High precision study of muon catalyzed fusion in D2 and HD gas

Abstract

Muon catalyzed dd fusion in D2 and HD gases in the temperature range from 28 to 350 K was investigated in a series of experiments based on a time-projection
ionization chamber operating with pure hydrogen. All main observables in this reaction chain were measured with high absolute
precision including the resonant and non-resonant ddμ formation rates, the rate for hyperfine transitions in dμ atoms, the branching ratio of the two charge symmetric fusion channels 3He + n and t + p and the muon sticking probability. The report presents the final analysis of the data together with a comprehensive comparison
with calculations based on recent μCF theories. The energy of the loosely bound ddμ state with quantum numbers J = 1, ν = 1, which is central to the mechanism of resonant molecule formation, is extracted with precision ɛ11(fit) = −1.9651(7) eV. in impressive agreement with the latest theoretical results ɛ11(theory) = −1.9646 eV.

Full text

Page 1

DRAFT September 12, 2007
HIGH PRECISION STUDY OF MUON CATALYZED FUSION
IN D2AND HD GASES
D.V. Balin, V.A. Ganzha, S.M. Kozlov, E.M. Maev, G.E. Petrov, M.A. Soroka,
G.N. Schapkin, G.G. Semenchuk, V.A. Trofimov, A.A. Vasiliev, A.A. Vorobyov, N.I. Voropaev.
Petersburg Nuclear Physics Institute (PNPI), Gatchina 188350, Russia.
C. Petitjean.
Paul Scherrer Institute, PSI, CH-5232 Villigen, Switzerland.
W.H. Breunlich, B. Gartner, B. Lauss, J. Marton, J. Zmeskal.
Stefan-Meyer-Institute for subatomic Physics (SMI), A-1090 Wien, Austria.
T. Case, K.M. Crowe, P. Kammel.
University of California Berkeley, UCB and LBNL, Berkeley, CA 94720, USA.
F.J. Hartmann.
Technical University of Munich (TUM), D-85747 Garching, Germany.
M.P. Faifman.
Russian Research Centre ”Kurchatov Institute”, Moscow 123182, Russia.
During 1994÷1996, a series of muon catalyzed fusion (µCF) experiments was performed in a high
intensity muon beam at PSI by the PSI-PNPI-IMEP-LBNL-TUM collaboration. These experiments
aimed at detailed studies of the muon catalyzed dd fusion (dµd fusion) in D2 and HD gases. A
time-projection hydrogen ionization chamber was used to detect the muon stops and the charged
products of the dd fusion reaction. The applied experimental technique allowed to determine with
high absolute precision the major parameters of the processes involved in the dµd fusion. This report
presents the results of final analysis of the experimental data. The obtained results are compared
with calculations based on recent µCF theories.
1. Introduction
1.1. Short historical overview
In this Section, we consider the milestones of the muon
catalyzed fusion (µCF) history related mainly to the
muon catalyzed dd fusion. Description of other aspects
of µCF studies one can find in several review articles [1].
The idea of the µCF process was first suggested by
C. Frank [2] in 1947 when he tried to find an alterna-
tive explanation of the tracks from cosmic rays observed
at that time by S. Powell and his group in the photo
emulsions exposed at high altitudes. This experiment
has detected some cosmic particles stopped in the pho-
toemulsion with emission of ∼5 MeV muons from the end
of the primary track. Note that by that time muons were
the only particles identified in the cosmic rays. This ex-
periment enforced the authors to assume existence of a
new particle, called later the π-meson, decaying into the
muon and neutrino.
In his alternative explanation, C. Frank pointed out
that the negatively charged cosmic muons stopped in the
photoemulsion may have some probability to be captured
by deuterons presented in the photoemulsion thus form-
ing small neutral objects, dµ atoms. Like neutrons, the
dµ atoms can travel in the matter easily penetrating in-
side the molecules. If, by chance, the dµ atom comes
close to a proton, this might lead to a fusion reaction
dµ+p→3He+µ + 5.5 MeV with the track picture simi-
lar to those observed by the Powell’s group. One can
see that, in the above explanation, the muon plays the
role of a catalyzer of the fusion reaction. Though this
interpretation of the Powell’s experiment was excluded
(the probability to form the dµ atoms in photoemulsions
is negligible) still the possibility of the muon catalyzed
fusion was thus formulated, and this idea proved to be
fruitful.
C. Frank did not consider any particular mechanism
of the fusion reaction. Such mechanism was first sug-
gested by A.D. Sakharov in 1948 [3]. The key point of
this mechanism is formation of the muonic molecules,
such as the ddµ molecule. This molecule is an analog
of the singly ionized ordinary deuterium molecule D+
However, due to the larger mass of the muon, the size
of the ddµ molecule is about 200 times less than that
of the D2molecule. Therefore, the two deuterons in the
ddµ molecule are enclosed in a small volume within a dis-
tance of ∼500 fm between them with a strongly reduced
2.

