The Corrected Asymmetry

In practice, differences in counter efficiency and deviations from perfect geometry complicate the above treatment. If the sensitivity of the individual positron counters are not the same, the values of the normalization constant $N^{\circ}$, the maximum decay asymmetry $A^{\circ}$and the random background levels $B^{\circ}$ will differ. The number of recorded events in each counter also depends on the solid angle they subtend at the sample. If the coverage of the solid angle is not maximized because of counter misalignments, there will be a decrease in the number of decay positrons detected. To account for these differences one can modify the above results by assuming:

$$\alpha = \frac{N_{j}^{\circ}}{N_{i}^{\circ}} \; \; , \; \; \b . . . . . . and}, \; \; \; \; \gamma = \frac{B_{j}^{\circ}}{B_{i}^{\circ}}$$ (29)

Experimentally one can determine these constants by examining data above T_c where the muon precesses around a spatially constant field. With the above considerations, the raw asymmetry A_ij(t) for the counters i and jcan then replaced by a corrected asymmetry A_corrected(t). This is obtained by substituting Eq. (3.29) into Eq. (3.20) and defining $A_{i}^{\circ} \equiv A^{\circ}$and $P_{i}(t) \equiv P_{x}(t)$:

$$A_{corrected}(t) = \frac{(1 - \alpha ) + A^{\circ} P_{x}(t) \ . . . . . . lpha \beta)}{(1 + \alpha ) + A_{raw}(t) \, (1 - \alpha \beta)}$$ (30)