% % Lecture presentation aid for the textbook: % % Laszlo P. Csernai: " Introduction to Relativistic Heavy Ion Collisions" % (John Wiley and Sons Ltd, Chicester, New York, Brisbane, Toronto, % Singapore, 1994; ISBN - 0-471-93420-8) % % Transparencies for Lecture 4 / Chapter 4 % % The following file is containing transparencies in plain LaTeX format. % Figures are usually not included, they can be pasted from the printed % textbook. Empty space is left for this purpose. Some LaTeX figures are % included in the files. References and Solutions of the assignments are % also not included! % \documentstyle[12pt]{book} % Transparencies in frame keretben \textheight=21.5cm \textwidth=19cm \oddsidemargin=10mm \evensidemargin=10mm %\voffset=-0.50cm \hoffset=-0.15cm % \newcommand{\beq}{\begin{equation}} \newcommand{\beqar}{\begin{eqnarray}} \newcommand{\eeq}[1]{\label{#1} \end{equation}} \newcommand{\eeqar}[1]{\label{#1} \end{eqnarray}} %\newcommand{\beq}{$$} %\newcommand{\eeq}{$$} \newcommand{\B}{$\bullet$ } \newcommand{\PB}{\phantom{$\bullet$} } \newcommand{\LT}{$\leadsto$ } %\renewcommand{\vec}[1]{{\bf #1}} \newcommand{\D}{\bigtriangledown} \newcommand{\etal}{{\it et al.,\/}} \newcommand{\lton}{$\raisebox{-.54ex}{$\stackrel{<}{\sim}$}$} \newcommand{\gton}{$\raisebox{-.54ex}{$\stackrel{>}{\sim}$}$} \newcommand{\gfm}{{\rm GeV/fm}^3} \newcommand{\ipint}{\int \frac{d^3 p}{p^0}} \newcommand{\dpi}{ \frac{d^3 p_i}{p_i^0}} \newcommand{\dpj}{ \frac{d^3 p_j}{p_j^0}} \newcommand{\dpk}{ \frac{d^3 p_k}{p_k^0}} \newcommand{\dpl}{ \frac{d^3 p_l}{p_l^0}} \newcommand{\dpkp}{ \frac{d^3 p'_k}{p^{'0}_k}} \newcommand{\dplp}{ \frac{d^3 p'_l}{p^{'0}_l}} \newcommand{\ip}{\int^{\infty}_{-\infty}dp} \newcommand{\pint}{\int d^3 p} \newcommand{\juttner}{ {1\over{(2 \pi \hbar)^3}} \exp \left( {{\mu - p^\mu u_\mu}\over{T}} \right) } \newcommand{\jut}{ e^{ \left( - {{ p^\mu u_\mu}\over{T}} \right) }} \newcommand{\pmum}{p^\mu u_\mu} \newcommand{\amev}{A$\cdot$MeV} \newcommand{\agev}{A$\cdot$GeV} \newcommand{\transparencyframe}[1]{ \phantom{a} \setlength{\unitlength}{1mm} \begin{picture}(10,10)(30,200) \thicklines \put(95,240){\oval(188,18)[t]} \put(95,20){\oval(188,18)[b]} \put(1,20){\line(0,1){220}} \put(189,20){\line(0,1){220}} \thinlines \put(95,240){\oval(186,16)[t]} \put(95,20){\oval(186,16)[b]} \put(2,20){\line(0,1){220}} \put(188,20){\line(0,1){220}} \put(95,8){\scriptsize\sf L.P. Csernai: Inroduction to Relativistic Heavy Ion Collisions (Wiley, 1994)} \put(10,235){\parbox[t]{170mm}{\Large\sf \parskip=0.3cm #1}}\end{picture} }%end transparencyframe % \begin{document} \pagestyle{plain} %\setlength{\unitlength}{1mm} %}%end tr-page %\newpage % transparency ===================================================== \transparencyframe{ \setcounter{chapter}{3} \vspace*{-1.9cm} \chapter{Equation of State} \B Two regions of the nuclear Equation of State (EOS): \\ - Energy and density region reachable in intermediate energy HI collisions (up to few \ GeV/nucl. energy), and\\ - Briefly: Quark - Gluon Plasma (QGP) phase transition. \B Intermediate energy region:\\ - Compressibility of nuclear matter and\\ - Liquid - gas phase transition\\ - Some basics on nuclear multifragmentation \B QGP:\\ - Only from the EOS view point\\ - Sensitivity to the nuclear compressibility\\ \B \B We concentrate on connections between experimental data and EOS\\ - - rather than on theoretical derivation of a particular EOS in the framework of a particular theoretical model. \B \B EOS: Limited information on static thermal equilibrium properties\\ - HI: Non-equilibrium processes are important:\\ - Transport properties \\ - Transport coefficients from exp. data\\ - E.g. [1,2]: Scaling properties of the data \LT Reynolds nu. (viscosity) }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.5cm} \section{Intermediate Energy EOS} \subsection{Bulk nuclear matter} \noindent{NUCLEAR COMPRESSIBILITY}\\ \B From conventional nuclear physics: \\ - equilibrium state at the normal nuclear density $n_0 =0.145-0.17 fm^{-3}$\\ - with compressibility (earlier: $K\approx180-240 \ MeV $ [5], and\\ - binding energy of $16 \ MeV/nucleon$\\ \B EOS at high densities and high temperatures:\\ - mostly theoretical estimates\\ * High density high temperature EOS \LT compression stage of HI collision\\ * Low density behavior \LT observables and mechanism of fin. expansion\\ \B Energetic ($100 \ MeV - 4 \ GeV/nucleon$) HI coll. \LT many light nuclear fragments, a few heavy fragments and a few mesons \\ \ \ \LT destruction of the ground state nuclear matter\\ \ \ $\rightarrow$ dilute gas ($n<200 \ MeV$ needed to reproduce pion multiplicity\\ - Basic problem (Maruhn a\& St\"ocker [10] - Pion multiplicities: measured at high energies only\\ - there is an energy gap btwn ground state and energy of data\\ \LT Uncertainty in EOS calculated from data using Rankine-Hugoniot relations\\ - To avoid the problem: measurement of $K$ around the ground state needed, or\\ - first and second derivative of $K$ (versus $n$ and $T$) at the ground state. } \setlength{\baselineskip}{20pt} \B \B {\it Sidewards Flow in High Energy Nuclear Collisions\/}:\\ - Shock waves \& ``bounce off'': predicted[11,12] long before expe.s[13]\\ - Now flow analysis: well-established method to extract EOS, and\\ - even transport properties of the hot nuclear matter\\[-2.0ex] \setlength{\baselineskip}{10pt} {\normalsize\sf Earlier described satisfactorily in FD, but recent transport theoretical models (VUU-BUU-LV) [14,15]: able to incorporate finite particle number \& non-eq. effects\\ Nuclear mean field potential is basic constituent of these models \LT the nuclear compressibility can be explicitly read off. } \setlength{\baselineskip}{20pt} \B Recent model calculations include:\\ - Momentum dependent interactions (important in initial, non-eq. stage)\\ - These calculations \LT $K=200-400 \ MeV$ to fit flow data.\\ - Momentum dependence allows for the lower compressibility values\\ - while non momentum dependent interactions \LT to stiffer EOS. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B \B {\it Supernova Explosions\/}:\\ - At late stages of star evolution a star of about $10 M_\odot$ may explode if its iron core is in the range of $1.3-1.35M_\odot$.\\ - Baron {\it et al.\/} [16] found: if EOS is sufficiently soft a successful prompt supernova explosion may occur\\ % - The compression modulus depends on the proton fraction % $Z/A$, which is smaller in supernova than in nuclei.\\ - EOS \LT explosion if $K(Z/A=1/3) = 138 \ MeV$, i.e., $K(1/2)=180 \ MeV$. \B \B {\it Neutron Stars\/}:\\ - Glendenning [7] used the same EOS [16] to calculate the maximum mass of a neutron star by solving the Tollman - Oppenheimer - Volkov equation.\\ \vspace*{-8.0mm} \setlength{\baselineskip}{10pt} {\normalsize\sf - The stiffer the EOS the heavier neutron star can be supported by it. He found that with $Z/A=1/3$ the maximum neutron star mass would be $1.25M_\odot$. - However, the neutron stars are more neutron rich, so $Z/A=1/5$ might be more appropriate value. In this case the maximum neutron star mass would be only $1M_\odot$. - Since there is a neutron star with $M=1.451 \pm 0.007 M_\odot$ (PSR1913+16), and another where the mass is less accurately measured with $M=1.85{{+0.35}\atop{-0.30}}M_\odot$, these indicate that the EOS may be more stiff than the supernova calculations predicted.