Page 2

width of the repulsive Coulomb barrier (Fig. 1). As a
result, the probability for quantum penetration through
the barrier proves to be so high that the fusion reaction
takes place with a rate much higher than the muon decay
rate. The striking feature of this mechanism is that, un-
like the thermo-nuclear fusion which requires very high
temperatures (∼ 50 · 106K), the muon catalyzed fusion
occurs at normal temperatures (“cold fusion”).
The next important step was to understand the mecha-
nism of the muonic molecule formation. A possible mech-
anism of this process was considered by Ya. Zeldovich in
1954 [4]. In his model, the formation of the ddµ molecule
after the muon stop in the D2medium occurs inside the
D2 molecule with the energy release upon binding the
ddµ molecule carried off by an ejected electron:
dµ + D2→ [(ddµ)de]++ e−.
Note that the exact value of the binding energy of the
ddµ molecule was not known at that time, and Ya. Zel-
dovich assumed it to be higher than the ionization po-
tential of the D2molecule (15 eV).
Experimentally, the first µCF reaction was discovered
by L. Alvarez and his coworkers in 1956 [5]. As a by-
product of their experiment with the Berkeley hydro-
gen bubble chamber, they observed the pd fusion reac-
tion catalyzed by negative muons stopped in the bub-
ble chamber, just the fusion channel first considered by
C. Frank: dµ + p →3He + µ. This accidental discovery
triggered intensive experimental and theoretical studies
of the µCF physics. In particular, the dµd fusion reaction
dµ +d → t+ p+ µ+ 4.03 MeV was observed [5], and its
rate was studied in the experiments with the liquid deu-
terium bubble chambers [6, 7]. The measured µCF rates
appeared to be quite low, especially the rate of the dµd
fusion. The ddµ molecule formation rate followed from
these measurements, λddµ≈ 0.1·106s−1, appeared to be
much less than the muon decay rate λ0= 0.455·106s−1.
These results were in fairly good agreement with the cal-
culations according to the Zeldovich model [8]. There-
fore, the main features of the µCF process seemed to be
understood at that time.
Some surprise, however, came from an experiment per-
formed at Dubna in 1966 [9]. In this experiment, the dµp
and dµd fusion was studied using a hydrogen/deuterium
filled diffusion chamber operating at 250 K. The puz-
zling result of this experiment was the unexpectedly large
ddµ molecule formation rate, by an order of magnitude
exceeding the value of λddµ found in previous exper-
iments performed at the liquid hydrogen temperature.
The problem was that neither the large value of λddµno
its strong temperature dependence could be explained
on the basis of the Auger process (1) considered in the
Zeldovich model. To resolve this problem, E. Vesman
suggested in 1967 [10] that the ddµ molecule may have
a very loosely bound state, so that the released energy
(1)
is small enough to go into vibration and rotation of the
resulting compound molecule:
dµ + D2→ [(ddµ)dee]∗.
(2)
This could be a resonant process increasing consider-
ably the ddµ molecule formation rate. Also, this rate
may be temperature dependent as the released energy is
the sum of the ddµ binding energy and the kinetic energy
of the dµ atom.
The existence of the loosely bound states in the muonic
molecules could be, in principle, shown by direct calcu-
lations of the energy levels in these molecules. However,
this turned to be a difficult task which required devel-
opment of high precision methods for description of the
three-body Coulomb interacting system. Only a decade
later L. Ponomarev and his collaborators [11] were able
to establish that such a loosely bound state does indeed
exist in the ddµ and dtµ molecules with the binding en-
ergies around 2 eV and 1 eV, respectively, well below the
dissociation energy of the D2molecule (4.5 eV). Then in
1977, applying the resonant molecular formation mecha-
nism of Vesman, S. Gerstein and L. Ponomarev [12] could
describe the observed temperature dependence of the dµd
fusion rate. Moreover, they predicted a very high dtµ
molecule formation rate which might exceed the muon
decay rate by orders of magnitude. Soon, in 1979, this
prediction was confirmed by a Dubna experiment [13] re-
porting λdtµ≈ 108s−1.
The discovery of the resonant muonic molecule forma-
tion mechanism triggered a new burst of interest to µCF.
New experimental programs were started in all labora-
tories possessing the muon beams, while the theoretical
efforts were even more widespread. Partly, these efforts
were motivated by the revived hopes to use the dµt fusion
for practical applications. On the other hand, the µCF
process is of great interest by itself allowing to study var-
ious aspects of mesoatomic, mesomolecular and nuclear
physics. From this point of view, the dµd fusion proved
to have important advantages. In contrast to the more
complicated process of dµt fusion, dµd fusion allows un-
ambiguous quantitative theoretical description. There-
fore, comparison of the experimental data on dµd fusion
with theoretical calculations is of primary importance for
understanding the µCF phenomena.
An important step in investigation of the dµd fusion
was done by the PSI-Vienna collaboration. This collab-
oration demonstrated that the ddµ molecule formation
rate strongly depends on the spin of the dµ atom, F= 3/2
or F= 1/2 - another convincing evidence of the resonant
formation of the muonic molecules. It was shown that
measurements of the temperature dependence of the ddµ
formation rates λ3/2
test of the various elements of the resonant ddµ forma-
tion mechanism. These experiments were performed in
a high intensity muon beam at PSI during 1979 ÷ 1990
ddµ(T) and λ1/2
ddµ(T) provide a sensitive

