 6 MeV T = 8 MeV T = 10 MeV Figure 14. nd Mass Fraction vs Baryon Density for point particles within T 2 [2; 10]MeV . 61 Finite Nonzero Radii Particles (Ri > 0 (i = n, p, d)) In this case, we assume that deuteron has a finite radius Rd = 1:2fm Table 2 Particle Pairs Volumes Particle Pair Volume (fm3) vnn = vpp 2.094 vpn = vnp 2.094 vdn = vnd 7.238 vpd = vdp 7.238 vdd 57.906 where vdn = vnd = vdp = vpd = 4 3 R3d = 7.238 fm3; (3.24) and vdd 4 3 (2Rd)3 = 57.906 fm3: (3.25) Using these terms we can introduce the rearrangement potentials into the number densities Ui = X j vij (ex) j V j ; (3.26) 62 hence Ui = X j vij (ex) j V j (ex) i = TV g(eff) i i 3i exp (ex) i mi Ui T ! : (3.27) Combining these last two equations, we express the potentials more simply as Ui = T X j vij gj 3i exp (ex) i mi Ui T ! = T X j vij nj V j : (3.28) And, using this, we can finally include volumeexclusion in our calculation of the particle densities ni = g(eff) i 3i exp (ex) i mi Ui T ! (3.29) = gi i 3i exp (ex) i mi Ui T ! ; (3.30) the results of which are shown in figures (15), (16), and (17). Comparing these to Figures (12), (13), and (14) for the point particles, we have a vastly different scenario. Having considered volume exclusion, Figures (15), (16), and (17) suggest that we get a decrease in deuteron mass fraction as the baryon density increases. And, as temperature increases deuteron mass fraction decreases. As in the previous case for point particles, the trend of neutrons and protons against the deuterons is reversed. But this time, the proton and neutron 63 0.00 0.05 0.10 0.15 0.20 nb(fm−3) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Xn = nn nB ExcludedVolume Neutron Mass Fractions vs Baryon Density for T 2 [2,10] MeV T = 2 MeV T = 4 MeV T = 6 MeV T = 8 MeV T = 10 MeV Figure 15. nn Mass Fraction vs Baryon Density for volumeexcluded particles within T 2 [2; 10]MeV . 64 0.00 0.05 0.10 0.15 0.20 nb(fm−3) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Xp = np nB ExcludedVolume Proton Mass Fractions vs Baryon Density for T 2 [2,10] MeV T = 2 MeV T = 4 MeV T = 6 MeV T = 8 MeV T = 10 MeV Figure 16. np Mass Fraction vs Baryon Density for volumeexcluded particles within T 2 [2; 10]MeV . 65 0.00 0.05 0.10 0.15 0.20 nb(fm−3) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Xd = 2nd nB ExcludedVolume Deuteron Mass Fractions vs Baryon Density for T 2 [2,10] MeV T = 2 MeV T = 4 MeV T = 6 MeV T = 8 MeV T = 10 MeV Figure 17. nd Mass Fraction vs Baryon Density for volumeexcluded particles within T 2 [2; 10]MeV . 66 mass fractions increase with baryon density, and also for increasing temperature. Future Work Hence, while the results presented in this chapter are for the special case of a symmetric mixture of nucleons (i.e. = 0) for three particle species, neutrons, protons and deuterons, this methodology should be applied and compared against actual data for different nuclei species to confirm its validity. Furthermore, more realistic consideration involving particle number conservation, nonconserved chemical potential, and energy loss or gain, and inclusion of relativistic physical situations should also be considered. However, while such an approach is beyond the scope of this study, the arguments and approach made herein can be extended and applied in a more general sense beyond a finite mixture of gases of simple nucleons  as we have shown here. 67 Chapter 4 APPLICATIONS TO BIG BANG NUCLEOSYNTHESIS We also note that this methodology  with the inclusion of relativity  can be applied to understanding Big Bang Nucleosynthesis. Additional but unrelated work to the one shown herein, was done with a collaboration with Prof. Spitaleri and his nuclear astrophysics group in Catania, Sicily as shown in Pizzone et al. (2014), the abstract of which is cited below. “Nuclear reaction rates are among the most important input for understanding the primordial nucleosynthesis and therefore for a quantitative description of the early Universe. An uptodate compilation of direct cross sections of 2H(d,p) 3H, 2H(d,n) 3He, 7Li(p, ) 4He and 3He(d,p) 4He reactions is given. These are among the most uncertain cross sections used and input for Big Bang nucleosynthesis calculations. Their measurements through the Trojan Horse Method (THM) are also reviewed and compared with direct data. The reaction rates and the corresponding recommended errors in this work were used as input for primordial nucleosynthesis calculations to evaluate their impact on the 2H, 3;4He and 7Li primordial abundances, which are then compared with observations.” 