} \vspace*{-2.0mm} \setlength{\baselineskip}{20pt} - According to Glendenning's calculations at least $K=200 \ MeV$ is necessary to account for the observed neutron star masses. \B \B {\it Giant Monopole Resonance\/}:\\ - New results for $K$ have been reported by the Groningen group who made precision measurements of the breathing mode of 5 $Sn$ and 4 $Sm$ isotopes [17,18].\\ \vspace*{-8.0mm} \setlength{\baselineskip}{10pt} {\normalsize\sf - These data were analyzed in conjunction with the already existing data on $^{208}Pb$ and $^{24}Mg$ nuclei. They determined the compressibility of infinite nuclear matter, the surface, the isospin and the Coulomb contribution to the data: $$ K_A = K_\infty + K_s A^{-1/3} + K_\tau ({{N-Z}\over{A}})^2 + K_C Z^2 A^{-4/3} . $$ } \vspace*{-2.0mm} \setlength{\baselineskip}{20pt} - The resulting $K_\infty$ was $299 \pm 25 \ \ MeV$, more than earlier\\ %The above mentioned cases are not complete, there are still other ways to %gain information about the EOS and the compressibility. The most accurate %measurements of course still apply to the ground state nuclear matter %(Giant Monopoles) or to the cold matter (Neutron Stars). The other data %deal with more dynamic situations and with hot and compressed matter so it %is not surprising that there is still room for improving the present %estimates. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \noindent{THERMODYNAMICAL VARIABLES} \B Definitions and notations of thermodynamical variables\\ - not completely unique in the literature. \B Thermodynamics: usually for macroscopic systems\\ - of given volume, $V$,\\ - and given particle number, $N$.\\ - * in HI: local quantities instead !!\\ - 1) {\em specific extensives}: dividing each extensive by $N$,\\ - 2) {\em extensive densities}: dividing each extensive by $V$.\\ \vspace*{-8.0mm} \setlength{\baselineskip}{10pt} {\normalsize\sf Here we will denote all specific extensives by Greek letters and all extensive densities by lower case latin letters. This notation is quite usual but we have to keep in mind that the energy density, denoted here by $e$ is frequently denoted by $\epsilon$. Here we use $\epsilon$ for the specific energy. The thermodynamics and the EOS can be formulated in all three formalisms. See Table 4.1.} \vspace*{-2.0mm} \setlength{\baselineskip}{20pt} \B Thermodynamical potential \LT all other quantities, by derivations\\ \B If microscopic properties known \LT {\em partition function}\\ \LT Thermodynamical potentials:\\ - Helmholz free energy (canonical partition function) \\ - Grand potential (grand canonical partition function) }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.5cm} {\normalsize\hspace*{-3.5mm} \begin{tabular}{||l|l|l||} \hline \multicolumn{3}{|c|}{\sf THERMODYNAMICAL VARIABLES} \\ \hline \hline Extensives & Specific Extensives$(1/N)$ &Extensive densities$(1/V)$ \\ $S=entropy$ & $N=particle \ number$ & $V=volume$ \\ & $ \sigma=S/N$, $\ \ \nu=V/N$ & $s=S/V$ \\ \hline $E(S,V,N)=TS-PV+\mu N$ & $ \varepsilon (\sigma,\nu)=T\sigma-P\nu +\mu$ & $ e(s,n)=Ts+\mu n -P$ \\ $dE=TdS-PdV+\mu dN$ & $ d\varepsilon =Td\sigma-Pd\nu$ & $ de=Tds+\mu dn$ \\ & & eg.: $ P=-e+se,_s +ne,_n $ \\ \hline \hline \multicolumn{3}{|l|}{One intensive replaces an extensive:} \\ \hline \hline \multicolumn{3}{||l||}{Enthalpy:} \\ \hline $H(S,P,N)=E+PV=TS+\mu N$ & $ \chi(\sigma,P)=\varepsilon +P\nu =T\sigma+\mu$ & $ w(s,n)=e+P=Ts+\mu n $ \\ $dH=TdS+VdP+\mu dN$ & $ d\chi=Td\sigma+ \nu dP$ & $ dw=Tds+\mu dn +dP$ \\ & & (redundant) \\ \hline \multicolumn{3}{||l||}{Helmholtz free energy:} \\ \hline & & \\ $F(T,V,N)=E-TS=$ & $ \Phi(T,\nu )=\varepsilon -T\sigma =\mu-P \nu$ & $ f(T,n)=e-Ts= $ \\ $=\mu N-PV$ & & $ =\mu n-P$ \\ $dF=-SdT-PdV+\mu dN $ & $ d\Phi =-\sigma dT-Pd \nu$ & $ df=-sdT+\mu dn$ \\ \hline \multicolumn{3}{||l||}{X - potential:} \\ \hline $X(S,V,\mu)=E-\mu N=TS-PV$ & $-$ & $x(s,\mu)=e-\mu n=Ts-P $ \\ $dX=TdS-PdV - N d\mu$ && $dx=Tds-nd\mu $ \\ \hline \hline \multicolumn{3}{|l|}{Two intensives replace two extensives:} \\ \hline \hline \multicolumn{3}{||l||}{Gibbs free energy:} \\ \hline $G(T,P,N)=E+PV-TS=\mu N$ & $ \mu(T,P)= -T\sigma +P \nu + \varepsilon$ & $-$ \\ $dG=- SdT+VdP+\mu dN$ & $ d\mu=-\sigma dT+\nu dP$ & \\ \hline \multicolumn{3}{||l||}{Grand potential $\Omega$:} \\ \hline $\Omega(T,V,\mu)=-PV= $ & $-$ & $z(\mu,T)=-P= $ \\ $ =E-TS-\mu N$ & & $ =e-Ts-\mu n $ \\ $d\Omega =-SdT-PdV-Nd\mu$ & & $dz=-sdT-nd\mu$ \\ \hline \multicolumn{3}{||l||}{Y - potential:} \\ \hline $Y(S,P,\mu) = E + PV -\mu N$ & & \\ $dY=TdS+VdP-Nd\mu$ & & \\ \hline \hline \multicolumn{3}{|l|}{Gibbs - Duhem relation} \\ \hline \hline $ E+PV-TS-\mu N=O$ & $ d\mu=-\sigma dT+ \nu dP $ & $ dP=sdT+nd\mu $ \\ $- SdT+VdP-Nd\mu=O $ & & \\ \hline \end{tabular} }%endsmall Table 4.1 {\it Thermodynamical potentials} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \noindent{A SIMPLIFIED EQUATION OF STATE} \B \B Most essential properties of the nuclear EOS below $n_0$ \LT\\ - A liquid gas phase transition is clearly predicted, with\\ - $T_c=15-20 \ MeV$ and \\ - $n_c =0.3-0.5 n_0$. \\ \B More accurate information \& details only from experimental research\\ \B Having defined the ``thermodynamical potential'' \\ - in terms of its proper variables (like $e(s,n)$, $F(T,V,N)$ or $\mu(T,p)$)\\ \LT all other thermodynamical variables obtained by differentiating it. \bigskip \B\B Example [19-27]:\\ \PB\B Analytic para\-met\-ri\-za\-tion for the nuclear equation of state:\\ \PB - thermodynamical potential $e=e(n,s)$ given by: $$ e(n,s) = e_c (n) + e^*_F(n,s) - e^*_F(n,0) , $$ \PB - where $e_c (n)$ is the ground state energy density, and\\ \PB - $e^*_F(n,s)$ is the energy density of an ideal Fermi-gas\\ }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B\B Parametrize the ground state energy density[25] as $$ e_c (n) = n_0 \sum_{i=2}^5 a_i \ ({{n}\over{n_0}})^{{{i}\over{3}}+1} , $$ - where $a_i=+21.1, -38.3, -26.7, +35.9 \ MeV$ for $i=2 ,..., 5$ \\ - This para\-met\-ri\-za\-tion \LT\\ \PB - a nuclear compressibility $K=210 \ MeV$ at $n_0 =0.15fm^{-3}$.\\ \PB - a binding energy $\varepsilon_0(n_0) \equiv \varepsilon (n_0) = e_0(n_0)/n_0 = -8 \ MeV$\\ \PB \ (instead of the usual infinite nuclear matter \\ \PB \ \ value of $-16$MeV to simulate finite size effects)\\ \B For small nuclear densities, $n<2n_0$ only:\\ \PB (at high densities the sound speed exceeds the speed of light.)\\ \PB - For the thermal part: non-relativistic ideal Fermi-gas.