In high energy physics and astro-particle physics it is very common to use the natural units where \(\hbar = c = 1\). The correspondence between natural units and physical units can be established by use of
\[\hbar = 6.58 \times 10^{-16} {\;\mathrm{GeV}\cdot \mathrm{ ns}} = 1\;\Rightarrow 1\;\mathrm{GeV} = 1.52 \times 10^{15}\;{\mathrm{ns}^{-1}}\]
\[c = 30.0\;\mathrm{cm/ns} = 1 \Rightarrow 30\;\mathrm{cm} = 1\;\mathrm{ns}.\]
In these units there is then only one fundamental dimension:
Larmor radius, or gyroradius, \(r_L\), is the radius of the orbit of a charged particle moving in a uniform, perpendicular magnetic field, obtained by simply equating the Lorentz force with the centripetal force. It is defined as: \[q v B = \frac{mv^2}{r_L} \rightarrow r_L = \frac{p}{ZeB},\]
where \(p\) has replaced \(mv\) in the classical limit. However, this also holds for the relativistic generalization by considering \(p\) to be the relativistic 3-momentum. There are several adaptations of this formula, tuned to units natural to various scenarios. One such is
\[r_L \simeq 1 \;\mathrm{ kpc} \left(\frac{p}{10^{18}\;\mathrm{ eV}\cdot{c}}\right)\left(\frac{1}{Z}\right)\left(\frac{\mu\mathrm{ G}}{B}\right).\]
In cosmic ray physics, one often sees references in the literature to the rigidity of a particle, defined as:
\[ R \equiv r_L B c = \frac{pc}{Ze}. \]
:rightfinger: Note that the rigidity, \(R\) has units of Volts.
Another concept that we will use frecuently in cosmic-ray physics is the superpositoin model. In the superposition model, a nucleus with mass \(A\) and energy \(E(A)\) is considered as \(A\) independent nucleons with energy \(E_0\). In a spallation process the energy per nucleon is approximately cosnserved therefore:
\[A + p \rightarrow A_1 + A_2\]
\[\begin{alignedat}{2} &E(A) &= &A E_0,\\ &E(A_1) &= &A_1 E_0,\\ &E(A_2) &= &A_2 E_0 \end{alignedat}.\]
The cross-section of a reaction is a very important parameter. It can be considered as the effective area for a collision between a target and a projectile. The cross-section of an interaction depends on interaction force, the energy of the particle, etc…
Cross-section is typically measured in surface, \(\mathrm{cm}^2\) or “barns”:
\[ 1 \mathrm{ barn} = 10^{-24} \mathrm{cm}^2\]
The unit barn comes from the expression “big as a barn” as in the past physisits saw with surprise that interactions were more frequent than expected, and they thought the nucleus was in fact bigger than they thought… big as a barn.
If a flux of projectile particles are crossing a volume of target particles with cross section \(\sigma_N\) then the disapperance of flux will be proportional to the initial number, the length travelled and number of target particles:
\[ \mathrm{d} I = -I n \sigma_N \mathrm{d} x ,\]
where \(n\) is the number density, ie, the number of particles per volume unit:
\[ n = \frac{N}{V} \]
note that the number density is related with the mass density as:
\[n = \frac{N_A}{M} \rho_m = \frac{\rho_m}{m_N}\]
where \(N_A\) is the avogadro number, \(M\) is the total mass of a mol and \(m_N\) is the mass of is the mass of a single particles N making up the volume. Solving the equation above we have:
\[I = I_0 e^{-\frac{x}{n\sigma_N}}\]
where we can define:
\[\lambda = \frac{1}{n\sigma_N},\]
as the interaction length. Likewise if projectile particles are travelling at speed \(v\), the length travelled can be expressed as \(\mathrm{d}x = v \mathrm{d} t\) giving a similar expression with a time constant:
\[\tau = \frac{1}{n v\sigma_N}\]
Known as the lifetime. If several processes are taking place, we need to replace \(n\sigma_N\) as \(\sum n_i\sigma_i\), which gives:
\[\frac{1}{\tau_{total}} = \frac{1}{\tau_1} + \frac{1}{\tau_2} + ... + \frac{1}{\tau_n}\]