Page 3

[14]. The muons were stopped in a D2liquid or gas tar-
get, and the time distribution of the neutrons produced
in the dd fusion reaction were measured by a set of neu-
tron detectors surrounding the target. The dµd fusion
leads to four reaction channels:
⎧
⎪
Detecting neutrons, one can study only the sum of the
first two channels. This limitation was overcome in the
experimental method developed at PNPI (Gatchina) al-
lowing to detect simultaneously all the dµd fusion chan-
nels. In this method, a deuterium-filled high pressure
ionization chamber was used as a sensitive target de-
tecting both the stopped muons and the charged fusion
products:
sion experiments was carried out by the PNPI group in
1982÷1988 [15] using the muon channel of the Gatchina
synchrocyclotron. These experiments allowed to mea-
sure for the first time the3Heµ sticking probability and
the yield ratio of the charge symmetric fusion channels
R = Y(3He+n)/Y(t+p). Surprisingly, it was found that
this ratio is quite different for dd fusion following reso-
nant and non-resonant formation of the ddµ molecule.
The Gatchina experiments demonstrated the ability of
the new experimental method to provide high absolute
precision in measurements of the main parameters of the
dµd fusion process.
1.2. Goals of the new µCF experiments
The Gatchina experiment suffered from rather low muon
beam intensity provided by the Gatchina synchrocy-
clotron. In particular, this was the reason why only the
steady-state λddµ(T) was measured in this experiment
without separation of the hyperfine components λ3/2
and λ1/2
investigation of the non-resonant ddµ formation mecha-
nism. That is why the PNPI, PSI, and Vienna groups
decided in 1987 to join their efforts and prepare new
µCF experiments, based on the Gatchina experimental
method, to be performed in the high intensity µE4 muon
beam at PSI. The new µCF collaboration included also
the experimental groups from Berkeley and Munich.
The first experiments performed at PSI by the µCF
collaboration in 1989 ÷ 1993 were direct observation of
the muon sticking probability in dµt fusion [16] and high
precision measurements of the µ3He nuclear capture rate
[17]. After that, the µCF collaboration returned to new
studies of the dµd fusion with the goal to investigate
with high precision both the resonant and non-resonant
µCF processes in D2 and HD gases in a wide temper-
ature range from 28 K to 350 K.The experiments were
performed in 1994 ÷ 1996. The preliminary data were
published elsewhere [18]. Below we present the final re-
ddµ →
⎪
⎪
⎪
⎩
⎨
3He + n + µ + 3.27 MeV
3Heµ + n + 3.27 MeV
t + p + µ + 4.03 MeV
tµ + p + 4.03 MeV
.
(3)
3He,3Heµ, and t + p. A series of dµd fu-
ddµ(T)
ddµ(T). Also, the possibilities were very limited for