\\ \PB - For the low density and temperature at the break-up \\ \PB \LT relativistic corrections are negligible \LT $$ e^*_F(n,s) = ({{n^{5/3}}\over{m}}) y(\sigma), $$ where $y$ is dimensionless;\\ - depends on the dimensionless specific entropy $\sigma$ (or $\mu/T$)\\ - $y(\sigma)$ given in integral form [28] or - analytic para\-met\-ri\-za\-tions [25,29] $$ \sigma(y) = 0.5213 + 1.5 \ln(y+0.7064) + {{1.809 y^{1/2}}\over {1 + 1.139 y^{1/3} + 1.417 y + 1.014 y^{3/2}}} . $$ }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B Other thermodynamical quantities: $$ T(n,s)=e,_s ={{n^{2/3}}\over{m}} y'(\sigma) , $$ $$ \mu(n,s)=e,_n={{1}\over{3}} \sum_{i=2}^5 (i+3) a_i ({{n}\over{n_0}})^{i/3} \ + \ {{5}\over{3n}} e^*_F(n,s) \ - \ T(n,s) \sigma , $$ $$ P(n,s) = P_c(n) + {{2}\over{3}} e^*_F(n,s) , $$ where $P_c(n) = {{n_0}\over{3}} \sum_{i=2}^5 i a_i ({{n}\over{n_0}})^{i/3+1}$ \\ \B EOS: a stable equilibrium only if the energy has a minimum:\\ - $M_{ik}= e,_{ik}$ (where $k,i = n,s$) is positive definite \LT\\ - Two independent constraints $$ c_\nu =T s(T,\nu),_T >0 , \ \ {\rm and} \ \ \kappa_T = -{{1}\over{\nu}} \nu(p,T),_p >0 , $$ - $\kappa_T$ is the isothermal compressibility, and\\ - $c_\nu$ is the isochoric specific heat (under constant specific volume) In nucl. phys. the compressibility: $$ K_\sigma = 9p(\sigma,n),_n, \ \ {\rm and}\ \ K_T = 9p(T,n),_n . $$ (\LT sound speeds: $u_\sigma > u_T > 0$) \B In the $[T,n]$ plane where $u_T^2<0$ we have an unstable region.\\ - In rapid processes (in a relativistic HI collision) the matter might penetrate into the unstable region [19,30]. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} PHASE COEXISTENCE BETWEEN LIQUID AND GAS PHASES In the unstable region, or even near this region: our system may split up into two phases. Consequence of stability requirements:\\ Allow for two co-existing liquid (L) and gas (G) phases \LT \\ one more free parameter in our thermodynamical problem:\\ the {\em volume fraction} of the phases $i=L,G$ $$ \lambda_i = V_i /V , $$ or equivalently the particle number fractions $$ \alpha_i = N_i /N . $$ Normalization: $\alpha_L + \alpha_G = 1$, $\lambda_L + \lambda_G = 1$,\\ relations among $\alpha_i$ and $\lambda_i$ $$ \lambda_i ={{n}\over{n_i}} \alpha_i , \ \ {\rm and}\ \ \lambda_L = {{n-n_G}\over{n_L - n_G}} . $$ The requirement of the energy minimum \LT \\ \B {\bf Gibb's criteria of phase equilibrium: }\\ $ P_L= P_G = P$, \ $T_L= T_G= T$, and $\mu_L = \mu_G = \mu$ Satisfied on one line in the $[n,s]$ plane: {\em Maxwell construction line}\\ (it lies in the stable region of the previous stability study, Fig. 4.1.) Outside the region confined by this line the matter is stable in one phase. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{8.0cm} Figure 4.1 {\it Phase diagram on the entropy density plane. Phase equilibrium is possible above the critical entropy too! $u_\sigma$ and $u_T$ are the adiabatic and isotherm sound speeds. The stable gas and liquid phases are separated from the metastable region by the Maxwell construction line. From [29] } \B Region between Maxwell construction line and boundary of $u_T^2<0$:\\ - metastable,\\ - the matter can be stable in this region if the other phase is not present. \B Superheating and supercooling:\\ - expected in fast relativistic heavy ion collisions. In HI reactions the phase mixture region can be always reached\\ at any energy in the final quasi-adiabatic expansion [21]\\ if the break-up density is sufficiently low. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} CRITICAL EXPONENTS Critical opalescence in liquid gas phase transition:\\ - observed more than a century ago\\ - 1940: Guggenheim [31]: several fluids behave similarly\\ around the critical point of the liquid-gas phase transition \LT Extended study of the critical exponents started in the 1960's [32] \B ``Order parameter'':= $n_L^{eq}(T)-n_G^{eq}(T)$, ($=n_L-n_G$)\\ \B Relative deviation from the critical temperature: $$ \varepsilon = (T-T_c)/T_c . $$ \vspace*{7truecm} Figure 4.2 {\it Determination of the critical behaviour. From [32]} Guggenheim's observation was that: just below the critical point $$ n_L - n_G \propto (-\varepsilon)^{\beta} , $$ where $\beta$ is a {\sl critical exponent} - found to be universally $\beta= 1/3$ Critical exponent $\delta$ is defined at $T=T_c$ by $$ P-P_c \propto (n-n_c)^\delta {\rm sign} (n-n_c) $$ }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} At the critical point the isotherm compressibility, $\kappa_T$, diverges\\ - $K_T$ tends to zero. (Fig. 4.2) - This divergence can be parametrized by a critical exponent also: $$ \kappa_T \propto (-\varepsilon)^{-\gamma^\prime}, \ \ {\rm if:} \ \ \varepsilon < 0 , $$ $$ \kappa_T \propto (\varepsilon)^{-\gamma}, \ \ {\rm if:} \ \ \varepsilon > 0 , $$ Similarly the specific heat around the critical point $$ c_\nu \propto (-\varepsilon)^{-\alpha^\prime}, \ \ {\rm if:} \ \ \varepsilon < 0 , $$ $$ c_\nu \propto (\varepsilon)^{-\alpha}, \ \ {\rm if:} \ \ \varepsilon > 0 . $$ - The critical exponents can be calculated for a given equation of state.\\ - In nuclear physics, however, critical exponents were seldom evaluated.\\ - In Table 4.2 (from [32]) some critical exponents are listed \begin{center} {\large \begin{tabular}{||l|r|r|r|r|r|r||} \hline &$ \alpha $&$ \alpha ' $&$ \beta $&$ \gamma $&$ \gamma ' $&$ \delta$ \\ \hline &$ T>T_c $&$ TT_c $&$ T0$ is the binding energy per nucleon\\ - $\mu$ is the non-relativistic chemical potential per nucleon\\ - related to the relativistic chemical potential by $\mu=\mu_{rel.}-m$\\ - if $\mu < -W_0$: the number density is exp. decreasing funct. of $A$\\ - if $\mu=-W_0$ the nuclei would coalesce, \& form uniform nuclear liquid Using Eq. (\ref{eos1}) for $p$ and $d$, and neglecting the binding energy difference, the deuteron to proton ratio is: $$ x \equiv n_d/n_p = {{3}\over{2}} \ 2^{3/2} e^{\mu/T} . \ \ \ \ \ \ \leadsto \ \ \ \ \mu/T=ln x - 1.445 $$ Now from $e=Ts+\mu n-P$, and Boltzmann ideal gas expressions \\ $e=n{{3}\over{2}}T$ and $P=nT$, \ \ we get\\ \hspace*{9cm} $\sigma = s/n = -\mu/T+2.5$\\ Using the expression of $\mu/T$ in terms of $x$ we can express the entropy by the $d/p$ ratio: {\LARGE $$ \sigma = 3.945 - \ln x . $$ } This result was first obtained by Siemens and Kapusta [41], and it served as the basis for experimental measurements of entropy later. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B At freeze out not only ground state nuclei\\ - but also nuclei in various excited states will be present \LT\\ $$ n_i(A) ={{g_i}\over{g_A}} n_g(A) e^{-E_i^*/T}, $$ - $g_i$ is the degeneracy of the excited state, and \\ - $E_i^*$ is its excitation energy above the ground state\\ From the total baryon density, $n$, \LT $\mu$ can be obtained: $$ n = \sum_A \sum_{g,i} A n_i(A). $$ $\exists$ large numerical models to calculate fragment mass distribution\\ - based on the law of mass action for ideal gases:\\ \B\B In (FREESCO) [36,42-45]: - all nuclear states with\\ - $A<16$ \& $\Gamma<1 \ MeV$ were included explicitly, and\\ - these levels for $A>4$ were supplemented by an effective level density formula for the higher lying states not known experimentally\\ - Approximate microcanonical event generator: exact microcanonical fragment distribution calculated recursively using grand canonical one-fragment inclusive distributions in each step\\ - Excluded volume approximation \LT non-ideal gas EOS\\ - - Pressure increases sharper at densities as $n \rightarrow n_0$\\ - - No first order phase transition because repulsive interactions only\\ \B\B In Quantum Statistical Model (QSM) [19,46] \LT\\ - grand canonical one-fragment inclusive fragment distribution functions\\ - some quantum statistical features included\\ - particle-stable and metastable nuclear states with $A<20$ are included\\ - repulsive interactions $\approx$ by the excluded volume approximation }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B\B Physical picture (for both models):\\ - 1 fast explosion creating light and medium mass fragments\\ - - according to the law of mass action\\ - 2 sequential evaporation from these products in a final decay step \B\B From both models [38]:\\ - 1 At $T=30-90\ MeV$: $\exists$ unique relationship between $X \Leftrightarrow \sigma$\\ - 2 At low $T$ \& $\sigma$ isotherms begin to deviate from universal curve\\ - 3 At high $\sigma$ and low $X$ the ``Siemens-Kapusta'' formula\\ \hspace*{3cm} $ \sigma = 3.945 - \ln X $ is a good approximation\\ - 4 From experiments at $400-1050\ MeV/nucleon$ [47]:\\ \hspace*{1cm} $ X \approx 0.48 - 0.68$ \LT According to both models $\sigma=3.45-3.9$ \B In intermediate energy HI reactions\\ - Temperatures from energy spectra of fragments with moving source fit\LT\\ - Temperatures are constant, independent of particle type [48] \LT\\ - This suggests thermal and phase equilibrium\\ These experiments [47] are of relatively high energy \LT\\ - nuclear liquid-gas phase transition is not expected before breakup\LT\\ - The above models (neglecting attractive interactions) are satisfactory \B At lower energies: - Light fragments show a relatively high entropy\\ - Intermediate fragments almost one unit smaller [49] \LT\\ - Above models are not satisfactory at lower energies, and other effects:\\ - nuclear liquid-gas phase transition [50], microcanonical statistics, attractive and Coulomb interactions should be considered. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \subsection{Finite systems and fragment abundances} PHASE TRANSITION IN FINITE SYSTEMS \B\B Finite number of particles \LT no phase transitions exists\\ - fluctuations can be important in HI systems [51] \B System with fixed temperature and pressure:\\ \B Density fluctuations of this system ?? The ratio of probabilities for a system to be at density $n_1$ or $n_2$ is $$ p(n_2 )/p(n_1 ) = \exp[-(G(n_2 ) - G(n_1 ))/T] $$ - $G(n)$ is the Gibbs free energy at $p$ and $T$ (fixed)\\ - For an infinite eq. system the density $n$ is determined by the EOS\\ - We want to know, however, the probability of having the system at a {\bf non-equilibrium density}! \B Therefore: Necessary to know $G(n)$ for densities not permitted by the EOS\\ - Landau theory [28]:\\ - $n$ is treated as an independent variable not restricted by $P$ and $T$\\ - Such an analysis for HI systems was carried out first in [51] }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} A simple nuclear EOS is studied: \beq P = -a_0 n^2 + 2a_3 n^3 + n T , \eeq{eos2} \setlength{\baselineskip}{10pt} {\normalsize\sf ($a_0=293 \ MeV fm^3$, \ $a_3 =666 \ MeV fm^6$, \ $n_0 =0.15 fm^{-3}$, \ $W_0=8 \ MeV$) which has a critical point at $n_c =a_0/(6a_3 )$, \ $T_c =a_0 n_c$ , \ $P_c = {{1}\over{3}} T_c n_c$ .}\\ \setlength{\baselineskip}{20pt} Expand the EOS (\ref{eos2}) around the critical point by introducing\\ \ \ \ $t=T-T_c$ and $\eta=n-n_c$ \ \ \ \LT \beq P-P_c = n_c t + t\eta + 2a_3 \eta ^3. \eeq{eos3} - Similar to the Van der Waals EOS\\ - For $t<0$ phase eq. points can be found by Maxwell construction: $$ \eta_L = -\eta_G = \sqrt{-t/2a_3} . $$ \B The essential feature of the Landau approach is the\\ - \ \ \ \ {\bf construction of free energy}\\ - in terms of a power series in the order parameter $\eta$. \LT $G(P,T,\eta)$ will be defined at non-eq. values of $\eta$ too ($P$ \& $T$ fix)\\ - In the neighborhood of the critical point the Gibbs free energy is: $$ G(P,T,\eta) = \hspace*{12cm} $$ \beq G_0(P,T,\eta) + \alpha(P,T)\eta + A(P,T)\eta^2 + C(P,T)\eta^3 + B(P,T)\eta^4 + ... \eeq{eos4} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B EOS (\ref{eos3}) can be used to obtain the coefficients in (\ref{eos4}):\\ - Eq. value of $\eta$ can be obtained from the requirement that\\ - - $G$ has an extremum in equilibrium: \beq {{\partial G}\over{\partial \eta}} = \alpha + 2A\eta + 3C\eta^2 + 4B\eta^3 = 0 \eeq{eos5} - this $\equiv$ EOS (\ref{eos3})\\ - Comparing (\ref{eos3}) and (\ref{eos5}) we obtain the coefficients:\\ - - $ \alpha = -(P - P_c - n_c t ) D$, \ $ A = {{1}\over{2}}tD$, \ $ B = {{a_3}\over{2}}D$, \ $ C = 0$, where $D = N/n^2_c$ and $N$ is the total number of nucleons in the system.\\ - This choice of $D$ gives the correct $G$ for equilibrium states \LT\\ - $G$ in the order parameter expansion is $$ G = G_0(P,T) + {{N}\over{n_c^2}}[-(P - P_c - n_c t)\eta + {{1}\over{2}} t \eta^2 + {{a_3}\over{2}} \eta^4] $$ The density, $\eta$, at phase equilibrium, $\eta_L$ \& $\eta_G$\\ are the solutions of the EOS (\ref{eos3}) if $P=P_c +n_c t$.\\ At this pressure, $P$, the probability distribution of the density: \beq R(n) = {{p(n)}\over{p(n_L)}} = \exp[-(G(P,T,\eta) - G(P,T,\eta_L))/T]. \eeq{eos6} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{14truecm} Figure 4.3 {\it The relative probability for the system to be at density $n$ compared to the thermodynamically favored values $n_L$ or $n_G$. The number of nucleons is $N=100$. The pressure is the equilibrium pressure. From [29]} \B For $T$ not too close to $T_c$ there are two well defined peaks\\ - corresponding to a separation of liquid and gas phases, i.e.,\\ - exhibiting a reasonably sharp first order phase transition\\ \B As $T$ approaches $T_c$ the valley separating the two peaks gets filled in\\ - distinction between liquid and gas gets washed out\\ - At $T_c$ the distribution is flat at the top $\sim$ critical opalescence\\ %\setlength{\baselineskip}{10pt} %{\normalsize\sf %To find the relative probability for a system composed of $N$ nucleons, %$N$ not necessarily $100$, one simply scales the results of Fig. %\ref{f3.3} to the power $N/100$, $R^{N/100}$, because in Eq. (\ref{eos6}) %the Gibbs free energy was taken to be proportional to the total number of %particles. For the density midway between $n_L$ and $n_G$, the relative %probability assumes the simple form $ R({{1}\over{2}}(n_L+n_G)) = %\exp(-0.75(T -T_c)^2N/(TT_c)) $. Thus a larger number of nucleons %sharpens the distinction between liquid and gas phases. %} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} DROPLET AND BUBBLE FORMATION \B Consider surface effects: - central HI collision - subsequent adiabatic expansion \LT\\ - at any entropy per baryon: the Maxwell curve is crossed\\ - either from the liquid or the fluid side\\ - a) from the liquid side \LT bubbles begin to form\\ - b) from the gas side \LT droplets begin to form\\ - - - we shall consider droplets in a gas now. \B Probability of droplet formation:\\ - from change in the Gibbs free energy when a droplet appears \vspace*{3cm} - spherical droplet containing $A$ nucleons in a gas of a $A+B$ nucleons \beq G_{no \ drop} = \mu_G (A+B) , \eeq{eos7} \beq G_{drop} = \mu_L A + \mu_G B + 4\pi R^2\sigma_s + T\tau \ln A . \eeq{eos8} - $\mu_G$ and $\mu_L$ are the nucleon chemical