2.1 Muons and Muon Beams

Muons are elementary particles, second-generation charged leptons about 207 times more massive than electrons. Two properties of muons are key to the $\mu{\cal SR}$technique: (a) they have an intrinsic spin angular momentum (of $\hbar/2$)and a magnetic moment (several times larger than the proton magnetic moment) which provides a means for the environment to couple with the spin, and (b) they are almost always created or destroyed in weak interactions which maximally violate parity symmetry. Thus the positive muons from $\pi^{+}$ decay are 100% spin-polarized and each decays anisotropically via the weak interaction to an energetic positron whose momentum is correlated with the muons angular momentum at the instant of decay. The physical properties of muons that are important to $\mu{\cal SR}$ are listed in Table 2.1.

 

Mass, m_µ 105.658389(34) MeV/c²

$$\tau_{\mu}$$ 2.19714(7) µs

Charge, |q| e

Intrinsic spin 1/2   h/2pi

Magnetic moment, µ_µ 4.4904514(15) x 10^-26 J/T = 8.890577(21) µ_N

Spin g-factor, g_µ 2.002331846(17)

Gyromagnetic ratio, g_µ µ_µ/h 135.69682(5) MHz/T

 

Muon spin rotation requires an intense beam of spin polarized muons obtained from the decay of charged pions ($\pi^{\pm}$). The pions are first produced by the collisions of energetic protons with the nuclei of a target, typically made of carbon or beryllium, in the proton beam of a particle accelerator. Charged pions with mass $m_{\pi^{\pm}}$ = 139.5669 MeV/c² then decay to produce one muon neutrino and one muon;

with a lifetime of 26.03 ns. Conservation of energy and momentum dictate the outcome of the two-body final state: muons and neutrinos always have momentum 29.7885 MeV/c; the muon always carries away the same kinetic energy of 4.119 MeV in the pion's rest frame. As a result of parity violation in the weak decay of the spinless pion, muon neutrinos are always left-handed (their spin angular momenta pointing in the opposite direction to their linear momenta), so that positive muons (anti-muons, to be precise) produced in this way must also always have their spin angular momenta pointed exactly backward along their momenta. Anti-neutrinos have their spins and momenta parallel, so negatively charged muons similarly produced have spins parallel to their momenta.

It was standard practice early in the development of $\mu{\cal SR}$ to collect positive and negative muons from pions that decayed while in flight down the secondary beamline. In order to stop the maximum number of muons in the thinnest samples it was preferable to select those muons with the lowest energy, so ``backward decay" muons with momenta (in the pions' frame) opposite to the pions' momenta (in the lab frame) were usually chosen for experiments; however these are not entirely spin polarized and still have a relatively high momentum of $\sim$40-120 MeV/c.

So far this discussion has been equally applicable to positive and negative particles, but this ends when either pions or muons are stopped in matter due to the different chemical nature of negatively and positively charged particles in matter. Negative pions that stop in the target behave like heavy electrons and rapidly cascade down to tightly bound orbitals where they almost always undergo capture by the nucleus instead of decaying to negative muons. Positive pions that have come to rest in solids take up interstitial positions between atoms so they are too far from nuclei to be captured; as far as $\mu{\cal SR}$ is concerned their lifetime is unaffected by any properties of the target material.

Perhaps the most important development in muon beam technology was the realization that by removing the windows that isolated the primary and secondary particle beamlines, and turning down the momentum tuning of the secondary channel, the low momentum positively charged so-called surface muons could be brought out to an experiment. [2,3] Those positive pions that happen to come to rest just within the surface of the pion production target decay to muons that need penetrate only a short distance (a fraction of a millimeter at most) to escape from the target into the beamline vacuum, with momenta up to the maximum of $p_{\mu}^{\rm max}$=29.8 MeV/c.

