Measuring the energy of particles

[aveline de grandpre]

If i have the momentum of the particle, could I measure their energy ? I'm talking about particles in a beam, they are moving in a relativistic speed.

 

[vanhees71]

If you know the momentum you also know the energy,
$$E=c \sqrt{m^2 c^2+p^2},$$
where $m$ is the mass (and mass is the invariant mass and nothing else!) of the particle, and $p=|\vec{p}|$ the three-momentum of the particle.

 

[ChrisVer]

And for very relativistic particles, that is $p^2 >> m^2$ you have:
$E = p \Big[ 1 + \mathcal{O}(\frac{m^2}{2p^2}) \Big]$
or that the energy is almost equal to the momentum...

So if you have an electron ($m=0.5MeV$) that has momentum $1GeV$, you can say that its energy is $1GeV$... the correction to the energy from the mass will only affect the decimals below $10^{-6} GeV=\frac{ MeV^2}{GeV}$ which you can check by actually putting numbers in the $\sqrt{\text{ }}$ expression given by vanshees:
$E=\sqrt{0.0005^2 + 1^2} GeV = 1.00000025 GeV$

 

[mfb]

If you don't know the particle type (and therefore the mass), measuring energy and momentum would allow calculating it - but that is rarely practical as the energy measurements are not precise enough. Measuring momentum and velocity does work, and it is the main idea how the LHCb detector identifies particles, for example. The energy can be calculated then.

 

[ZapperZ]

If i have the momentum of the particle, could I measure their energy ? I'm talking about particles in a beam, they are moving in a relativistic speed.

In a particle accelerator beam, the momentum is in fact the dynamical variable that you actually measure directly, not energy. This is true when you use, say, a dipole magnet to extract the "energy" and "energy spread" of the particle beam using a magnetic spectrometer. See Pg. 18 of William Barletta's lecture here:

http://uspas.fnal.gov/materials/09UNM/Unit_2_Lecture_4_Beam_Properties.pdf

It is only with the identification of the type of particle (electron, proton, etc... to obtain the rest mass) can you then extract the energy of the particle, using the equations that have been mentioned in this thread, or look on Pg. 17 of the same lecture notes.

Zz.