potentials, at $P$ and $T$\\ - 3rd term in Eq. (\ref{eos8}) is the surface free energy for a droplet\\ - with surface tension $\sigma_s = \sigma_s(T)$\\ - last term (Fisher [53]): the droplet surface closes on itself \LT\\ - which reduces the total entropy associated with surface fluctuations \B Critical exponent $\tau$\\ - related to critical exponent $\delta$ as $\tau = 2 + 1/\delta$\\ - in mean field theories $\delta=3$, \LT $\tau \equiv \ 2 + 1/\delta\ = \ 7/3$ . }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B Probability of formation of the droplet is $\propto \exp(- \Delta G/T)$,\\ - ($\Delta G$ is the diff. btwn Eqs. (\ref{eos8}) and (\ref{eos7}))\\ \B The yield of a fragment of mass $A$ is \beq Y(A) = Y_0 \exp[{{\mu_G-\mu_L}\over{T}}A - {{4\pi r_0^2\sigma_s}\over{T}} A^{2/3} - \tau \ln A]. \eeq{eos9} - $Y_0$ is a normalization constant\\ - $r_0$ is related to the droplet radius by $R=r_0 A^{1/3}$\\ - and to the density by $n_L^{-1}=4\pi r_0^3/3$ \B Surface effects: first by the Purdue-Fermilab group [54-56]\\ - applied to high energy, $80-350 \ GeV$, proton-nucleus reactions\\ - Mass \& charge distributions for $A$ up to $30$ were measured\\ - possible because of the use of an in-beam gas jet target\\ \LT a power law $A^{-2.65}$ fits the data (better than $e^{-\alpha A}$) \B Interpretation: target nucleus was instantaneously heated by the $p$\\ - subsequently the heated nucleus expanded in size\\ - until passed through the critical point, $T=T_c$ and $n=n_c$\\ - At this point the distribution of droplets is: $ Y(A) = Y_0 A^{-\tau} $\\ because\\ - in Eq. (\ref{eos9}) $\mu_G =\mu_L$ and $\sigma_s =0$ at the critical point\\ \LT the volume and surface free energy terms vanish\\ }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B $\exists$ \ \ \ \ \ \ 2 DIFFICULTIES with the above interpretation:\\ - 1) why should one be so lucky to hit the critical point\\ - - accidentally with proton energies ranging from $80$ to $350 \ GeV$\\ - - and with targets so different in size as krypton and xenon?\\ - 2) according to Fisher's version of the droplet model, $2<\tau<2.5$,\\ - - whereas the data were outside this range. \B Mass distributions were remeasured\\ - depending on the proton energy in the range of $E_p=1-20 \ GeV$ [57]\\ - Eq. (\ref{eos9}) \LT parametrization in terms of $x$ and $y$: $$ Y(A) = Y_0 \ x^{A^{0.6386}} \ y^A \ A^{-\tau}. $$ - if the critical point approached $x$ \& $y \ \longrightarrow 1$ [see (\ref{eos8})]\\ - $x$ and $y$ determined by fitting the experimental mass yields - \B\B found: $x$ and $y$ tend to 1 monotonically from below (above)\\ - At $E_p=2 \ GeV$, $x=0.2, y=1.4$ \\ - at $10 \ GeV$ both reach 1 \\ - At this fit $\tau$ was kept constant at $2.2$\\ \B These data indicate: - the path in the thermodynamical space is energy dependent\\ - it gets in the vicinity of the critical point only at high proton energies\\ - when $\tau$ was independently fitted to the data:\\ \LT convergence to $\tau \approx 2.1$ from below. \B These results are consistent with the liquid gas phase transition picture\\ - with increasing bombarding energy multifragmentation occurs\\ - first in the mechanical instability region\\ - then in the supersaturated vapor region and\\ - finally at energies above $10 \ GeV$ in the critical region. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{0.8cm} \B Nuclear fragmentation models are in development - The connection between the nuclear fragmentation models and the nuclear EOS should be firmly established before a final conclusion about the nuclear liquid- gas phase transition can be drawn. \setlength{\baselineskip}{10pt} {\normalsize\sf EOS underlying the statistical models was seldom calculated (apart from some simple cases [36]). In statistical fragmentation models the evaluation of the EOS is in principle possible.} \setlength{\baselineskip}{20pt} - On the experimental side the problems are to separate central heavy ion collisions and eliminate geometric effects arising in peripheral reactions. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \section{The Nuclear EOS and Quark Gluon Plasma} \B EOS: theoretical work invested in the study of\\ - pure SU(N) Yang-Mills theory on the lattice. \\ - These calculations, however, are restricted to zero net baryon density\\ - - - or zero chemical potential\\ - For low or ``zero'' baryon density matter in the deconfined phase\\ - - - one needs extremely high energy \B Lattice QCD calculations will be discussed Chapter 10\\ \B Simple ``phenomenological'' theories:\\ \LT EOS in the phase transition region for cold matter [59-61]\\ - for zero baryon charge at finite temperature [62] \\ - in the complete phase space for finite density and temperature [63-68] \B Phenomenological EOS studies can yield:\\ - good qualitative insight into the phase transition problem\\ - easily incorporated into phenomenological phase transition models }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.5cm} \subsection{Hadronic Equation of State} \B Attention to high temperature and high density behavior is necessary In sect. 4.1.1 we introduced a simple para\-met\-ri\-za\-tion of the EOS\\ - Other para\-met\-ri\-za\-tions: - E.g. energy density $e$ in terms of $n$ \& $T$: \beq e = n[ m - W_0 + K (n/n_0 - 1)^2 /18 + 3T/2 ], \eeq{eos10} - $m$ is the nucleon rest mass\\ - $W_0>0$ is the binding energy\\ - third term is the compressional energy $e_c$, usually called ``quadratic"\\ - last term is the thermal energy (Boltzmann ideal gas)\\ - at high temperatures pion pairs should also be taken into account:\\ - by neglecting their rest mass: $$ e_m=g_1(\pi^2/30)T^4, \ \ P_m=e_m/3, \ \ s_m={{4}\over{3}}e_m/T, $$ - $g_1$ is the degeneracy of states\\ - for pions only, $g_1=3$ and so: $$ e_\pi=\pi^2T^4/10,\ \ P_\pi=\pi^2T^4/30,\ \ s_\pi=4\pi^2T^3/30. $$ - at high temperatures some nucleons can be excited \LT ! $\Delta$s\\ - total baryon charge is conserved, $n = n_N+ n_\Delta$\\ - in the Boltzmann approximation the delta to nucleon ratio is given by $$ n_\Delta /n_N=4(m_\Delta/m_N)^{3/2} \exp(-[m_\Delta-m_N]/T) . $$ \LT change in $e$: the mass term is $n_\Delta m_\Delta + n_N m_N$ \B Sum of two or more of the above mentioned combinations provides the total hadronic (h) EOS, e.g.:\\ $e_h= e_n + e_\pi$. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.1cm} COMPRESSIONAL PART OF THE NUCLEAR EOS\\ \B Causality \LT constraints on the EOS\\ - very {\em hard} EOS may \LT superluminal speed of sound [70]\\ \B However, this is not a problem if the acausality:\\ - occurs in the mixed or plasma phase\\ - because the phase transition softens the matter\\ \B Specific energies $\varepsilon = e/n$ at high densities \begin{itemize} \item ``Linear'' and ``Quadratic'' [71] \[ \varepsilon_L(n;K) = E_B + \frac{K}{18} \frac{(n-n_0)^2}{n n_0}, \]\[ \varepsilon_Q(n;K) = E_B + \frac{K}{18} \frac{(n-n_0)^2}{n_0^2}. \] \item ``Sierk--Nix'' [20] \[ \varepsilon_{SN}(n;K) = E_B + \frac{2K}{9} \left( \sqrt{ \frac{n}{n_0} } - 1 \right) ^2. \] \item ``Grant--Kapusta'' [70] \[ \varepsilon_{GK}(n;K,a) = \varepsilon_{SN}(n;K) + a \left( \sqrt[3]{ \frac{n}{n_0} } - 1 \right) ^3. \] \end{itemize} The corresponding density of the free energy is \[ f_{compr}(n;K) = n \varepsilon_{compr}(n;K). \] - All of these parametrizations are acausal at sufficiently high densities.