These muons have a range of about 140 mg/cm² in water so they conveniently penetrate several thin windows but still stop in small samples, while the spins of the muons remain almost completely polarized. Surface muon beams are not mono-energetic since muons will come from pions decaying at various depths into the pion production target; those that start out deeper will spend more of their range and lose more energy on their way out. The resulting muon spectrum rises with momentum, then drops sharply at the ``surface muon edge" at $p_{\mu}^{\rm max}$.Secondary beamlines, usually with magnetic steering elements and positron separators ($\vec{E} \times \vec{B}$velocity selectors which remove positron contamination from the muon beam) are tuned to transport muons in a narrow momentum range $\Delta p_{\mu}/p_{\mu}$ of a few percent, with $p_{\mu}$ usually chosen to be just below the surface muon edge, in order to achieve the greatest beam intensity. Efforts to make still lower-energy polarized muon beams continue to this day, the motivation being the desire to deposit muons into extremely thin samples or with controlled depths into the surface layers of a sample. Compared to backward decay muons, surface muons are easier to collimate and focus into a clean, well-defined spot on a thin sample, minimizing the background due to muons that miss the sample. Most important, they arrive at the sample virtually 100% spin polarized.

The positive muon decays via the parity-violating weak interaction to produce an energetic positron and two neutrinos:

$$\mu^{+} \rightarrow e^{+} + \nu_e + \bar{\nu}_{\mu}$$ (1)

with a lifetime of $\tau_{\mu}$=2.19714(7) ${\mu}$s that is unaffected by sample properties or experimental conditions. In this decay the three body final state allows for a spectrum of positron energies from 0 to $E_{e^+}^{\rm max}$=52.3 MeV, since all combinations of neutrino and positron momenta that conserve energy and momentum are allowed. The decay rate also depends on the inner product ${\vec \sigma} \cdot {\vec p}$of the positron spin ${\vec \sigma}$ and momentum ${\vec p}$in such a way that the positron is emitted preferentially in the direction of the muon spin at the instant of decay. The probability per unit time of being emitted in a direction at an angle $\Theta$ to the spin is given by  

$$\frac{d^2W(\Theta)}{d\epsilon \, d(\cos \Theta)} = \frac{G^2... ...5}{192 \pi^3} \Lambda(\epsilon)[1 + a(\epsilon) \cos(\Theta) ]$$ (2)

where the asymmetry of the decay $a(\epsilon)$ is a function of the reduced positron energy $\epsilon = E_{{\rm e}^+}/E_{{\rm e}^+}^{\rm max}$ and is given by

$$a(\epsilon) = \frac{2 \epsilon - 1}{3 - 2\epsilon},$$ (3)

while the normalized spectrum of positron energies resulting from the available phase space is

$$\Lambda(\epsilon) = 2(3-2\epsilon)\epsilon^2.$$ (4)

These functions of positron energy are shown in Fig. 2.1, along with their product. The resulting angular distribution of decay positrons is shown in Fig. 2.2 for several values of positron energy. Since both the number of positrons and their decay asymmetry rise with energy, the asymmetry of the ensemble angular distribution is largely due to those positrons with energies above about 2/3 of the maximum energy.

  

Figure 2.1: Muon decay asymmtery $a(\epsilon)$, energy spectrum $\Lambda(\epsilon)$ and the product $a(\epsilon)\Lambda(\epsilon)$ ploted against the reduced positron energy $\epsilon$.

  

Figure 2.2: A polar-coordinate plot of the rate of positron emission from muon decay as a function of angle from the muon spin $\Theta$, at various energies $\epsilon$. The distribution has axial symmetry about the muon spin polarization direction, which points toward the right in this plot.

We cannot be certain in which direction a single muon spin was pointing from its single decay positron. However, we can determine the ensemble average polarization by measuring the angular distribution of positrons emitted in the decay of a large number of muons. Fast scintillators and phototubes give nanosecond timing resolution but do not yield any information about the positrons' energies. If positrons of all energies are detected with equal efficiency, we must integrate over the positron energy spectrum to obtain a theoretical ensemble average asymmetry $\langle a \rangle_{\epsilon} = 1/3$.In practice, this theoretical asymmetry is never achieved due to the use of positron detectors that cover quite a large solid angle, averaging over a range of $\theta$ in Eq. (2.4) which reduces the observed asymmetry considerably. With a few cm of absorber one can eliminate the low energy positrons, which have $a(\epsilon) < 0$and actually detract from the ensemble asymmetry, to increase the measured asymmetry and improve the signal-to-noise ratio. If the positrons with energy $\epsilon \leq 0.5$are absorbed, for example, the asymmetry of the remaining ensemble rises to 0.435. There is usually some material such as cryostat parts and sample mounts between the sample and positron detectors that will stop some of the low energy positrons. Overall, the maximum initial asymmetry measured by most spectrometers is typically about 0.25.