3.2.4 Four-Counter Geometry and the Complex Polarization

Now consider the complete set of four positron counters in Fig. 3.1. Ignoring geometric misalignments and differences in counter efficiency, the x-component of the muon polarization P_x(t) [monitored by the L and R counters] differs from the y-component of the muon polarization P_y(t) [monitored by the U and D counters] by a phase of $90^{\circ}$. The two components of the muon polarization can be combined to form a ``complex'' polarization function

$$\tilde{P}(t) = P_{x}(t) + i \, P_{y}(t) \, ,$$ (25)

where

$$P_x (t) = G(t) \int_{0}^{\infty} \, n(B) \cos ( \gamma_{\mu} Bt + \theta ) dB \, ,$$ (26)

and

The complex asymmetry for the four-counter setup is defined as

where A_x(t) and A_y(t) are the real and imaginary parts of the complex asymmetry, respectively. The number of counts per second in the i^th counter (i = L, R, U or D) is  

$$N_{i}(t) = N_{i}^{\circ} e^{-t/ \tau_{\mu}} \left[ 1 + A_{i}(t) \right] + B_{i} \, ,$$ (27)

where $A_{i}(t) = A_{i}^{\circ} P_{i}(t)$ is the asymmetry function for the i^th raw histogram. Rearranging Eq. (3.29) gives  

$$A_{i}(t) = e^{t/ \tau_{\mu}} \left[ \frac{N_{i}(t) - B_{i}}{N_{i}^{\circ}} \right] - 1 \, .$$ (28)

In terms of the individual counters, the real asymmetry A_x(t) and the imaginary asymmetry A_y(t) are

In this thesis the real and imaginary parts of the asymmetry were fit simultaneously, assuming a phase difference of $90^{\circ}$ between them.