Difference between revisions of "Thermodynamical Properties of Elementary Particles"
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<div id="лейбел"></div> | <div id="лейбел"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | <div id="ter_1"></div> | ||
<div style="border: 1px solid #AAA; padding:5px;"> | <div style="border: 1px solid #AAA; padding:5px;"> | ||
=== Problem 1 === | === Problem 1 === | ||
− | + | Find the energy and number densities for bosons and fermions in the relativistic limit. | |
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
<div class="NavHead">solution</div> | <div class="NavHead">solution</div> | ||
<div style="width:100%;" class="NavContent"> | <div style="width:100%;" class="NavContent"> | ||
− | <p style="text-align: left;"> | + | <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_1"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 2 === | ||
+ | |||
+ | |||
+ | |||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;"> </p> | ||
</div> | </div> | ||
</div></div> | </div></div> |
Revision as of 17:52, 1 October 2012
In the mid-forties G.A.~Gamov proposed the idea of the "hot" origin of the World. Therefore thermodynamics was introduced into cosmology, and nuclear physics too. Before him the science of the evolution of Universe contained only dynamics and geometry of the World. 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 2