Thermodynamical Properties of Elementary Particles
In the mid-forties G.A. Gamov proposed the idea</br> of the "hot" origin of the World. Therefore thermodynamics</br> was introduced into cosmology, and nuclear physics too. </br> Before him the science of the evolution of Universe </br> contained only dynamics and geometry of the World.</br> A.D.Chernin.
| Particles | Mass | Number of states | $g$ (particles and anti-particles) | |
|---|---|---|---|---|
| spin | color | |||
| Photon ($\gamma$) | 0 | 2 | 1 | 2 |
| $W^+,W^-$ | $80.4\, GeV$ | 3 | 1 | 6 |
| $Z$ | $91.2\,GeV$ | 3 | 1 | 3 |
| Gluon ($g$) | 0 | 2 | 8 | 16 |
| Higgs boson | $>114\, GeV$ | 1 | 1 | 1 |
| Bosons | 28 | |||
| $u,/,\bar{u}$ | $3\, MeV$ | 2 | 3 | 12 |
| $d,/,\bar{d}$ | $6\, MeV$ | 2 | 3 | 12 |
| $s,/,\bar{s}$ | $100\,MeV$ | 2 | 3 | 12 |
| $c,/,\bar{c}$ | $1.2\,GeV$ | 2 | 3 | 12 |
| $b,/,\bar{b}$ | $4.2\, GeV$ | 2 | 3 | 12 |
| $t,/,\bar{t}$ | $175\,GeV$ | 2 | 3 | 12 |
| $e^+,\,e^-$ | $0.511\, MeV $ | 2 | 1 | 4 |
| $\mu^+,\,\mu^-$ | $105.7\,MeV$ | 2 | 1 | 4 |
| $\tau^+,\,\tau^-$ | $1.777\, GeV$ | 2 | 1 | 4 |
| $\nu_e,\,\bar{\nu_e}$ | $<3\,eV$ | 1 | 1 | 2 |
| $\nu_\mu,\,\bar{\nu_\mu}$ | $<0.19\,MeV $ | 1 | 1 | 2 |
| $\nu_\tau,\,\bar{\nu_\tau}$ | $<18.2\, MeV $ | 1 | 1 | 2 |
| Fermions | 90 | |||
Problem 1
Find the energy and number densities for bosons and fermions in the relativistic limit.
The energy density equals to $$ \rho = \left\{ \begin{array}{c} \frac{\pi^2}{30}gT^4 \quad \; for\;bosons\\ \frac{7}{8}\frac{\pi^2}{30}gT^4 \quad for \;fermions \end{array} \right. $$ Here $g$ equals to the number of internal degrees of freedom for the particle. For photon (with spin $S=1$) $g=2$ due to absence of longitudinal polarization for the zero-mass particles. The number density is the following: \[ n = \left\{ \begin{array}{c} \frac{\zeta (3)}{\pi ^2 }gT^3 \quad \;for\;bosons \\ \frac{3} {4}\frac{\zeta (3)}{\pi ^2 }gT^3 \quad \; for\;fermions \end{array} \right. \] Here $\zeta (x)$ is the Riemann $\zeta$-function; $\zeta (3) \approx 1.202$.
Problem 2
Find the number densities and energy densities, normalized to the photon energy density at the same temperature, for neutrinos, electrons, positrons, pions and nucleons in the relativistic limit.
\[ \frac{n_i}{n_\gamma} = \left\{ \begin{gathered} \frac{1}{2}g_i \quad for\; bosons\\ \frac{3}{8}g_i \quad for\; fermions\\ \end{gathered} \right.;\quad \frac{\rho_i}{\rho_\gamma} = \left\{ \begin{gathered} \frac{1}{2}g_i \quad for\; bosons\\ \frac{7}{16}g_i \quad for\; fermions\\ \end{gathered} \right. \] \[ \begin{gathered} \left(\frac{n_i }{n_\gamma},\frac{\rho_i}{\rho_\gamma} \right)_{\nu_{\mu ,e}} = \left( \frac{3}{8},\frac{7}{16} \right);\;\;\left( \frac{n_i}{n_\gamma}, \frac{\rho_i}{\rho_\gamma} \right)_{e^\pm} = \left( \frac{3}{4},\frac{7}{8} \right);\\ \left( \frac{n_i}{n_\gamma},\frac{\rho_i}{\rho_\gamma} \right)_\pi = \left( \frac{1}{2},\frac{1}{2} \right);\;\;\left( \frac{n_i}{n_\gamma},\frac{\rho_i}{\rho_\gamma} \right)_{n,p} = \left( \frac{3}{4},\frac{7}{8} \right) \\ \end{gathered} \]
Problem 3
Calculate the average energy per particle in the relativistic and non-relativistic limits.