sults together with some comparison with the µCF the-
ory.
2. µCF kinetics in D2/HD/H2gas mixtures
Figure 2 shows the kinetics scheme of the µCF processes
in D2, D2+H2, and HD gases used in the analysis of our
experimental data. The µCF process starts with slowing
down of the muon entering the hydrogen gas target to
an energy of ∼ 10 eV when it is captured by a hydrogen
molecule forming a pµ or dµ atom in excited state. The
slowing down process takes less than 10−11s. Accordingly
to the theoretical considerations [19], the excited muonic
atoms have a wide primary distribution over principal
quantum number from n=8 up to n=40 with a maximum
at n=13, and they have some initial kinetic energy with
the mean energy around 0.9 eV and 0.5 eV for the pµ
and the dµ atoms, respectively.
The de-excitation of the muonic atoms proceeds mainly
by ejection of electrons from the target molecules (ex-
ternal Auger effect), by radiative transitions and by
Coulomb collisions. This process is quite fast in a dense
hydrogen target. In particular, the 1s-ground state will
be reached in less than 1 ns at the gas densities used in
our experiment. The theory predicts some increase of
the kinetic energy of the muonic atoms during the cas-
cade process due to Coulomb collisions [20, 21], and there
is some experimental evidence supporting this prediction
[22]. So one can expect that in our experimental condi-
tions the initial kinetic energy of the pµ(1s) and dµ(1s)
atoms will be peaked around (1÷2) eV with possible tails
up to 200 eV. In case the muon has formed a pµ atom,
it will be quickly transferred to a dµ atom from either
the excited or the ground state of the pµ atom [23]. In
this case the dµ atoms receive kinetic energy up to 43 eV
according to the reaction:
pµ(n) + d → dµ(n) + p +135
n2eV.
(4)
The rate of this reaction at our experimental conditions
is (3.5 ÷ 7) · 108s−1[24] depending on the deuterium
concentration in the gas mixture. This means that all
muons stopped in the target should be transferred to
the dµ atoms in less than 10 ns. Then the dµ atoms
start to loose their kinetic energy in elastic scattering
on deuterons (protons) and, at the energies below 1 eV,
on the hydrogen molecules. At our experimental condi-
tions, full thermalization of the dµ atoms is expected to
be reached after ∼100 ns in the D2gas and after ∼200 ns
in the HD gas [25].
It is anticipated that the two dµ spin states F=3/2
and F=1/2 are populated after the de-excitation cas-
cades according to their statistical weights (η3/2=2/3 and
η1/2=1/3). So the initial state of the dµ atoms entering
the process of formation of the ddµ molecules appears
to be well defined, and this is of primary importance for
quantitative consideration of the µCF process (note the
different situation in the dtµ cycle).

