Bubble chamber

Consider a chamber with a uniform magnetic field through it, such that ${\displaystyle {\vec {B}}=B{\hat {e}}_{z}}$. From Newton's second law and the Lorentz force on a particle with mass ${\displaystyle m}$ and charge ${\displaystyle q}$, the equation of motion of the particle can be derived as follows:[4]

${\displaystyle {\frac {d{\vec {p}}}{dt}}=q{\vec {v}}\times {\vec {B}}}$

, where ${\displaystyle {\vec {p}}=\gamma m{\vec {v}}}$, the momentum of a particle with velocity ${\displaystyle {\vec {v}}}$ and ${\displaystyle \gamma }$ is the Lorentz factor. Note that ${\displaystyle \gamma }$ is a constant for a moving charge in a static magnetic field as no work is done by the field and as a result, the magnitude of the total velocity ${\displaystyle v}$ does not change.[5]

${\displaystyle {\frac {d{\vec {v}}}{dt}}=q{\vec {v}}\times {\vec {B}}={\vec {v}}\times {\vec {\Omega }}}$

, where ${\displaystyle {\vec {\Omega }}={\frac {q{\vec {B}}}{\gamma m}}}$. This leads to three equations defining the motion of the moving charge in 3 dimensional Cartesian space.

${\displaystyle {\begin{array}{lcl}{\dot {v}}_{x}=\Omega v_{y},\\{\dot {v}}_{y}=-\Omega v_{x},\\{\dot {v}}_{z}=0\implies v_{z}=const.\end{array}}}$

To solve the coupled differential equations defining the transverse motion of the particle, we can let ${\displaystyle {\check {v}}=v_{x}+iv_{y}\implies {\dot {\check {v}}}=\Omega v_{y}-i\Omega v_{x}=-i\Omega {\check {v}}}$.

Solving gives ${\displaystyle {\check {v}}=v_{\perp 0}e^{-i(\Omega t+\delta )}\implies {\begin{cases}v_{x}=v_{\perp 0}cos(\Omega t+\delta )={\frac {dx}{dt}},\\v_{y}=-v_{\perp 0}sin(\Omega t+\delta )={\frac {dy}{dt}}.\end{cases}}}$

${\displaystyle \delta }$ is a time-independent phase. The solution for ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle z}$ is then:

${\displaystyle {\begin{array}{lcl}x=x_{0}+{\frac {v_{\perp 0}}{\Omega }}sin(\Omega t+\delta ),\\y=y_{0}+{\frac {v_{\perp 0}}{\Omega }}cos(\Omega t+\delta ),\\z=z_{0}+v_{z}t.\end{array}}}$

This is a particle moving with a helical trajectory in the ${\displaystyle z}$ direction and circular transverse motion centred about the coordinates ${\displaystyle (x_{0},y_{0})}$. Note the radius is then given by ${\displaystyle r={\frac {v_{\perp 0}}{\Omega }}={\frac {\gamma mv_{\perp 0}}{Bq}}={\frac {p_{\perp }}{Bq}}}$, returning the well-known relationship between the measured radius of curvature in the known field and the particle's transverse momentum.

Notable discoveries made by bubble chamber include the discovery of weak neutral currents at Gargamelle in 1973,[6] which established the soundness of the electroweak theory and led to the discovery of the W and Z bosons in 1983 (at the UA1 and UA2 experiments). Recently, bubble chambers have been used in research on weakly interacting massive particles (WIMP)s, at SIMPLE, COUPP, PICASSO and more recently, PICO.[7][8][9]