In the 2-counter setup of Fig. 3.11,
information is lost
by failing to detect decay positrons emitted in the *up* and *down*
directions. A more efficient setup is the 4-counter arrangement depicted
in Fig. 3.13, where additional
counters are placed on the *y* and
-*y* axes. The **L** and **R** counters monitor the *x*-component
of the polarization *P*_{x}(*t*) as before, while the **U** and **D**
counters measure the *y*-component of the polarization *P*_{y}(*t*).
Ignoring geometric misalignments and differences in counter efficiency,
*P*_{y}(*t*) differs from *P*_{x}(*t*) only by a phase of
.
In
terms of the field distribution *n*(*B*), *P*_{x}(*t*) and *P*_{y}(*t*)are defined as:

(40) |

P_{y}(t) |
= | ||

= | (41) |

where, . For an applied field in the -direction, the muon spin polarization transverse to the magnetic field

The real Fourier transform of the *complex muon polarization*
approximates the field distribution *n*(*B*_{z})
with statistical
noise due to finite counting rates
[34]. At the later times in the asymmetry spectrum, most of
the muons have already decayed. With few muons left, the statistics
at these later times are low, resulting in increased noise at the end
of the asymmetry spectrum. Using the fact that
is defined only for positive times ,
the field distribution
*n*(*B*_{z}) may be written:

where the gaussian

The *complex asymmetry* for the 4-counter setup is defined as:

where

where is the asymmetry function for the

In terms of the individual counters depicted in Fig. 3.13, the real asymmetry

(47) |

(48) |

In the present study, the real and imaginary parts of the asymmetry are fit simultaneously. For the vortex state of , the real asymmetry can be fit with Eq. (3.39). The imaginary part of the asymmetry can be fit with the same function, but with a phase difference of .

2001-09-28