Page 4

The kinetics scheme shown in Fig. 2 involves the fol-
lowing processes:
2.1. Formation of the ddµ molecule and back-decay
The ddµ molecule contains five bound states defined by
the rotational (J) and vibrational (ν) quantum num-
bers (Table I). The least bound state (J=1, ν=1) can be
formed by the resonant formation, the other states can
be populated either by de-excitation of the (1,1) state
or by direct ddµ formation via the non-resonant mecha-
nism. The ddµ molecule can be formed from the F=3/2
or F=1/2 hyperfine states of the dµ atoms. In case of
resonant ddµ formation, this process is accompanied by
back-decay of the muonic molecular complex:
(dµ)F+ D2→ [(ddµ)dee]∗→ (dµ)F? + D2.
Owing to the exchange symmetry of the ddµ molecule,
the de-excitation rate of the complex [(ddµ)dee]∗is com-
paratively slow. Therefore, the back-decay can occur be-
fore the dd fusion reaction takes place. Note that the
experimentally determined rates Λ3/2
the kinetics scheme shown in Fig. 2 include the back-
decay components. In other words, these are the effective
ddµ formation rates leading directly to dd fusion.
(5)
ddµand Λ1/2
ddµentering
TABLE I: Binding energies of the bound states in the ddµ
molecule [26]
(J, ν)
Binding energy, eV 325.074 35.844 226.682 1.96482 86.494
(0,0)(0,1) (1,0) (1,1)(2,0)
2.2. Spin flip in dµ atoms
The energy difference between the dµ hyperfine states
F=3/2 and F=1/2 is ∆ = 0.0485 eV [27]. The spin flip
occurs in inelastic collisions with the D2(HD) molecules,
the rates of the two hyperfine transitions being related
by the detailed balance equation:
Λ12= 2 · e−∆/kT· Λ21,
(6)
∆ = 0.0485 eV,k = 8.62 · 10−5eV/K,
where Λ21 and Λ12 are the rates of the hyperfine tran-
sitions (3/2 →1/2) and (1/2 → 3/2), respectively. In
case of resonant formation of the ddµ molecule, there is
also some contribution to the spin flip rates due to the
back-decay reaction (5). In this case the experimentally
determined rate Λ21(shown in Fig. 2) is in fact the sum
of two rates: Λ21= Λsc
the inelastic scattering and Λbd
2.3. dd fusion
The dd fusion rate λf depends on the quantum numbers
of the ddµ state from which fusion occurs. In particular,
theory predicts λf= 0.44·109s−1for the (1,1) state and
λf= 1.5·109s−1for the (1,0) state [28]. In the symmetric
21+ Λbd
21, where Λsc
21to the back-decay.
21is related to
ddµ molecule, the ∆J=1 transitions are forbidden, apart
of small relativistic effects. The calculated ∆J=0 transi-
tion rate between the (1,1) and (1,0) states is rather low:
Γdex= 0.02 · 109s−1[29]. Therefore, after resonant ddµ
formation, fusion takes place nearly exclusively from the
(1,1) state.
The dd fusion leads to four decay channels shown in
Fig. 2 with the branching ratio of the charge symmet-
ric channels R= β/(1 − β) =Y(3He+n)/Y(t+p) depend-
ing on the angular momentum (J) of the ddµ state from
which the fusion takes place. Our experiment allows to
measure this branching ratio. This is important for un-
ambiguous analysis of the experimental data.
2.4. Muon sticking to3He
Our experimental method provides a unique possibility
to perform direct measurements of the probability ωdof
muon sticking to the3He nucleus during the dd fusion
process by measuring the yields of the channels3Heµ + n
and3He + n + µ.
2.5. Background reactions
One of the background reactions is the pdµ fusion
through its decay channel pdµ →3He+µ + 5.5 MeV.
This background is negligible for the D2 runs, but it
becomes visible in the HD and H2+D2 runs. However,
in this experiment, the pdµ →3He + µ channel is de-
tected and separated well from the dd fusion reaction.
In addition, the triton from the muon capture reaction
pdµ →3Heµ → t + ν is detected as well that serves as a
control for the background from this reaction.
The main (though quite small) background comes from
the charged products of the nuclear muon capture by the
gas impurities such as N2, H2O. Fortunately, our experi-
mental method allows to detect the muon capture prod-
ucts with a known efficiency ωr. This makes it possible
to determine the impurity concentration in the hydro-
gen gas and to apply the necessary corrections for this
background.
3. Experiment
3.1. Experimental setup
The experimental setup is shown in Fig. 3. The basic
element of this setup is a high pressure cryogenic hydro-
gen ionization chamber (IC) operating as an active tar-
get detecting both the incoming muons and the charged
products from the dd fusion and from the nuclear muon
capture reactions. A remarkable feature of this detector
is that it allows to select clean muon stops in the IC sen-
sitive volume, well separated in space from the chamber
electrodes, and provides 100% efficiency in detection of
the dd fusion events. The IC body was surrounded by an
array of plastic counters for detection of the 2.45 MeV
neutrons from the fusion reaction dd→3He+n. This ar-
ray contained 20 neutron counters with five 2.5 mm thick
detectors in front of them for elimination of signals pro-
duced by the muon decay electrons. The detection effi-
ciency of the neutron detector was 18%. This detector
played a complementary role in the experiment. Being

