4-Counter Geometry and the Complex Polarization

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 $90^{\circ}$. In terms of the field distribution n(B), P_x(t) and P_y(t)are defined as:

$$P_{x}(t) = e^{- \sigma_{eff}^{2} t^{2} /2} \int_{- \infty}^{+ \infty} n(B) \cos \left[ \gamma_{\mu} Bt + \theta \right] dB$$ (40)

$$e^{- \sigma_{eff}^{2} t^{2} /2} \int_{- \infty}^{+ \infty} n(B) \cos \left[ \gamma_{\mu} Bt + \theta - \pi /2 \right] dB$$ P_y(t) =

$$e^{- \sigma_{eff}^{2} t^{2} /2} \int_{- \infty}^{+ \infty} n(B) \sin \left[ \gamma_{\mu} Bt + \theta \right] dB$$ = (41)

where, $G_{x}(t) = G_{y}(t) = e^{- \sigma_{eff}^{2} t^{2} /2}$. For an applied field in the $\vec{z}$-direction, the muon spin polarization transverse to the magnetic field B_z may be described by the complex quantity [81]

 

$$P^{\textstyle +}(t) P_{x}(t) + i \, P_{y}(t)$$ =

$$\equiv e^{- \sigma_{eff}^{2} t^{2}/2} \int_{- \infty}^{+ \infty} n(B_{z}) e^{i \left( \gamma_{\mu} B_{z} t + \theta \right)} dB_{z}$$ (42)

 

The real Fourier transform of the complex muon polarization $P^{\textstyle +}(t)$ 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 $P^{\textstyle +}(t)$is defined only for positive times $t \geq 0$, the field distribution n(B_z) may be written:

$$n(B_{z}) \simeq 2Re \left\{ \frac{\gamma_{\mu}}{2 \pi} \int_{ . . . . . . a_{\mu} B_{z}t - \sigma_{eff}^{2} t^{2}/2 \right)} dt \right\}$$ (43)

where the gaussian apodization parameter $\sigma_{A}$ is chosen to provide a compromise between statistical noise in the spectrum and additional broadening of the spectrum which such a procedure introduces.

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

 

$$A^{\textstyle +}(t) A^{\circ} P^{\textstyle +}(t)$$ =

$$A^{\circ} P_{x}(t) + i \, A^{\circ} P_{y}(t)$$ =

$$A_{x}(t) + i \, A_{y}(t)$$ = (44)

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 = R (right), L (left), U (up) or D (down)], may be written:

$$N_{i}(t) = N_{i}^{\circ} e^{-t/ \tau_{\mu}} \left[ 1 + A_{i}(t) \right] + B_{i}$$ (45)

where $A_{i}(t) = A_{i}^{\circ} P_{i}(t)$ is the asymmetry function for the i^th raw histogram. Rearranging Eq. (3.45):

$$A_{i}(t) = e^{t/ \tau_{\mu}} \left[ \frac{N_{i}(t) - B_{i}}{N_{i}^{\circ}} \right] - 1$$ (46)

In terms of the individual counters depicted in Fig. 3.13, the real asymmetry A_x(t) and the imaginary asymmetry A_y(t) may be written:

$$A_{x}(t) = \frac{1}{2} \left[ A_{R}(t) - A_{L}(t) \right]$$ (47)

$$A_{y}(t) = \frac{1}{2} \left[ A_{U}(t) - A_{D}(t) \right]$$ (48)

In the present study, the real and imaginary parts of the asymmetry are fit simultaneously. For the vortex state of $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{6.95}$ }$, 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 $90^{\circ}$.