Introduction

The formation of muonium from a positive muon and an electron

$$\mu + e \rightarrow Mu$$ (1)

is usually described by the first order kinetic equation

$$\displaystyle\frac{dn}{dt}=-kn.$$ (2)

Here n(t) is the muonium formation rate and k is the reaction rate [1].

The resulting muonium spin state is evenly divided between the S-state and the triplet state. The polarization of those muons in the S-state is completely lost during the muonium formation process [2] and the muon polarization can be described in zero magnetic field as

$$P_{Mu} (t) = 1 - \displaystyle\frac{1}{2}\int\limits_0^t n(t')dt' \;.$$ (3)

From (2) and (3) we have

$$P(t) \propto A+B e^{-kt}.$$

The interaction between a muon and an electron in liquid helium can be described as a Coulomb attraction of two charges moving in the viscous regime with mutual mobility b. Let the radial density distribution function between muon and electron be a Gaussian law

$$W(r) = \displaystyle\frac{1}{\Delta^3 \pi^{3/2}} \exp[-(r/\Delta)^2].$$ (4)

In this case the polarization function P(t) should be described by the formula

$$P(t) = 1 + \displaystyle\frac{2}{\sqrt{\pi}}x \cdot \exp(-x^2)- {\rm erf}(x),$$ (5)

where parameter x is determined by the expression

$$x=(3 b e t)^{1/3}/\Delta.$$

Here $\Delta$ is the scale of the Gaussian law Eq.(4), b - mutual mobility, e - electric charge of muon and electron, and t is time.

The behavior of Eq.(5) is close to exponential $\exp(-kt)$. This explains why the muonium formation process is often be described as a chemical reaction.

However in some cases (specifically in superfluid helium), the approximation (5) is inadequate [3]. The standard description of the complicated muonium formation process as a superposition of fast and slow subprocesses is known to be a very crude and inadequate model.

In superfluid helium a positive charge forms a `snow ball' with mass $M_{+} \simeq 40 - 50$ He atoms and the electron is localized in a cavity with hydrodynamic mass $M_{-} \simeq 200$ He atoms (see the review of V.B.Shikin [4]). This is the physical reason why Mu formation in superfluid helium is a rather long process compared to other substances [5].

In zero magnetic field Eq.(3) can be presented as a sum of two components

$$P_{Mu}(t) = \displaystyle{ \frac{1}{2}\int\limits_0^t n(t') dt' + \int\limits_t^\infty n(t')dt',}$$ (6)

which have an obvious physical sense. The first integral in this formula describes of Mu atoms in triplet state ( $\uparrow\uparrow$), which were formed until time t, and the second one corresponds to free muons, which have not yet combined with electrons to form Mu atoms.

We will now look at the muon polarization in weak transverse magnetic field (wTF), where the field is sufficiently small that the precession of the free muons is negligibly small during the muon lifetime of $2.2 \cdot 10^{-6}$ s. This is possible because the gyromagnetic ratio of muonium in the triplet state $\gamma_{Mu} = 1.404$ MHz/Oe is approximately 105 times larger than $\gamma_\mu$ for the bare muon. Typical values of magnetic field are about $H \leq 0.4~Oe$ for liquid helium. The spins of those Mu atoms which were born in time t' will precess with the Larmor frequency

$$\omega_{Mu} = \gamma_{Mu} H$$

and have a phase delay $\omega_{Mu}t'$.

The polarization function will have two components

$$P(t) = \displaystyle\frac{1}{2}\int\limits_0^t n(t') \cos[\omega_{Mu}(t-t')]dt'+ \int\limits_t^{\infty}n(t')dt',$$ (7)

where the first one describes the spin precession of Mu atoms with Larmor frequency $\omega_{Mu}$ with time delay t', and the second one describes the polarization of free muons.

Typically measurements are fit using some analytic model for P(t) using a least squares method (LSM) [2]. The number of fit parameters is of the order of 10. The success of this standard parametric procedure is mainly determined by whether or not correct the theoretical function (usually analytic) was chosen for P(t) or n(t) respectively. This is the main drawback of the standard procedure.

In this paper we propose a new nonparametric algorithm for recovering the muonium formation rate n(t) from the experimental $\mu SR$ data by using the program package RECOVERY for restoration of signals from noisy data which is based on the maximum likelihood method.