** Next:** 3.2.2 The Corrected Asymmetry
** Up:** 3.2 Measuring the Internal
** Previous:** 3.2 Measuring the Internal

Consider a pair of opposing counters--*e.g.* *L* and *R* in
Fig. 3.1 with a magnetic field applied parallel to the
axis.
The ``raw asymmetry'' of the
histograms *N*_{L} (*t*) and *N*_{R} (*t*) is defined as

| |
(7) |

where

| |
(8) |

| |
(9) |

so that

| |
(10) |

The reason for introducing is to eliminate the
muon lifetime (which is a well known quantity)
and the random backgrounds and . Ideally,
the two counters in question are identical
to one another, so that the histograms recorded by the two counters differ
only by a phase. In this idealistic situation the
difference in phase between the histograms
is due solely to the geometry
of the positron counters with respect to the sample.
The polarization of the muon spin which is seen by
each counter is

| |
(11) |

| |
(12) |

where and are the initial phases of the muon spin
polarization vector in counters *L* and *R*, respectively.
If the counters are aligned precisely opposite one another, the
difference between these phases is
, so that
| |
(13) |

The counters *L* and *R* measure the projection of the muon polarization
on the axis so that
| |
(14) |

Thus Eq. (3.7) becomes
| |
(15) |

If and then Eq. (3.15) reduces to
| |
(16) |

If ,
then using Eqs. (3.7), (3.8),
(3.9) and (3.16) the *x*-component of the
polarization can be written as
| |
(17) |

** Next:** 3.2.2 The Corrected Asymmetry
** Up:** 3.2 Measuring the Internal
** Previous:** 3.2 Measuring the Internal