View source for Thermodynamical Properties of Elementary Particles

[[Category:Thermodynamics of Universe|1]]





<p align="right">
''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.''</p>




<div id="tab:ter_sm">
{| cellpadding="5" cellspacing="0" border="1" align="center"
|+ |The Standard Model particles and their properties.
! rowspan="2" | Particles 
!rowspan="2" |Mass 
!colspan="2" |Number of  states
! rowspan="2" width="100" |$g$ (particles and anti-particles)
|-
|spin
|color
|- align="center" |
|Photon ($\gamma$) || 0 || 2 || 1 || 2
|- align="center" |
|$W^+,W^-$ || $80.4\, GeV$ || 3 || 1 || 6
|- align="center" |
|$Z$ || $91.2\,GeV$ || 3 || 1 || 3
|- align="center" |
|Gluon ($g$) || 0||2||8||16
|- align="center" |
| Higgs boson ||$>114\, GeV$||1||1||1
|- align="center" |
|Bosons 
| colspan="3" |
| 28
|- align="center" |
| $u,/,\bar{u}$ || $3\, MeV$ || 2||3||12
|- align="center" |
||$d,/,\bar{d}$||$6\, MeV$||2||3||12
|- align="center" |
|$s,/,\bar{s}$||$100\,MeV$||2||3||12
|- align="center" |
|$c,/,\bar{c}$||$1.2\,GeV$||2||3||12
|- align="center" |
|$b,/,\bar{b}$||$4.2\, GeV$||2||3||12
|- align="center" |
|$t,/,\bar{t}$||$175\,GeV$||2||3||12
|- align="center" |
|$e^+,\,e^-$|| $0.511\, MeV $||2||1||4
|- align="center" |
|$\mu^+,\,\mu^-$|| $105.7\,MeV$||2||1||4
|- align="center" |
|$\tau^+,\,\tau^-$|| $1.777\, GeV$||2||1||4
|- align="center" |
|$\nu_e,\,\bar{\nu_e}$|| $<3\,eV$||1||1||2
|- align="center" |
|$\nu_\mu,\,\bar{\nu_\mu}$|| $<0.19\,MeV $||1||1||2
|- align="center" |
|$\nu_\tau,\,\bar{\nu_\tau}$|| $<18.2\, MeV $||1||1||2
|- align="center" |
|Fermions
| colspan="3" |
| 90
|}
</div>


__NOTOC__



<div id="ter_1"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
Find the energy and number densities for bosons and fermions in the relativistic limit.
<div class="NavFrame collapsed">
  <div class="NavHead">solution</div>
  <div style="width:100%;" class="NavContent">
    <p style="text-align: left;">
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;  $\zeta (3) \approx
1.202$.</p>
  </div>
</div></div>



<div id="ter_2"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== 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.
<div class="NavFrame collapsed">
  <div class="NavHead">solution</div>
  <div style="width:100%;" class="NavContent">
    <p style="text-align: left;"> 
\[
\frac{n_i}{n_\gamma} = \left\{ 
\begin{gathered}
  \frac{1}{2}g_i \quad  for\; bosons\\
  \frac{3}{8}g_i \quad for\; fermions\\
\end{gathered}  
\right.;\quad 
\frac{\rho_i}{\rho_\gamma} = \left\{ 
\begin{gathered}
  \frac{1}{2}g_i \quad   for\; bosons\\
  \frac{7}{16}g_i \quad for\; fermions\\
\end{gathered}  \right.
\]
\[
\begin{gathered}
  \left(\frac{n_i }{n_\gamma},\frac{\rho_i}{\rho_\gamma} \right)_{\nu_{\mu ,e}}  = \left( \frac{3}{8},\frac{7}{16} \right);\;\;\left( \frac{n_i}{n_\gamma},
\frac{\rho_i}{\rho_\gamma} \right)_{e^\pm}  = \left(
\frac{3}{4},\frac{7}{8} \right);\\
  \left( \frac{n_i}{n_\gamma},\frac{\rho_i}{\rho_\gamma} \right)_\pi   = \left( \frac{1}{2},\frac{1}{2} \right);\;\;\left( \frac{n_i}{n_\gamma},\frac{\rho_i}{\rho_\gamma} \right)_{n,p}  = \left( \frac{3}{4},\frac{7}{8}
 \right) \\
\end{gathered}
\]
</p>
  </div>
</div></div>



<div id="ter_3"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 3 ===
Calculate the average energy per particle in the relativistic and non-relativistic limits.
<div class="NavFrame collapsed">
  <div class="NavHead">solution</div>
  <div style="width:100%;" class="NavContent">
    <p style="text-align: left;">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:
\[
\left\langle \varepsilon  \right\rangle  = m + \frac{3}{2}T \approx m
\]</p>
  </div>
</div></div>



