# What is the temperature of an LHC Bunch?

From various places, I've found that an LHC bunch seems to be about 100mm long, 0.1mm wide, and contains about $10^{11}$ protons or anti-protons. The de Broglie wavelength is about $10^{-18}$ meters, so this is close to a plane wave for the purposes of using the S-matrix formalism. Each bunch is separated from the next bunch by about 10 meters (the beam is about 27km long, there are are 2808 bunches).

If I was co-moving with a bunch, what is the temperature (at the center of the bunch, say)? This I have not been able to find. A reference if possible.

• this is an awesome question. I once asked a bunch of LHC beam physicists this, and they just said something about emittance, how it was related to temperature, shrugged. – Shep Feb 14 '15 at 19:41

• Your second link gives $\beta^*=0.55\,\rm m$ and $\epsilon_n = 3.75\,\rm \mu m$, but is a little unclear on the meaning of the "normalized" emittance. This is nearly enough information to answer the question. – rob Feb 14 '15 at 20:54
• I have from the Moehl lecture, $k_BT\approx m_p(\beta\gamma c)^2\epsilon/\beta_x\approx m_pc^2\beta\gamma\epsilon_n/\beta_x$, where $m_pc^2\beta\gamma$ is very nearly the beam energy per proton, so $T\approx\mbox{(7e12eV)}*\mbox{(1.6e-19joule/eV)}*\mbox{(2.5e-6m)}/\mbox{(0.44m)}/\mbox{(1.38e-23joule/K)}$ $=\mbox{4.6e11K}$, taking numbers from rdemaria.web.cern.ch/rdemaria/www/hllhc11. I wasn't expecting something quite that high!? – Peter Morgan Feb 16 '15 at 18:48
• @DarioP thanks. So I find somewhere that the energy spread is about $\mbox{1e-4}*\mbox{7TeV}=\mbox{700MeV}=m_p\gamma (\Delta v)^2$, where $\gamma\approx \frac{\mbox{7TeV}}{\mbox{1GeV}}=7000$, so $m_p(\Delta v)^2\approx 0.1MeV$ and $T\approx \mbox{(1e5eV)}*\mbox{(1.6e-19joule/eV)}/\mbox{(1.38e-23joule/K)}$ $\approx \mbox{1e9K}$. Within a couple of orders of magnitude of the transverse temperature, slightly smaller than the transverse temperature at the IP where there's greater transverse pressure but slightly larger elsewhere. – Peter Morgan Feb 20 '15 at 16:08