68 APPENDICES 69 APPENDIX A CALCULATION OF bnn 70 CALCULATION OF bnn From eq. (2.22) p(vi;cl) = (vi;cl)(T; V; (vi;cl) n ) V = T 3 n gnz(vi;cl) n + b(cl) nn z(vi;cl) n 2 ; (4.1) and p(vi;qu) = (vi;qu)(T; V; (vi;qu) n ) V = T 3 n gnz(vi;qu) n + b(qu) nn z(vi;qu) n 2 ; (4.2) we have nn = @p(vi;cl) @ n = 1 3 n gnz(vi;cl) n + 2b(cl) nn z(vi;cl) n 2 ; (4.3) and nn = @p(vi;qu) @ n = 1 3 n gnz(vi;qu) n + 2b(qu) nn z(vi;qu) n 2 : (4.4) Simplifying this, we arrive at the relations nn = 1 3 n gnz(vi) n + 2bnn z(vi) n 2 ; (4.5) nn 3 n = gnz(vi) n + 2bnn z(vi) n 2 ; (4.6) 71 leading to z(vi) n 2 + z(vi) n gn 2bnn nn 3 n 2bnn = 0; (4.7) which has the solution: z(vi) n = 1 4bnn p g2n + 8bnnnn 3 n gn : (4.8) Hence, inserting into eqs. (4.1) and (4.2) respectively p(vi) = T 3 n gnz(vi) n + bnn z(vi) n 2 ; = T 3 n gnz(vi) n + bnn nn 3 n 2bnn z(vi) n gn 2bnn ; = T 3 n gnz(vi) n + 1 2 nn 3 n gnz(vi) n ; = T 2 3 n h gnz(vi) n + nn 3 n i ; = T 2 3 n h nn 3 n + gn p g2n + 8bnnnn 3 n gn i ; = T 2 3 n h nn 3 n g2n p g4n + 8g2n bnnnn 3 n i : (4.9) 72 To produce real values for p(vi) we must have the condition (constraint to bnn) g4n 8g2n bnnnn 3 n; (4.10) or g2n 8 3 nnn bnn: (4.11) For the classical case for neutrons, this is: b(cl) nn (T) = g2n 3 n 4 3 R3 nn; (4.12) or g2n 8 3 nnn + g2n 3 n 4 3 R3 nn: (4.13) Thus, the limiting density is g2n 8 3 nnn g2n 3 n 4 3 R3 nn; (4.14) 73 or 1 8nn 4 3 R3 nn: (4.15) In terms of the neutron number density (and recalling that Rnn 1fm) 3 32 1 R3 nn nn 0:029842 fm3; (4.16) the excluded volume is vnn = 1 2 4 3 R3 nn; (4.17) leading to 3 32 1 R3 nn ! 1 16vnn nn 0:029842 fm3: (4.18) Noticeably, this is temperature independent and is only constrained to the the volume of the sphere in question  i.e. volume dependent  as it should be. 74 For the quantum mechanical case b(vi;qm) (4.19) nn (T) = lX=1 l=0 b(l) nn(T); thus, using b(l) (4.20) nn(T) = (2l + 1)g2n 3 n 3 nn Z 1 0 exp k2 2 nnT d (ij) l dk ; we obtain b(vi;qm) nn (T) = lX=1 l=0 b(l) nn(T); (4.21) where g2n 8 3 nnn lX=1 l=0 b(l) nn(T): (4.22) Defining the integration result as f(T), we have b(vi;qm) nn (T) = lX=1 l=0 (2l + 1)g2n 3 n 3 nn f(T) = (2l + 1)g2n 3 n 3 nn lX=1 l=0 f(T); (4.23) since g2n 8 3 nb(vi;qm) nn (T) nn; (4.24) 75 we get g2n 8 3 n g2n 3 n 3 nn nn X (2l + 1)f(T); (4.25) or g2n 3 nn 8 3 ng2n 3 n nn X (2l + 1)f(T); (4.26) that is 3 nn 8 6 n P (2l + 1)f(T) nn: (4.27) From an examination of the terms, we observe that P (2l + 1)f(T) requires that the overall integrand result be unitless, so that it can be thought of as a single function dependent on T. Hence at a given temperature and nucleon density, we can write 3 nn 8 6 nnn X (2l + 1)f(T): (4.28) 76 REFERENCES Beth, E., & Uhlenbeck, G. E. (1937). The quantum theory of the nonideal gas. ii. behaviour at low temperatures. Physica, 4 (10), 915–924. Carroll, S. (2007). Dark matter, dark energy: The dark side of the universe (guidebook part 2). The Teaching Company. Greiner, W., Neise, L., & Stöcker, H. (1999). Thermodynamics and statistical mechanics. Springer. Huang, K. (2001). Introduction to statistical physics. Boca Raton: CRC Press. Huang, K. (2010). Introduction to statistical physics (2nd edition) (second ed.). Boca Raton: CRC Press. Pizzone, R., Spartá, R., Bertulani, C., Spitaleri, C., La Cognata, M., Lalmansingh, J., . . . Tumino, A. (2014). Big bang nucleosynthesis revisited via trojan horse method measurements. The Astrophysical Journal, 786 (2), 112. Typel, S., Röpke, G., Klähn, T., Blaschke, D., & Wolter, H. (2010). Composition and thermodynamics of nuclear matter with light clusters. Physical Review C, 81 (1), 015803. Typel, S., Röpkec, G., Klähnd, T., Blaschked, D., Woltere, H., & Voskresenskayab, M. (2010). Clusters in dense matter and the equation of state. Nuclei in the Cosmos, 1 , 39. 77 Uhlenbeck, G. E., & Beth, E. (1936). The quantum theory of the nonideal gas i. deviations from the classical theory. Physica, 3 (8), 729–745. 78 VITA Jared Lalmansingh received his Bachelor of Science in Physics from South Carolina State University in 2011. He next pursued his Master of Science in Physics at Texas A&M UniversityCommerce, where he is expected to graduate in 2015. Permanent address: Department of Physics and Astronomy, Texas A&M UniversityCommerce Commerce, Texas 75429 Email: jlalmansingh@leomail.tamuc.edu 
Date  2015 
Faculty Advisor  Newton, William G 
Committee Members 
Bertulani, Carlos A Montgomery, Kent 
University Affiliation  Texas A&M UniversityCommerce 
Department  MSPhysics 
Degree Awarded  M.S. 
Pages  92 
Type  Text 
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Language  eng 
Rights  All rights reserved. 



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