Curl up of charged particles in the Solenoidal Magnet

Figure 1 is a summary of when tracks will spiral in the solenoidal field. The line labeled p vs $\theta$ corresponds to tracks which have a radius of curvature of 0.30 meters. This corresponds to a transverse momentum of $p_{\perp}=0.201 GeV/c$. This means that if charged tracking extends out to a radius of 0.60m, these tracks will spiral in the solenoidal magnet. All tracks which lie above this curve have a radius of curvature larger than 0.30m and will presumably be lost when the encounter detector elements outside of the charged tracking. This line is defined by equation 1.  

$$p \sin (\theta ) < e B r_{max} \,\,\, ( = 0.201 GeV/c )$$ (1)

The nearly horizontal bars indicate where tracks make a specific number of turns before coming out of the downstream end of the solenoid. All particles above the $\frac{1}{2}$ turn line make less than a half a turn. The particles above the 1 turn line, but below the $\frac{1}{2}$ turn line make between $\frac{1}{2}$ and 1 turn before exiting the solenoid. This continues until we reach 90 degrees where in principal the particle will stay forever. Equation 2 defines the condition of these nearly horizontal lines. z_max is the length in z to exit the solenoid and is taken as $z_{\max}=3.65 m$. B is the magnetic field strength of 2.24 T and n is the number of turns made by the track.  

$$p \cos (\theta ) = \frac{z_{max} e B}{2\pi n} \,\,\, ( = \frac{0.39038 GeV/c}{n} )$$ (2)

The intersection condition between equations 1 and 2 is given by equation 3. The intercepts for several values of n are given in Figure 1.  

$$\tan (\theta ) = \frac{2\pi n r_{max}}{z_{max}} \,\,\, ( = \frac{0.201 n}{0.39038} )$$ (3)

For tracks that curl around more than once, the momentum measurement is made only from the first loop. However, the time-of-flight measurement is likely to be made from the entire flight path. This actually complicates things due to the energy loss, and corresponding slowing down of the lower momentum particles. Essentially, there will be a class of slow particles for which no time-of-flight or DIRC measurement will be likely in full B-field operation.

 

Figure:  Summary of spiraling tracks in the 2.24T field.