Page 5

fast and without any dead time after the muon stop, the
neutron detector was used to control the initial part of
the time distributions of the fusion events in the 0.4 µs
interval after the muon stop. This fine gap was not ac-
cessible by the IC because of the pile up of the fusion
signals with the preceding muon signals. In addition, the
neutron detector was used for determination of the “zero
time” in the time distributions measured with the IC.
Figure 3 shows also the muon beam telescope consisting
of two thin plastic counters (S1 and S2) in coincidence
and a collimator (C) with a 2 cm2hole in the center.
3.2. Hydrogen Ionization Chamber
The cryogenic hydrogen ionization chamber was designed
and manufactured at PNPI. It is a gridded ionization
chamber with the cathode-to-grid and grid-to-anode dis-
tances of 13.0 mm and 1.0 mm, respectively. The grid
wires are 25 µm in diameter, and the wire pitch is 200 µm.
The anode has a multi-pad structure as shown in Fig. 4.
The central pads (B1÷B5) define the useful muon stop
area of (25 × 40) mm2. It is surrounded by three veto
pads (A1÷A3) and by five entrance pads (C1÷C5). The
total anode area is (40 × 60) mm2.
(60 × 75) mm2, and the cathode is a disc of 110 mm in
diameter. Such a geometry provides a uniform electric
field in the cathode-grid space over the anode pads.
The IC operates as a time-projection chamber. Muons
enter the chamber nearly parallel to the electrode sur-
face and stop in the space between the cathode and the
grid. The ionization electrons drift towards the grid and
are collected on the anode pads. Each anode pad had
independent electronics and amplifier circuits. If a sig-
nal above 120 keV threshold was sensed, the charge of
this anode signal was digitized by 8-bit flash ADCs for
a period of 10 µs and recorded by the computer. The
signal shape analysis provided the arrival time T (front
edge), the duration F, the charge integral proportional to
the deposited energy E, and the mean time <t> of each
signal. The fusion signals are detected similarly. As an
example, Fig. 4 shows a sequence of signals on the pads
C3−B1−B2−B3−B4produced by a muon stopped at pad
B4and followed by a fusion signal on the muon-stop pad.
The arrival times are measured with respect to the
muon trigger time Ttr provided by the beam telescope
S1·S2. The difference ∆Tµ= Tµ– Ttr(where Tµis the
arrival time of the muon signal on the muon-stop anode
pad) determines the vertical coordinate of the muon stop
in the cathode-grid space: z = ∆Tµ/w where w is the
electron drift velocity. This is a valuable information
allowing to eliminate the muon stops in the vicinity of
the cathode and grid surfaces. The time difference ∆Tf
= Tf - Tµ (where Tf is the arrival time of the fusion
signal on the muon-stop anode) is used to determine the
fusion time after the muon stop.
The IC is designed to work at hydrogen gas pressures
up to 120 bar. At such gas pressures, the electron-ion
recombination of the initial ionization becomes essential.
The grid area is