\\ - Fortunately the acausality occurs\\ - - well within or beyond the mixed or plasma phase \\ - - for all the parametrizations except the ``Quadratic''. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} The basic parameter is the (isothermal) compressibility $K$: \[ K \equiv 9 \left( \frac{\partial P}{\partial n} \right) _{n=n_0, T=0}. \] \B $K$ can be determined in several different ways:\\ - usually $K \approx 100-400 \ MeV$\\ - but depends on the parametrization\\ - e.g. the ``Sierk--Nix'' parametrization with $K=550\ MeV$ \\ - corresponds to $K=275\ MeV$ in ``Quadratic'' parametrization!\\ - The two parametrizations yield essentially the same EOS [70]\\ - for densities of interest \B the nuclear EOS strongly influences the phase transition\\ - and the phase diagram\\ - compressional energy is particularly important:\\ - when neglected [68] \LT phase diagram with pathological behavior:\\ - - a) the matter at $n_0$ \& $T=0$ is in the mixed phase\\ - - b) $\exists$ 1st order phase transition only: $\root 4 \of {B}= 149-154 \ MeV$ [68] \B Possible method for compressional energy in hadronic phase:\\ - via the excluded volume approximation [65] \\ - standard way of treating nuclear matter in relativistic nuclear collisions }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{QGP Equation of State} \B QCD Lagrangian \LT EOS which describes both the HM \& QGP\\ - due to nonlinear interactions it is not easy to find the EOS\\ - at high energy densities the coupling tends to zero\\ - - the ``asymptotic freedom'' sets in\\ \LT QGP EOS would correspond to a non-interacting gas of\\ - - $N_f$ flavor quarks that come in $N_c$ colors, and\\ - - $(N^2_c-1)$ spin 1 gluons \LT \\ - EOS is Stefan-Boltzmann gas expression: $$ e_{SB}(T,\mu) = {{\pi^2}\over{15}}(N^2_c-1+{{7N_cN_f}\over{4}})T^4 \ + \ {{ N_cN_f}\over{2}}(T^2\mu^2 + {{\mu^4}\over{2\pi^2}}) , $$ $$ P_{SB}(T,\mu)={{1}\over{3}} e_{SB}(T,m) , $$ $$ n_{SB}(T,\mu)={{N_cN_f}\over{9\pi^2}}(\mu^3 +\pi^2T^2\mu) , $$ - $T$ \& $\mu=\mu_q$: quark temperature \& chemical potential\\ - - ($\mu_b =3\mu_q$)\\ - $n_{SB}=n_b$ is the {\sl baryon charge} density in the quark phase }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.5cm} \B The vacuum in which the ideal gas of quarks and gluons exist\\ - differs from our everyday vacuum\\ - we do not see the quarks and gluons in our physical vacuum \LT\\ - this vacuum should have lower energy than the QCD/perturbative one\\ - - where $q$ and $g$ can exist. \B We take this effect into account by adding a constant\\ - $Bg^{\mu\nu}$ to the energy momentum tensor of the ``quark world'' \LT\\ $$ e_q(T,\mu) = e_{SB}(T,\mu) + B, \ \ \ P_q(T,\mu) = P_{SB}(T,\mu) - B, $$ - $B$ is called the bag constant\\ - this EOS is the ``Bag Model'' EOS \B Usually we can restrict ourselves to \\ - two flavors ($u$ and $d$) in the quark-gluon phases, \\ - so $N_f =2$ and $N_c =3$ (for $u,\ d,\ s$ quarks $N_f=3$) \LT \\ - $P_q$ in terms of $T$ and $\mu_b$ $$ P_q= 37 \pi^2 T^4/90 + \mu_b^2 T^2/9 + \mu_b^4/162 \pi^2 - B, $$ - $\mu_b$ is the chemical potential associated with the baryon charge \B Frequently used. In some cases 1-loop or 2-loop perturbative corrections are also included. The introduction of these perturbative terms leads to a 10-20\% increase of the critical temperature and to a similar decrease of the critical densities $n_{cq}$ and $n_{ch}$. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.0cm} PHASE MIXTURE \B Hadronic + QGP EOS \LT complete EOS by Maxwell construction\\ - containing the 2 pure phases and\\ - a region where the above two phases coexist.\\ \B Zero baryon charge: 2 of Gibb's criteria, $P_q=P_h, \ T_q = T_h$\\ \B Baryon-rich plasma: additional - $ 3\mu_q = \mu_h $ For baryon free plasma $P_q= 37\pi^2T^4/90 - B$\\ For pion gas $P_h = 3\pi^2T^4/90$\\ \LT from $P_q(T_c)=P_h(T_c)$: critical temperature $T_c$ (1st ord. Ph.T.) $$ T_c^4 = 90B/(34 \pi^2) $$ - E.g. if $B$ $=$ $\Lambda_B^4/(\hbar c)^3$ $=$ $0.397$ GeV/fm$^3$\\ \LT $T_c=169\ MeV$, $P_c$ $=$ $35\ MeV /fm^3$ \& critical $e$s at $T_c$:\\ $e_h(T_c)$ $=$ $106\ MeV/fm^3$, and $e_q(T_c)$ $=$ $1.695\ GeV/fm^3$ \vspace*{9cm} Figure 4.4 {\it The pressure is reduced by the phase transition (in the mixed phase). It increases again only when the pure QGP is reached. Thus phase transition softens the EOS. From [77]} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B For baryon-rich matter the Maxwell construction: - $P_q(\mu,T)$ is given - the chemical potential of the hadronic phase is known\\ - - i.e., Boltzmann approximation $$ \mu_b= T \ln ({{n_b C}\over{dT^{3/2}}})+\ m_N+\ W_0+\ K (n_b -n_0)(3n_b -n_0)/(18n_0^2) $$ - $C=(2\pi(\hbar c)^2/(mc^2))^{3/2}$\\ - $d$ is the degeneracy of the nucleon gas, $d=4$ \LT phase eq. at $T_c$ from the Gibbs criteria\\ - - by solving a single equation (numerically) for $n_b$ (Fig. 4.4-5) \vskip 11.5truecm Figure 4.5 {\it Phase diagram of the nuclear matter quark matter phase transition from a simplified phenomenological model. From [29]} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{QGP phase transition and nuclear compressibility} \B Phase diagram is sensitive to both nuclear and plasma parameters\\ - increase of $K$ \LT decrease of critical $n_{cq}$ \& $n_{ch}$\\ - increase of $B$ \LT increase of the critical temperature, and densities\\ - hadronic resonances \\ - - have negligible effect on phase diagram at $T= 0$ or $\mu = 0$\\ - - push phase boundaries to higher $n$ \& $T$ in intermediate region\\ - - increases eq. pressure at fixed chemical potentials\\ - the equilibrium pressure is higher at $T=0$ than at $T_c$\\ - - which is an interesting feature first observed in [67] \B mixed phase formation becomes possible:\\ - at $1-2\ GeV/fm^3$ at finite densities\\ - below $1\ GeV/fm^3$ when the density tends to zero \B pure QCD plasma reached - at $2-6\ GeV/fm^3$ energy density at finite densities\\ - at $1-4\ GeV/fm^3$ at $n=0$ For comparison: the normal nuclear matter has $e= 0.134 \ GeV/fm^3$. \subsection{Dependence of phase transition on the nuclear EOS} \B ``Sierk--Nix'' parametrization $(T=0)$ \LT \\ - baryon-rich plasma: $n_{cH} = 1.39$ \& $n_{cQ} = 3.80\ fm^{-3}.$\\ - corresponding energy densities $e_{cH} = 2.04$ \& $e_{cQ} = 7.53\ GeV/fm^3$ The parameters of EOS influence the phase diagram (Fig. 4.6)\\ }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{16cm} Figure 4.6 {\it Shock adiabats and phase boundaries for different equations of state. From top to bottom a, b, c, d. [77]} \begin{description} \item[Bag parameter:] - Decreasing $B$ \LT (4.6b) $1\rightarrow .2$GeV/fm$^3$\\ - easier to reach perturbative vacuum\\ \LT decrease in both the critical temperature at zero baryon density\\ - - \& critical baryon densities at zero temperature\\ \LT decrease of the ($n,T$) domain of hadronic matter \end{description} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \begin{description} \item[Compressibility:] - Decreasing $K$\\ - gives a ``softer'' EOS\\ - increases critical baryon densities in cold matter\\ - Fig. 4.6c: ``Sierk--Nix'' EOS with $K=350\ MeV.