<div id="ter_4"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 4 ===
Find the number of internal degrees of freedom for a quark.
<div class="NavFrame collapsed">
  <div class="NavHead">solution</div>
  <div style="width:100%;" class="NavContent">
    <p style="text-align: left;">For quark one obtains the following:
\[
g = 2(\, spin) \times 3(\, particle) \times 2(\, particles + \, antiparticle) = 12
\]</p>
  </div>
</div></div>



<div id="ter_5n"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 5 ===
Find the entropy density for bosons end fermions with zero chemical potential.
<div class="NavFrame collapsed">
  <div class="NavHead">solution</div>
  <div style="width:100%;" class="NavContent">
    <p style="text-align: left;">In the considered case one gets:
     \[
s = \frac{\rho  + p}{T};\quad p = \frac{1}{3}\rho
\]
Using the results of  [[#ter_1|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.</p>
  </div>
</div></div>



<div id="ter_5"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 6 ===
Find the effective number of internal degrees of freedom for a mixture of relativistic bosons and fermions.
<div class="NavFrame collapsed">
  <div class="NavHead">solution</div>
  <div style="width:100%;" class="NavContent">
    <p style="text-align: left;">\[
\rho  = \sum\limits_{i = bosons} g_i \frac{\pi^2}{30} T^4 +
\sum\limits_{j = fermions} \frac{7}{8}g_j\frac{\pi^2}{30}
T^4 \frac{\pi^2}{30}g^* T^4,
\]
where $g^* $ is the effective number of degrees of freedom:
\[
g^*  = \sum\limits_{i = bosons} g_i + \frac{7}{8}\sum\limits_{j
= fermions}g_j.
\]
    </p>
  </div>
</div></div>



<div id="therm5"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== 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 $T$.
<div class="NavFrame collapsed">
  <div class="NavHead">solution</div>
  <div style="width:100%;" class="NavContent">
    <p style="text-align: left;">\[
g^*  = \sum\limits_{i = bosons}g_i \left(\frac{T_i}{T} \right)^4  + \frac{7}{8}\sum\limits_{j = fermions}g_i \left( \frac{T_j}{T}\right)^4 
\]      </p>
  </div>
</div></div>



<div id="ter_6"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 8 ===
Find the effective number of internal degrees of freedom for the Standard Model particles at temperature $T>1TeV$.
<div class="NavFrame collapsed">
  <div class="NavHead">solution</div>
  <div style="width:100%;" class="NavContent">
    <p style="text-align: left;"> At temperature $1\, TeV$ all the Standard Model
particles are relativistic (see [[#tab:ter_sm|table]]). Therefore
\[
g^*  = 28 + \frac{7}{8} \times 90 = 106.75.
\]     </p>
  </div>
</div></div>



<div id="ter_7"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 9 ===
Find the change of the number of internal degrees of freedom due to the quark
hadronization process.
<div class="NavFrame collapsed">
  <div class="NavHead">solution</div>
  <div style="width:100%;" class="NavContent">
    <p style="text-align: left;">  
At temperature $T \approx 0.2\,  GeV$ quarks and gluons transform into colorless hadrons (protons and neutrons). Therefore
\[
\Delta g = 16\,(gluons) + 12 \times 6\,(quarks) -
8\,(nucleons) = 80.
\]
 </p>
  </div>
</div></div>


<div id="ter_8"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 10 ===
Find the relation between the energy density and temperature at
$10^{10}\, K$.
<div class="NavFrame collapsed">
  <div class="NavHead">solution</div>
  <div style="width:100%;" class="NavContent">
    <p style="text-align: left;">  
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 \[\alpha  =
\frac{\pi^2}{15}\frac{k^4}{(\hbar c)^3} \approx 7.56 \cdot
10^{-16}\frac{ J}{ m^3K^4}.\]
 </p>
  </div>
</div></div>


<div id="ter_9"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 11 ===
Find the ratio of the energy density at temperature $10^{10}\, K$ to that at $10^8\, K$.
<div class="NavFrame collapsed">
  <div class="NavHead">solution</div>
  <div style="width:100%;" class="NavContent">
    <p style="text-align: left;">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:
\[
\rho  = \left( 1 + \frac{7}{4} \right)\rho _\gamma   =
\frac{11}{4}\rho _\gamma \Rightarrow
  \frac{\rho\left(T_2 = 10^{10} K \right)}{\rho \left( T_1 = 10^8 K \right)} = \frac{18}
{11}\left(\frac{T_2}{T_1}\right)^4\simeq1.6\cdot10^8.\]
 </p>
  </div>
</div></div>


<div id="razm40"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 12 ===
Determine Fermi energy of cosmological neutrino background (CNB) (a gas).
<div class="NavFrame collapsed">
  <div class="NavHead">solution</div>
  <div style="width:100%;" class="NavContent">
    <p style="text-align: left;">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
$$E_F  \simeq 0.4 \times 10^{-3} \, eV.$$
 </p>
  </div>
</div></div>