Figure 5 displays the results of special measurements of
the recombination of the ionization produced by 4.78
MeV α-particles in D2 gas at various pressures.
can see from this figure that it is not possible to elimi-
nate completely the recombination at high gas pressures
even applying the highest achievable electric fields (30
kV/cm in our case). However, as it was shown in prelim-
inary studies and confirmed by the present experiment,
even considerable losses of the ionization charge due to
recombination (30% ÷ 40% of the initial charge) do not
deteriorate seriously the energy resolution. The effect
of recombination is just some shifts of the energy peaks
without much distortion of the peak shapes. Therefore,
complete elimination of the recombination is not needed.
Moreover, the recombination effect proved to be even use-
ful, as it allows to separate the3He and3Heµ peaks thus
providing a unique method for measuring the muon stick-
ing probability. Note that the singly charged3Heµ par-
ticles have a factor of three longer range than the doubly
charged3He particles, and, as a consequence, the ioniza-
tion charges produced by the3Heµ particles suffer much
less recombination effect. Most of the measurements in
the present experiment were performed at around 45 bar
pressure (measured at room temperature) with about 25
kV/cm electric field in the IC drift zone and 40 kV/cm
in the grid-anode space. Under these conditions, the re-
combination effect for 0.82 MeV3He and 0.80 MeV3Heµ
particles was 25% and 5%, respectively.
Figure 6 shows the electron drift velocities measured in
D2and H2gases at various pressures at room tempera-
ture. One can see that the drift velocity is determined by
the ratio E/P of the electric field to the gas pressure. In
our experiment, the drift velocity was around 0.7 cm/µs
in the drift zone and 1.0 cm/µs in the grid-anode zone.
The maximal collection time of the ionization electrons
was 2.0 µs. As we observed in the present experiment,
the drift velocity at a given ratio E/P is not changed no-
ticeably in the explored temperature range (30 ÷ 300)K
under condition that the gas density is constant.
3.3. IC cooling and temperature control system
The primary goal of our experiment was high precision
study of µCF in D2 gas at various temperatures. Due
to high sensitivity of the resonant ddµ formation pro-
cess to gas temperature, a requirement was that the IC
temperature could be fixed at any point in the explored
temperature range, stabilized and measured with abso-
lute precision better than ±0.5 K. To meet this require-
ment, a special cooling and temperature control system
has been developed at PNPI which was able to stabilize
the IC temperature with absolute precision ±0.15 K in
the temperature range from 28 K to 350 K.
The IC engineering design had some specific features
dictated by the conditions of our experiment. In particu-
lar, a critical part of the design was manufacturing a high
voltage input connector which could stand in the hydro-
gen atmosphere gas pressures up to 120 bar and transmit
One