Average energy per particle reads$\left\langle \varepsilon \right\rangle = \rho /n$. For $T \gg m$ \[ \left\langle \varepsilon \right\rangle = \left\{ \begin{gathered} \frac{\pi ^2}{30\zeta (3)}T\quad \; for\; bosons\\ \frac{7\pi ^4}{180\zeta(3)}T\quad for\; fermions\\ \end{gathered} \right. \] In the non-relativistic limit $(T \ll m)$ one obtains the following: \[ \left\langle \varepsilon \right\rangle = m + \frac{3}{2}T \approx m \]
Problem 4
Find the number of internal degrees of freedom for a quark.
For quark one obtains the following: \[ g = 2(\, spin) \times 3(\, particle) \times 2(\, particles + \, antiparticle) = 12 \]
Problem 5
Find the entropy density for bosons end fermions with zero chemical potential.
In the considered case one gets: \[ s = \frac{\rho + p}{T};\quad p = \frac{1}{3}\rho \] Using the results of problem of the present Chapter one obtains the following: \[ s = \frac{2\pi ^2}{45}g^* T^3, \] where $g^* = 1$ for bosons and $g^* = 7/8$ for fermions.
Problem 6
Find the effective number of internal degrees of freedom for a mixture of relativistic bosons and fermions.
\[ \rho = \sum\limits_{i = bosons} g_i \frac{\pi^2}{30} T^4 + \sum\limits_{j = fermions} \frac{7}{8}g_j\frac{\pi^2}{30} T^4 \frac{\pi^2}{30}g^* T^4, \] where $g^* $ is the effective number of degrees of freedom: \[ g^* = \sum\limits_{i = bosons} g_i + \frac{7}{8}\sum\limits_{j = fermions}g_j. \]
Problem 7
Generalize the results of the previous problem for the case when some $i$-components have temperature $T_i$ different from the CMB temperature $T$.
\[ g^* = \sum\limits_{i = bosons}g_i \left(\frac{T_i}{T} \right)^4 + \frac{7}{8}\sum\limits_{j = fermions}g_i \left( \frac{T_j}{T}\right)^4 \]
Problem 8
Find the effective number of internal degrees of freedom for the Standard Model particles at temperature $T>1TeV$.
At temperature $1\, TeV$ all the Standard Model particles are relativistic (see table). Therefore \[ g^* = 28 + \frac{7}{8} \times 90 = 106.75. \]
Problem 9
Find the change of the number of internal degrees of freedom due to the quark hadronization process.
At temperature $T \approx 0.2\, GeV$ quarks and gluons transform into colorless hadrons (protons and neutrons). Therefore \[ \Delta g = 16\,(gluons) + 12 \times 6\,(quarks) - 8\,(nucleons) = 80. \]
Problem 10
Find the relation between the energy density and temperature at $10^{10}\, K$.
Represent the expression for energy density of relativistic particles in the following form: \[ \rho = \left( \sum\limits_{i = bosons} \frac{g_i}{2} + \sum\limits_{j = fermions} \frac{7}{16}g_j \right)\rho _\gamma. \] At the temperature under consideration the relativistic particles are presented by photons, electron, positrons and neutrinos. Therefore \[ \rho = \left( 1 + \frac{7} {4} + \frac{7}{4} \right)\rho _\gamma = \frac{9}{2}\rho _\gamma = \frac{9}{2}\alpha T^4, \] where the radiation constant \[\alpha = \frac{\pi^2}{15}\frac{k^4}{(\hbar c)^3} \approx 7.56 \cdot 10^{-16}\frac{ J}{ m^3K^4}.\]
Problem 11
Find the ratio of the energy density at temperature $10^{10}\, K$ to that at $10^8\, K$.
Unlike the case considered in the previous problem, now only photons and neutrinos remain relativistic, because $10^8 K \sim 10^{ - 2} \, MeV$, and electron mass is $0.511 \, MeV$ is considerably greater. Therefore in the case under consideration one obtains the following: \[ \rho = \left( 1 + \frac{7}{4} \right)\rho _\gamma = \frac{11}{4}\rho _\gamma \Rightarrow \frac{\rho\left(T_2 = 10^{10} K \right)}{\rho \left( T_1 = 10^8 K \right)} = \frac{18} {11}\left(\frac{T_2}{T_1}\right)^4\simeq1.6\cdot10^8.\]
Problem 12
Determine Fermi energy of cosmological neutrino background (CNB) (a gas).
For a relativistic Fermi-gas the Fermi energy reads the following $E_F = \hbar c\left( 3\pi ^2 n \right)^{1/3} ;$ $\hbar c \simeq 197.3 \, MeV\cdot \, fm$, $n \simeq 300 \, cm^{-3}$, so $$E_F \simeq 0.4 \times 10^{-3} \, eV.$$
