Spin Precession

The surface muons arrive at the target highly polarized with their spin anti-parallel to their momentum direction. If some net average polarization S remains after the muons have stopped in the target, then the spin will precess with angular frequency $\omega$ along an axis parallel to any existing magnetic field B. The precession frequency is proportional to B and to the muon's g-factor. For a free muon, the precession rate $\omega/B = 8.5\times 10^8$ rad/sT, which corresponds to a revolution frequency of 13.5 kHz per gauss. In the case of free muonium, i.e. a muon bound to an electron, in a spin-triplet state, the precession rate is much larger, $\omega/B =1760.8\times 10^8$ rad/sT. However, we do not expect muonium to form at any appreciable level in the stopping target we will use and, as outlined below, faster precession is better.

If S and $\omega$ are both non-zero, the precession of the muon spins will produce a change in the angular distribution of the decay positrons, and, consequently, a change with time of the number of counts registered in any element of a segmented detector. This precession will alter the expected pure exponential shape of the decay and is therefore a potential source of systematic error. The detector asymmetry, magnetic field, and residual muon polarization can never be guaranteed to vanish individually at the ppm level. To understand how spin precession affects the extraction of $\tau_{\mu}$, we divide the contribution according to the three major items which will influence the result. They are:

1.

The asymmetry in the detector.

2.

The residual average polarization of an individual stopped muon.

3.

The residual ensemble-averaged polarization of the muons at the end of the accumulation period.

With respect to a fixed orientation of the muon spin, the detector can be expected to be asymmetric due to alignment, energy acceptance i.e. threshold and intrinsic efficiency. All of these are ultimately related to the angular and energy distribution of the decay positrons; both depend on the angle $\alpha$ between the momentum of the decay positron and the muon spin. With a threshold at $y = \xi$ and an average remnant polarization of the muons with value S, the detected fraction of events $dP(\alpha)/d\Omega$ is

$$\begin{array}{rl} \frac{dP(\alpha)}{d\Omega} & = \int\limi... ...{1}{3}+\frac{2}{3}\xi^3-\xi^4\right)\cos\alpha \cr \end{array}$$ (12)

which shows that the modulation amplitude approaches $\frac{1}{3}S$ for a nearly 100% energy acceptance and a completely uniform detector. The modulation amplitude goes to zero by adding the rates of tile pairs which are diametrically opposed in order to correct for the shift, assuming identical geometrical and intrinsic efficiency. This stresses the importance of a measurement system with point-symmetry such as the µLan detector with its 90 tile pairs.

For the lifetime analysis, consider the sum of the two tile pairs as a single counter. The individual tiles in this pair are defined to have overall geometrical and intrinsic efficiencies of $\eta_\alpha$ and $\eta_{\overline{\alpha}}$. Considering their respective thresholds separately as energy-dependent differences introduces the additional factors of $\xi_\alpha$ and $\xi_{\overline{\alpha}}$. For a non-zero magnetic field, a small modulation will be observed whose amplitude A is given by

$$A = S\left[ \frac{1}{3}\left(\eta_\alpha - \eta_{\overline... ..._{\overline{\alpha}}\xi_{\overline{\alpha}}^4 \right) \right]$$ (13)

A typical value for the detection threshold is 2 MeV corresponding to $\xi$ = 0.04. The first term on the right hand side therefore dominates the amplitude, i.e. $A \approx \frac{1}{3} S (\eta_\alpha - \eta_{\overline{\alpha}})$. This states that it is important to keep the geometric and intrinsic differences small between the pairs of tiles. Integrated over the whole detector, the intrinsic efficiency component in this term will largely vanish. However, a global alignment asymmetry can certainly be expected. Therefore, we take the value for the detector-related dilution of the asymmetry to be no smaller than $(\eta_\alpha - \eta_{\overline{\alpha}}) = 0.02$.

Muons enter the target with nearly 100% polarization. The use of a sulfur target is known to depolarize the muons during the stopping phase to a level below approximately 3%. Using this figure, the residual polarization average for a single muon, at the time of stopping, has the value <S> = 0.03.

At PSI, the ``pulsed'' beam will be made by simply chopping out approximately 90% of the beam. The 10% accumulation period should have no structure in the rate of incoming muons. Therefore, muons arriving early in this period will have precessed more than those arriving late. This implies that the muon spins will be automatically dephased with respect to one another, depending on the size and direction of an external magnetic field. For a sample of muons stopped in the target with their spin vectors initially aligned antiparallel to the beam direction, the residual ensemble-averaged polarization at the end of the accumulation period is $\kappa$. For example, a field strength which precesses the spin by $2\pi$ during a full accumulation period results in a depolarized distribution, i.e., $\kappa = 0$. The cancellation of the asymmetry will not be complete because some of the earliest arriving muons decay. A plot of $\kappa$ versus the length of the accumulation period, T_acc, is shown in figure 12 for a magnetic field strength of 75 G. It has minima when the product of the magnetic field and the accumulation period combine to rotate the muon spin in integer multiples of $2\pi$. At this field, the first minimum occurs for a $1\mu s$ accumulation period. We have tested small permanent rare-earth magnets which are capable of making such a field easily over the area of the disk-like target. Figure 13 illustrates the effect for a series of 15 muons arriving during the accumulation period of $1~\mu s$ and a B field of 75 G. The average polarization of the ensemble is shown as the dark line in the figure and remains close to zero during the measuring period.

Figure 12: The dephasing factor $\kappa$ versus the length of the accumulation period, T_acc, is shown for a magnetic field strength of 75 G. A 100% polarized muon beam is assumed to arrive uniformly in time and stop in a target centered in a magnetic field oriented transverse to the beam axis. The minima correspond to integer numbers of spin revolutions during a full accumulation period.

Figure: Dephasing illustrated for a series of 15 muons arriving during the accumulation period of $1~\mu s$ and a B field of 75 G. The average polarization of the ensemble is shown as the dark line in the figure and remains close to zero during the measuring period.

Taken together, the detector symmetry, the depolarization during stopping and the dephasing reduces the influence on $\tau_{\mu}$ well below the necessary 1 ppm level. Figure 14 shows the shift in the measured $\tau_{\mu}$ versus applied external magnetic field for a series of simulations which progressively includes the factors discussed above. The top curve corresponds to a single muon, a single counting tile, and a non-depolarizing target. Next the summed response from a diametrically opposed tile pair is used assuming the tile pair efficiency is matched to no better than 2%. The next curve adds in the depolarization effect due to the use of a sulfur target and the final curve includes the dephasing process. We have used relatively conservative factors in this simulation. For a magnetic field near 75 G, the shift in $\tau_{\mu}$ is a few tenths of a ppm.

Figure 14: Expected shift in the measured lifetime as a function of the magnetic field. From top to bottom, the curves correspond to: single muon, single tile, no depolarization; single muon, tile pair ( $\Delta \eta =0.02)$, no depolarization; single muon, tile pair, depolarization (<S>=0.03); dephased muon ensemble $(\kappa =0.1)$, tile pair, depolariztion.