... mixture[*]
When referring to a a mixture of hydrogen isotopes, a notation X/Y is used in this thesis, where X, Y are protium (1H), deuterium or tritium. It is usually assumed that the mixture has an equilibrium molecular composition, i.e. , X2:XY:Y2= cX2:2cXcY:cY2, where cX and cY denote the relative atomic concentrations. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... molecule[*]
To be precise, this should be called a muonic molecular ion, but we shall simply denote it as a muonic molecule according to the convention in the field. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...tex2html_comment_mark[*]
Recall that for the principle quantum number n, the orbital radius rn, the energy En and the orbital velocity vn are: $r_n=\frac{n^2\hbar ^2}{\overline{m} Z e^2}$, $E_n=-\frac{Ze^2}{2r_n} = -\frac{\overline{m} Z^2 e^4}{2n^2 \hbar ^2}$ and $v_n = \frac{Ze^2}{n\hbar}$, where $\overline{m}\sim m_{\mu}$ is the reduced mass of the muon and the nucleus with the charge Z. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... electron[*]
Note that the corresponding velocity of the electron differs from that of the muon by the factor $v_e/v_{\mu} \sim \sqrt{m_{\mu}/m_{e}}$ 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... fast[*]
This is expected from a naive argument that the muonic atom with higher n is much more extended in size, hence has larger cross sections, but the bigger effect presumably is the long range behavior of the potential for the excited states. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... effect[*]
A qualitative remark of the possibility of epithermal formation was given earlier by Rafelski [73]. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... D/T[*]
Markushin et al. also performed a Monte Carlo analysis of the triple mixture H/D/T [75,27] 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... scattering[*]
This is due in part to the existence of a virtual (nearly bound) state near $\mu p + p$ threshold. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... century[*]
It is said that there were over 800 papers published on the subject between 1750 and 1920s [86]. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... resources[*]
See recent calculations of muonic and other three-body systems, [88,89,36,90,91]. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...[*]
Taking out the factor 1/R from the definition of $\chi _(R)$ is a standard procedure which simplifies the form of the operator. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... integro-differential[*]
Integration comes from the operator Uij. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... threshold[*]
These should be compared to, for example, the $[\mu t]_{2s,p}$-d threshold which is $\sim 2$ keV higher. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... involved[*]
We note that an alternative term Hyperradius approach has been suggested by Alexander Matveenko as a more general name. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...$\nu _{f},K_{f})$[*]
There are recent studies on the [($d\mu t$)dee] energy level splitting (of a few meV level) due to the $d\mu t$interactions with the host molecular complex including the quadrupole finite-size corrections [135,136,137]. This would greatly increase the number of levels in the final state spectra (hence further complicating the notation!), but it is neglected here. Due to the Doppler effect the resonance profiles in the $\mu t$ lab frame are already broad (see the discussion later in this section), so presumably these splittings would not have much effect in our measurement of epithermal molecular formation, though more care may be necessary at low energy. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... footnote[*]
This possibility of what I call the ``resonant excitation'' of a D2 molecule by $\mu t$ is not apparent in the papers of Faifman et al. [133,134,71]. In fact, we point out that for kinematic reasons, this occurs only for $\nu _f \geq 4$ for most Kf (for all Kf, if the Kf distribution is relaxed to low temperature equilibrium). 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... footnote[*]
Padial, Cohen and Walker promised to extend their calculations to vibrational transitions [154], and we await their results eagerly. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... have[*]
We sometimes refer to these rates as the total formation rates, as opposed to the effective rates described below. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...tex2html_comment_mark[*]
In Ref. [134,71], the dependence on S in the fusion probabilities $W^{SF}_{\nu _f} (K_f)$ (Eq. 2.38) and $W^{SF}_{\nu _f}$ (Eq. 2.41) is not explicitly indicated, but they do indeed depend on S. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... unclear[*]
Note that their conclusion contradicts at least partly Adamczak's slow thermalization model [167], so they cannot both be right. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... MHz[*]
It is, however, possible to change this structure. For a measurement of pion lifetime, for example, the beam cycle was reduced by a factor of five [172]. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... delay[*]
When MWPCs were used, INH was also activated by a high voltage supply failure WCINH. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... events[*]
This was derived from comparing the number of triggers going into the Starburst (EVTR), and the number it actually accepted (STAR, a scaler counting Starburs events). 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... signal[*]
For the purpose here, this can be any signal uncorrelated to the muon trigger. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... detector[*]
Canberra, model FD/S-600-29-150-RM 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... detectors[*]
Canberra, model FD/CY-2000-37-300-RM 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... target[*]
< 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... target[*]
< 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... target[*]
Deviations of the capture ratio from the concentration ratio were reported for pionic hydrogen and deuterium (for a review, see [207]), but this does not affect our simulations, because of very small tritium concentrations used in the measurements. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... factor[*]
See Section 6.4 for the description of the Huff factor. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... Markushin[*]
Now at Paul Scherrer Institute, Switzerland. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...g[*]
The subscript g is given here to stress it is the width only of the Gaussian part of the beam, as opposed to that of the entire beam. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...ground[*]
The validity of assuming a constant background will be discussed later. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... hydrogen[*]
A fit with M=3 was also performed to investigate a possible existence of the copper component in the time spectrum, with two of the three lifetimes fixed to those of Au and Cu from Ref. [215], but this did not affect the derived hydrogen stopping fraction. Fixing of two lifetimes was necessary to avoid a fitting problem due to a strong correlation between them. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... results[*]
The time scales for the RF cycle and the gold signal are rather similar, but the background amplitude is several orders of magnitude smaller, so the effect should be negligible. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... mode[*]
In this mode, the detector can accept only one event per incident muon, as opposed to the multi hit mode in which multiple events per incident muon can be recorded. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...%[*]
One could conceive the background to have a form , where $\epsilon$ is the detection efficiency and the signal lifetime, if the background is an accidental one. In fact, , derived from the above, is close to the combined efficiency for the 1st electron obtained in Table 6.17 in page [*]. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... hydrogen[*]