$ \item[Parametrization of compressional energy:] parametrization\\ - influences the transition region\\ - E.g. almost identical behaviour up to $n=3-4n_0$\\ - - ``Quadratic'' with $K=275\ MeV$, or \\ - - ``Sierk--Nix'' with $K=550\ MeV$ (4.6a) or \\ - - ``Grant--Kapusta'' with $K=200\ MeV$ and $a=400\ MeV.$\\ - If compression energy is neglected:\\ - - critical baryon density of HM is lowered to just about $2n_0$\\ - - bag parameter is restricted to $B=149-154\ MeV/fm^3.$ \item [$\Delta$-resonances:] Including $\Delta$-resonances\\ - no changes at low temperatures or baryon densities\\ - chemical potential is lowered slightly at intermediate $n$ and $T$\\ \LT pushes the phase boundary outward (see Fig. 4.6d). \item [``Massive'' pions:] Lowers the pressure in the HM\\ - reduces the critical temperature by 0.5 MeV \\ - - (at $T_c=215.9\ MeV$) \item[Nucleon--anti-nucleon pairs:] Increases the pressure in HM \\ - increases the critical temperature at $n=0$ by about one $MeV$ \item [Relativistic nucleon gas:] small increase of pressure of HM \\ - \& the critical temperature at $n=0$ (approximately $0.5\ MeV$) \end{description} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \section{EOS from microscopic theory} \setlength{\baselineskip}{10pt} {\normalsize\sf In elementary statistical physics: see calculation of the partition function and the thermodynamical potentials for a statistical microscopic system. An introduction to this approach can be found e.g. in Chapters 6, 7, 9, and 10 of the textbook of Reif. [52]. We present here this approach based on an example of nuclear matter. [78] In Chapter 10 further applications of this approach will be mentioned related to the EOS of quark gluon plasma. An introduction to the calculation of EOS in quantum field theories can be found in the book of Kapusta [79].} \setlength{\baselineskip}{22pt} The most general Hamiltonian\\ - for an $A$-nucleon system\\ - with two-body interaction $v_{ij}$,\\ - three-body interaction $v_{ijk}$, etc. : \beq H = \sum_{i=1}^{A} \frac{p_{i}^{2}}{2m} + \frac{1}{2} \sum_{i,j=1}^{A} v_{ij} + \frac{1}{6} \sum_{i,j,k=1}^{A} v_{ijk} + \ ... \ , \eeq{Hpart} - $m$ is the nucleon mass\\ \B Non-relativistic phase space distribution function: $f({\vec{r},\vec{p}})$\\ \LT In terms of $f$ the total energy of the system: \beqar E = & \left.\int \right. d^3r d^3p \ {p^2 \over {2m}} f({\vec{r},\vec{p}}) \nonumber \\ & + {1 \over 2} \left.\int \right. d^3r d^3p \ d^3r' d^3p' f({\vec{r},\vec{p}}) f(\vec{r}\,',\vec{p}\,') \ v_{ij}(\vec{r},\vec{r}\,',\vec{p},\vec{p}\,') + \ldots \nonumber \\ & = \left.\int \right. d^3r d^3p \ {p^2 \over {2m}} f({\vec{r},\vec{p}}) + V[f] . \eeqar{H} \setlength{\baselineskip}{10pt} {\normalsize\sf Clearly, the one-body term of the Hamiltonian depends on one distribution function, while the two-body term contains two distribution functions, etc. In general, the potential energy, $V[f]$, (in addition to its complicated momentum dependence) is explicitely nonlocal, as a consequence of the single-particle distribution functions appearing in it.} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} In the {\em local approximation}:\\ - pot. energy density, $u[f]$, contains $\delta$-functions for $\vec{r}$\\ \LT only one nontrivial integral over one position coordinate, $r$\\ - - is left in the potential energy, $V[f]$: \beq V[f] = \int d^3r \ u[f({\vec{r},\vec{p}}),f(\vec{r},\vec{p}\,'),\ ...\ ] \, . \eeq{pedensl} E.g., if we assume a momentum-independent 2-body interaction, \beq v_{ij}(\vec{r},\vec{r}\,',\vec{p},\vec{p}\,') = v_{ij}^{(0)}(\vec{r}-\vec{r}\,') , \eeq{G1} then \beqar V & = {1\over 2} \left.\int \right. d^3r d^3p \ d^3r' d^3p' \ f(\vec{r},\vec{p}) f(\vec{r}\,',\vec{p}\,') \ v_{ij}^{(0)}(\vec{r}-\vec{r}\,') \nonumber \\ & = {1\over 2} \left.\int \right. d^3r d^3r' \ n(\vec{r}) n(\vec{r}\,') \ v_{ij}^{(0)}(\vec{r}-\vec{r}\,') . \eeqar{example0} Furthermore, with a local (contact) two-body interaction, \beq v_{ij}^{(0)}(\vec{r}-\vec{r}\,') = {a\over{n_0}} \delta(\vec{r}-\vec{r}\,') , \eeq{G2} - $n_0$ is the normal nuclear matter density\\ - $a$ is a constant of energy dimension \LT the potential energy: \beqar V & = & {1\over 2} \int d^3r d^3p \ d^3r' d^3p' \ f(\vec{r},\vec{p}) f(\vec{r}\,',\vec{p}\,') \ {a\over{n_0}} \delta(\vec{r}-\vec{r}\,')\nonumber \\ & = & \int d^3r \ {a \over 2} {{n^2(\vec{r})}\over{n_0}} . \eeqar{example} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B Local approximation in the followings\\ - parameters of the nucleon distribution function, $f(\vec{r},\vec{p})$\\ - may be position-dependent, e.g. the local temperature, $T = T(\vec{r})$\\ \B $f$ for a gas with a momentum-dependent interaction \\ - may contain $\vec{r}$-dependent parameters, e.g. $\Lambda = \Lambda(\vec{r})$ eq. (\ref{symu}). \B In the local approximation the exchange part of 2-body interaction \LT\\ - a momentum-dependent term\\ - which expresses the effective nonlocality due to Pauli principle\\ - For a Yukawa 2-body interaction: term from the exchange part [81] \beq v_{ij} = v_{ij}^{(0)} (\vec{r} - \vec{r}\,') + \frac{2c}{n_0} \frac{\delta(\vec{r} - \vec{r}\,')} {1+ (\frac{\vec{p}-\vec{p}\,'}{\Lambda})^2}\,, \eeq{symu} - $c$ is a constant of energy dimension \LT full energy: \beqar E & = & \left.\int \right. d^3r d^3p \ {p^2 \over {2m}} f(\vec{r},\vec{p}) \ \ \ \nonumber \\ & + & {1\over 2}\left.\int\right.d^3r d^3p \ d^3r' d^3p' f({\vec{r},\vec{p}}) f(\vec{r}\,',\vec{p}\,') \nonumber \\ & \times & \left[ v_{ij}^{(0)}(\vec{r}-\vec{r}\,') + \frac{2c}{n_0} \frac{ \delta(\vec{r}-\vec{r}\,') } {1+(\frac{\vec{p}-\vec{p}\,'}{\Lambda})^2}\ \right] \nonumber \\ & + & \ldots, \eeqar{H3} - the momentum dependent part of the 2-body interaction is separated\\ - $v^{(0)}_{ij}$, 3-body and further terms of the Hamiltonian:\\ - - are assumed to be momentum independent }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.5cm} \B For simplicity consider the 2-body and higher order terms as: \beqar E & = & \int d^3r \left[ \int d^3p {{p^2}\over{2m}} f(\vec{r},\vec{p}) \right. \nonumber \\ & + & {a \over 2} {{n^2}\over{n_0}} + {b \over {\sigma + 1}} {n^{\sigma+1} \over n_0^{\sigma}} \nonumber \\ & + & \left. \frac{c}{n_0} \int d^3p d^3p'\frac{ f(\vec{r},\vec{p}) f(\vec{r},\vec{p}\,')} {1+(\frac{\vec{p}-\vec{p}\,'}{\Lambda})^2} \right] \, , \eeqar{H4} - the expression in square brackets is the pot. energy density, $u[f]$\\ - choosing $\sigma > 1 \\ \sim $ effective representation of 3- and more-nucleon interactions }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B The {\em mean field approximation}\\ - can be derived from the potential energy density, $u[f]$\\ - by taking the variational derivative\\ - - with respect to the distribution function, $f$: \beq \varphi \equiv \delta u[f]/ \delta f(\vec{r},\vec{p}) \, . \eeq{mf} \LT the momentum-dependent part of the mean field \beq \varphi_{mom}(\vec{r},\vec{p}) = \frac{2c}{n_0} \int d^3p' \frac{f(\vec{r},\vec{p}\,')}{1+(\frac{\vec{p}-\vec{p}\,'}{\Lambda})^2} \, , \eeq{momdepfi} - the