Muon stopping

The actual eigenstates are equal admixtures of the two unperturbed pure spin states when the field is exactly at the value at which the crossing would have occurred (v,m = 0). Since initially (when the muon stops) the system is in a well defined muon spin state, i.e., one of the two unperturbed pure spin states, the system oscillates at the frequency vT between the muon spin being along and opposite to the field, as implied by Eqs. 10 and 11. Thus, upon time averaging the positron counts the forward-backward asymmetry is reduced. 

Bilokon H, Castagnoli GC, Castellina S, Piazzoli BD, Mannocchi G, Meroni E, Picchi P, Vernetto S (1989) Flux of vertical negative muons stopping at depths 0.35-1000 hg/cm2. J Geophys Res 94 12,145-12,152... 

Figure 5.7 Depth dependence of cosmogenic 3He in a drilled core in Haleakala volcano. The diagram shows a very large deviation from an exponential depth dependence (i.e., mean absorption free path / = 165gcnT2) below 170gcnT2. The curve that yields a better fit to the experimental points is a combination of a simple exponential and the effect of muon-produced 3He via 6Li(n, a)3H. The dashed line is the assumed depth dependence of the muon stopping rate. After Kurz (1986).

A negative muon can participate in a variety of atomic and molecular processes. A muonic atom is formed when a muon stops in matter replacing an electron. A muonic atom interacting with ordinary atoms or molecules can form a muonic molecule. The latter in turn can result in fusion reactions between the nuclei if the target consists of hydrogen isotopes, a phenomenon known as muon catalyzed fusion (pCF) [6]. 

A single muon stopped in a target of deuterium-tritium mixture can catalyze more than 100 fusions, but this number is limited by two major bottle-necks. One is the rate at which a muon can go through the catalysis cycle before its decay (cycling rate), and another is a poisoning process called p-a sticking in which, with a probability u)s < 0.01, the muon gets captured after the fusion reaction to atomic bound states of the fusion product 4He, and hence lost from the cycle (see Section 5). 

Taking the fact that at 16 hPa 4% of all muons reach the 2S state and assuming a conservative number for the slowing down probability of pp(2S) initially formed at energies above 0.31 eV, we can give a preliminary number of 1.5% for the fraction of metastable np(2S) per muon stopped in hydrogen. 

In the third stage the 708 nm light is converted to 6.02 pm via third Stokes Raman shift in a high pressure hydrogen cell. The 0.5 mJ, 7 ns long pulses are injected into a multipass cavity inside the gas target in order to effectively illuminate the muon stop region. 

Fig. 8. The components of the laser system. The high power XeCl excimer laser pulse triggered my the muon entrance detector is converted in two steps to a high quality 7 ns long pulse of 708 nm which is shifted to the desired 6 pm light inside the multipass Raman cell. The light is injected into a multipass cavity to effectively illuminate the muon stop volume inside the PSC solenoid. High resolution frequency selection is provided by injection of a cw Ti Sa laser...

We now have the opportunity to perform a laser resonance experiment to measure the Lamb shift in muonic hydrogen. Taking into account the muon stop rate, entrance detector efficiency, long-lived metastable 2S population, laser transition probability, solid angles and detection efficiency of the X-ray detector we estimate our event rate on resonance to be 9 events per hour. 

From a simple cascade calculation model It can be shown that about 60% of the muons stopped in the target will pass through the D levels of course, the natural emission of X-rays, during the cascade of the iT, represents our main physical background, from which we have to sort out the small increase due to the laser-stimulated emission. 

The muon rate can be predicted from test measurements [31] to reach about 100 /i /(cm keV mA s) in the PSC solenoid, corresponding to about 600 muon stops per second in a 10 hPa H2 target, 15 cm long, for a beam size 5 x 15 mm. ... 

The technique of muon spin rotation ( SR) is described, with examples of its application in the fields of chemistry and solid state physics. It is shown how the raw experimental data contains information about the evolution of the spin polarization of muons stopped in matter. Fourier transformation provides a means of extracting the precession frequencies characteristic of various muonic species. Some manipulation of the raw data is essential to ensure accurate representation of the frequency information, and further techniques are often used to improve the final spectrum. These are discussed, and some examples are given of their effects. This is followed by descriptions of specific applications of Fourier transform / SR in the study of the light hydrogen isotope muoniim (Mu = /i e"), muonium-substituted free radicals, and paramagnetic states of the /A in sol ids. 

For continuous muon beam data, there is also a short time after the muon stop t = 0) for which the data are distorted and cannot be used. Often called initial dead time , this period is usually less than 30 ns, and with some care in setting up the counters and electronics can be reduced to as little as 5 ns. While often of little consequence in many p.SR experiments, the size of the dead time limits the maximum relaxation rate that can be measured, and disordered magnetic states with rare-earth moments often generate muon relaxation times of less than 10 ns, so initial dead time is an important consideration in the experiments described in this review. 

Fig. 12. Top High-pressure cell (CuBe) used in the studies of rare-earth metals and intennetallic compounds up to 0.9 GPa (9 kbar). Bottom ZF-(xSR spectrum of FM gadolinium metal inside the CuBe high-fvessure cell. The oscillating signal (see also inset) is the spontaneous spin precession pattern of Gd. The Gaussian relaxation spectrum comes from muons stopped in the cell walls. After Schreier et al. (1997) and Kalvius et al. (20001).

Collimation of the beam hardly works beyond a reduction of 50% in illuminated area, The collimator must be placed some distance in front of the sample to avoid that forward or backward detectors pick up positrons stemming from muons stopped in the collimator, The divergence of the muon beam and especially scattering of muons by the collimator and also in the start counter, the cryostat windows etc. break up the collimated beam again in the case of surface muons. 

Fig. 19. Angular dependence of the muon spin precession frequency Vjj of a single crystal of paramagnetic CeBj in transverse field. Such data can be used to fix the muon stopping site (see text). v i is the spin precession frequency for a bare i. From Amato et al. (1997a).

The earlier pSR studies were concerned with the peculiar temperature dependence of the spontaneous precession frequency caused by the interplay of the isotropic contact field and the anisotropic dipolar field at the muon site. The latter is clearly dependent on the spatial alignment of the Gd spins. Also, the muon stopping site needed to be determined. These data will be discussed in sect. 4.2.2 below. Later studies included more detailed investigations of the dynamical critical behavior in the paramagnetic state (see following section). 

Fig. 30. Critical spin fluctuations in Gd ftom ZF- iSR measurements. Left Anisotropy of ftie relaxation rate A just above the Curie point. After Hartmann et al. (1990a). Right Comparison of the temperature dependence of the relaxation rate measured in perpendicular geometry with the prediction of mode-coiq>ling ftieory for two muon stopping sites. The rates of 1989 were re-analyzed, changing slightly their absolute values (compared to those plotted in the left-hand panel). Tq and 7) mark the points where dipolar and uniaxial anisotropies start to play a significant role. After Hennebeiger et al. (1997).

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