The frame for the foil is rather thick, so the foils could be bent such that the view from the side is blocked by the foil frame. We have observed evidence for such effects in the November 1993 runs with the MWPC imaging system. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... electrons[*]
Recall that any event, including the delayed electron event, can be collected only when the Event Gate is open. Thus, if for example, a Si event occurred at 9 $\mu $s, there is only about a 1 $\mu $s window for the delayed electrons to be detected to fulfill the Del cut, as opposed to the nominal Del window (t2 - t1) of $\mu $s. The Del time efficiency is constant as long as , where TEV is the event gate width. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... ch[*]
Recall that 1 ch $\sim$ 1 keV. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... 1671-83[*]
Target ID = II-9. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... 1709-30[*]
Target ID = II-13, II-14 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... place[*]
For Run D below, the standard emission target without overlayer (target ID=II-7) was used. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... ambiguity[*]
In addition to lack of independent energy calibration, we note that the simulation does not explicitly include the resolution of the detector, though the simulated physics processes in the detector partly account for it. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... target[*]
Target ID I-8 in Table 4.2 in page [*]. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... frames[*]
Recall that muons bound in the atomic states of heavy elements disappear quickly (within a few 100 ns) due to a high rate of nuclear muon capture via the semi-leptonic weak interaction. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... target[*]
Target ID I-8 in Table 4.2 in page [*]. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... frames[*]
Recall that muons bound in the atomic states of heavy elements disappear quickly (within a few 100 ns) due to a high rate of nuclear muon capture via the semi-leptonic weak interaction. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... parameters[*]
See Section 6.1 for the notation of beam parameters. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... target[*]
Target ID = I-1 in Table 4.2 in page [*]. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... run[*]
Target ID = I-8. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... energy[*]
Recall that where t is the time of flight over the drift distance l. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...%[*]
Target ID = I-7. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... tritium[*]
The specific activity of tritium is 2.58 Ci/cm3 at STP, hence 1 T$\cdot l$ corresponds to about 3.4 Ci. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... tritium[*]
The specific activity of tritium is 2.58 Ci/cm3 at STP, hence 1 T$\cdot l$ corresponds to about 3.4 Ci. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... tritium[*]
The specific activity of tritium is 2.58 Ci/cm3 at STP, hence 1 T$\cdot l$ corresponds to about 3.4 Ci. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... gate[*]
Here I am referring to the master gate for the ADC; the use of a gate for individual channels was redundant in our case. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... discrimination[*]
LLD was specified by a CAMAC command, and set a threshold, below which energy only zero was recorded in the ADC. This could save read out time. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... IV)[*]
For our purpose, an event where the silicon detector was triggered due to noise on the logic signal line can be included in [b]. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... detectors[*]
This background is removed in future runs by placing a thin Cu foil to shield the Si from the beam. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... measurements[*]
The notation MOD is given because of its sensitivity to the $\mu t$ moderation process in the US D2 layer 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... distribution[*]
Flat-top Gaussian mm was assumed for the fusion radial profile in these simulations 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... negligible[*]
This approximation introduces an error of no more than 0.5% in the final correction. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... other[*]
In a very crude statistical estimate, 6 cases out of 16 data point have the deviation between two energy cuts which is larger than the uncorrelated error (i.e., without normalization errors). This is perfectly consistent with statistical fluctuation. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... introduced[*]
The detection of $\mu t$ using the fusion signal is assumed to be constant in this model. This is not precisely correct due to the energy dependence of the molecular formation rates. In addition, the effect of muon decay is obviously not included in the model. The fraction of the $\mu t$ reaching DS before muon decay changes if the energy of the transmitted $\mu t$ changes. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... each[*]
The former was estimated by doing similar $\chi ^{2}$ minimization with fits from varied MC geometry, but with much reduced number of fit points, while the latter was estimated by observing the shift in mean time-of-flight with one fixed SE. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... processes[*]
The energy scaling method in Ref. [234] is slightly different from ours described here, but this difference alone would not resolve the apparent discrepancy between the two results. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... rate[*]
Assuming the Doppler width given by Faifman. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... factorized[*]
See discussion of the factorization approximation in the appendix B. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... frame[*]
This may be possible, for example, in the case of molecular formation in a solid at energies low compared to its Debye temperature, where recoilless processes analogous to the Mossbauer effect may dominate. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... equations[*]
Scrinzi et al. give an example of such rate equations in Ref. [12], though for different purposes. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... reduced[*]
Note that resonant deexcitation (e.g., ) is possible for high temperature targets, but is negligible at low temperature. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... elements[*]
I am using the language of the perturbation theory here with some reservation, recognizing the controversy in the formulation of the MMC formation and back decay processes. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... arrangement[*]
Emission target with ct=0.1% with a 14 T$\cdot l$ D2 moderation layer in the US, and 3 T$\cdot l$ D2 in the DS. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... emission[*]
Here we are approximating these loss processes as an exponential function with a simple ``effective'' rate. The real situation could be more complex. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... emission[*]
Here we are approximating these loss processes as an exponential function with a simple ``effective'' rate. The real situation could be more complex. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... lifetime[*]
In this thesis, histograms were collected occasionally with a 10 ns bin size, which gives for k= gold ( ns), clearly requiring the integrated function Eq. C.7 for accurate fits. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...bkgd[*]
In a single-hit detection mode (as opposed to a multi-hit mode), the detection efficiency is time dependent, hence the recorded histogram will have a slope even though the accidental background was constant in time. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.