momentum-dependent part of the single-particle energy: \beq \varepsilon_{mom}(\vec{r},\vec{p}) = {p^2\over{2m}} + \varphi_{mom}(\vec{r},\vec{p}) \, . \eeq{speg} With the momentum-independent part of the mean field[15,81-82]\\ - the full single-particle energy: \beq \varepsilon(\vec{r},\vec{p})= {p^2\over{2m}} + a {n\over{n_0}} + b({n\over{n_0}})^\sigma + \frac{2c}{n_0} \int d^3p' \frac{f(\vec{r},\vec{p}\,')}{1+(\frac{\vec{p}-\vec{p}\,'}{\Lambda})^2} \, . \eeq{spefull} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \subsection{Momentum dependent interaction} \B Simplified momentum-dependent mean field [78,82] - obtained by replacing $\vec{p}\,'$ in eq. (\ref{symu}) by its average, $<\vec{p}\,'>$: \beqar \varphi_{mom}({\vec{r},\vec{p}}) & = & \delta u[f]/ \delta f({\vec{r},\vec{p}}) \nonumber \\ & = & c \frac{n}{n_0} \left[ \frac{1}{1+(\frac{{\vec{p}}-<\vec{p}\,'>}{\Lambda})^2} \right] \nonumber \\ & + & c \frac{n}{n_0} \left \langle \frac{1}{1+(\frac{{\vec{p}}-<\vec{p}\,'>}{\Lambda})^2} \right \rangle_{p} \eeqar{nonsym} \B the manifest symmetry between $\vec{p}$ and $\vec{p}\,'$,\\ - (reflecting the symmetry of nucleon-nucleon interaction (\ref{symu}))\\ - is lost in this approximation\\ \B (\ref{nonsym}) is acceptable for HI collisions at high energies \B This momentum dependent mean-field potential \LT\\ - N-N attraction for the same or similar N-momenta\\ - Pot. energy: \beq u[f] = a {{n^2}\over{2n_0}} + b {{n^{\sigma+1}}\over{(\sigma+1)n_0^\sigma}} + c {n\over{n_0}} \int d^3p \ {f({\vec{p}}) \over {1+({{{\vec{p}}-{\vec{p}}_0}\over{\Lambda}})^2}} . \eeq{ped} - ${\vec{p}}_0$ is the local mean momentum\\ - - In the LR frame of the matter ${\vec{p}}_0 = 0$\\ - $n_0$ is the standard nuclear matter density\\ - $\Lambda$ describes the width of the momentum distribution }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \subsection{Momentum distribution} \B If $\exists$ a momentum independent mean field (in classical limit)\\ \LT the Maxwell-Boltzmann momentum distribution: \beq f_0(p) = {n \over g \ (2 \pi m T)^{3/2}} \exp{[-p^2/2mT]} . \eeq{mb} - $n$ is the baryon density of nuclear matter\\ - $g$ is the degeneracy of nucleons, $g=4$ \B With a momentum-dependent mean field\\ - the exponent of the momentum distribution will contain \\ - the momentum-dependent part of the single-particle energy (\ref{speg}) \\ \LT no longer simple Maxwellian momentum distribution. The single-particle energy corresponding to eq. (\ref{ped}): \beqar \varepsilon(\vec{r},\vec{p}) = {p^2\over{2m}} + a {n\over{n_0}} + b({n\over{n_0}})^\sigma &+& c{n\over{n_0}} {1 \over {1+({{\vec{p}}-{\vec{p}}_0 \over \Lambda})^2}} \nonumber \\ &+& c{n\over{n_0}}\Biggl<{1 \over {1+({{\vec{p}}-{\vec{p}}_0 \over \Lambda})^2}} \Biggr>_p . \eeqar{spe} - For static nuclear matter $\vec{p}_0$ vanishes \\ \LT at $T=0$ the integral in the last term of (\ref{ped}) can be carried out: \beq \varepsilon^0_{MD}(\vec{r};n) = 3c {n\over{n_0}} \Bigl({{\Lambda}\over{p_{F}}}\Bigr)^3 \Bigl[ {{p_{F}}\over{\Lambda}} - \tan^{-1}({{p_{F}}\over{\Lambda}}) \Bigr]. \eeq{momint} where $p_{F}$ is the Fermi momentum. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} The momentum-dependent part of the interaction\\ \LT the following distribution (see assignment [4.a]): \beq f(p) = N \exp\left[-{1 \over T} \left({p^2\over{2m}} -c({n\over{n_0}}){p^2\over{p^2+\Lambda^2}}\right)\right] \eeq{distr} - $N$ is the value of the distribution function at zero momentum\\ (Because the momentum-independent term is absorbed into the momentum-independent part of the potential.) \B {\em approximate} the full distribution with a Maxwellian\\ - introduce an `equivalent mass', $m_{eq}^{MB}$, \\ - which can be used at given $T$ \& $n$ in a Maxwell-Boltzmann distribution\\ - with the closest correspondence to the real momentum-distribution \beq f_{eq}^{MB}(p) = N \exp{[-p^2/2m_{eq}^{MB}T]} \ , \eeq{eqmb} - $ N = n/[(2\pi m_{eq}^{MB} T)^{3/2} \ g]$.\\ - equivalent mass depends on the temperature and the density\\ - but it is momentum-independent, $m_{eq}^{MB}= m_{eq}^{MB}(n,T)$ \B When $f(\vec{p})$ is approximated by the equivalent MB distribution,\\ - $m_{eq}^{MB}$ is fixed by the condition $$ f(p=0) \ = \ f_{eq}^{MB} (p=0) . $$ - This yields an equivalent mass of $$ m_{eq}^{MB}(n,T) = $$ \beq {1 \over{2\pi T}} \left\{ \int d^3p \exp\left[ -{{ p^2}\over{T}} \left( {1 \over{2m}} - c \left({n\over{n_0}} \right) \frac{1}{p^2+\Lambda^2}\right) \right] \right\}^{2/3} . \eeq{mass_eq} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{The partition function} \B momentum dependent interaction $\sim $ EOS\\ - the thermal part of EOS: modified by momentum dependent interaction\\ - because the equilibrium momentum distribution is also changed\\ - - E.g. the incompressibility will increase more rapidly with $T$ \B thermodynamic perturbation theory will be applied [28] \LT\\ - the free energy of the system: $$ F = F_0 + + {1\over{2T}} \left< (V-)^2 \right> , $$ - we take the kinetic energy to determine the 0th order, $F_0$\\ - only the first nonvanishing correction will be evaluated\\ - given by the average of the interaction over the phase space, $$ \B Take the kinetic energy only as the single particle contribution\\ - $E = \sum_{i=1}^{A} p_i^2 / 2m$, then\\ - the 0th order of the free energy: \beqar F_0 = -T \ln Z_0 &=& -T \ln \left\{ \int ' d\Gamma e^{-E/T} \right\} \nonumber \\ &=& -A T + AT \ln \left[ {{n(2\pi \hbar)^{3}}\over{g\ (2\pi m T)^{3/2}}} \right] , \eeqar{eos11} - $Z_0$ is the canonical partition function\\ - $ \int ' d\Gamma \ \ \sim $ the proper Boltzmann counting in the phase space integral }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B The average of potential energy over the phase space: \beq = {{\int ' d\Gamma \ V }\over {\int ' d\Gamma }} = e^{(F_0-E)/T} \ \int ' d\Gamma \ V \\ . \eeq{f_def} The EOS can be obtained by evaluating the canonical partition function or the free energy, $F$, (see [78] and assignment [4.b]). \bigskip \begin{center} \begin{tabular}{|rcl|c|} \hline\hline Thermodynamical & & Partition & Statistics \\ potential & & function & \\ \hline $S(E,V,N) $ &=& $\ln Z_{mc}(E,V,N) \ [=\ln\Omega(E)]$& micro canonical \\ $F(T,V,N) $ &=& $\ln Z_{c}(T,V,N)$ & canonical \\ $\Omega(T,V,\mu)$ &=& $\ln Z_{gc}(T,V,\mu)$ & grand canonical \\ \hline\hline \end{tabular} \end{center} Table 4.3 {\it Connection between the partition functions and thermodynamical potentials } \bigskip The thermodynamic potentials and the partition function are connected to each other as shown in Table 4.3. This way, by calculating the partition function in a microscopic theory one can obtain the EOS. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{1.5cm} \section{Assignment 4} \begin{description} \item[4.a] Calculate the distribution function, $f(p)$, if the single particle energy is given by eq. (\ref{spe}). \item[4.b] Calcualte the nuclear EOS for the momentum dependent interaction given by eq. (\ref{spe}), using the thermodynamic perturbation theory (up to first order). \end{description} }%end tr-page % transparency ================================================================ \end{document}