Page 6

the high voltage up to 40 kV without micro-discharges
and which could allow the temperature variations from
30 K to 400 K. Fortunately, SVETLANA company in
St. Petersburg proved to be able to work out the technol-
ogy and to fabricate specially for our experiment a small
series of such connectors satisfying above specifications.
Some precautions were taken in the choice of materials.
For example, the grid in the IC was made of stainless steel
wires soldered on a covar frame. This combination of ma-
terials helped to keep the wire tension in the acceptable
limits during the cooling procedure. For our experiment,
it was planned to use hydrogen gas with various relative
H2/D2/HD concentrations including pure HD gas. To
keep these concentrations unchanged during the run, one
should exclude some materials, like Ni, which may cat-
alyze dissociation of the HD molecules. Note that the
covar grid frame proved to be such a catalyzer. However,
this effect was eliminated after the frame was covered
with an electro-deposited layer of silver. The final tests
of the chamber filled with the HD gas showed that the
HD concentration remained stable within 0.5% during at
least several days.
Figure 7 presents the IC layout. The ionization cham-
ber was mounted with a cooling jacket inside a high-
vacuum vessel to allow cooling of the chamber and high
voltage supply to the cathode and to the grid of the ion-
ization chamber. The vacuum in the vessel was about
10−7mbar at room temperature and lower under cool-
ing conditions. The chamber body was a stainless steel
cylinder 126 mm in diameter and 100 mm in height with
6 mm wall thickness. The thickness of the stainless steel
top and bottom flanges was 18 mm. The muons entered
the chamber through a 4 mm thick 25 mm in diameter
beryllium window. The ionization chamber volume was
1.2 L, and the total weight with the cooling jacket was
27 kg.
The chamber was cooled with two liquid helium heat
exchangers HE1and HE2which had direct thermal con-
tact with the top and bottom chamber flanges. The prin-
cipal scheme of the cooling system is presented in Fig. 8.
The temperature of both IC flanges was measured with
two platinum thermometers Pt 1001 and Pt 1002 spe-
cially calibrated in the temperature range (12 ÷ 330) K
with absolute precision ±0.01 K. The liquid helium flux
was regulated and stabilized by the pressure in the he-
lium tank, the pressure being measured with a reference
pressure sensor. The outlet helium gas was heated to the
room temperature in a heater, and its flux was controlled
by the mass-flow controllers CV1 (at T≤ 70 K) or CV2
(at T≥ 70 K).
Besides cooling, the heat exchanger could be also
heated by electric currents with controllable power dissi-
pation. In the experiment, both systems operated simul-
taneously. The heating system being much less inertial
than cooling, it helped to reduce considerably the setting
time of the required temperature. The temperature set-
ting procedure starts with setting the helium flux slightly
higher than needed for the given temperature, the cool-
ing power excess being compensated by heating. Then,
knowing the introduced heating power, a correction for
the helium flux was applied with corresponding reduc-
tion of the heating power. Finally, the heating system
was used for the fine tuning and stabilization of the re-
quired temperature. The whole process was completely
automated with a micro-processor and a PC-computer
with special software. The allowed maximal rate of the
temperature change was set to be 5 K per minute, and
the temperature difference between the top and bottom
flanges during the temperature setting procedure did not
exceed 3 K. With these limitations, the time required to
reach the preset temperature and to stabilize it within
±0.15 K was about two hours.
3.4. Gas filling and purity control
The ionization chamber was filled with highly purified
gases (D2, HD, or H2+D2) at 45 to 78 bar pressure at 300
K. The minimal pressure was about 4 bar at 30 K. The
pressure was measured with three digital monometers
with pressure ranges up to 100 bar (0.2 bar precision), up
to 50 bar (0.1 bar precision), and up to 20 bar (0.02 bar
precision). Purification of the D2and H2gases was per-
formed using a specially constructed purification system.
This system included a set of purification columns with
Zeolite (CaA) operating at the LN2 temperature. The
hydrogen purity was controlled by a chromatograph with
a sensitivity at the 10−8level for N2and O2. Some mea-
surements were performed with the H2+D2gas mixture
both in the equilibrium and also in the non-equilibrium
states. The equilibrium H2+2HD+D2 mixture was ob-
tained using a column with Ni catalyzer. The H2/D2/HD
concentrations were measured with a chromatograph and
with a mass-spectrometer at a 1% level of accuracy. A
special setup was constructed for production of pure
HD gas in the reaction LiD+H2O → LiOH+HD. The
achieved concentration of the HD molecules in the gas
was 97% with only 1% admixture of D2and 2% admix-
ture of H2.
4. Measurements and data processing
The experiments were carried out during 1994÷ 1996 in
the µE4 muon beam at PSI. The parameters of the muon
beam were as follows:
dimensions:(40 × 25) mm2FWHM; pµ
50) MeV/c; ∆pµ/pµ = 4% FWHM; intensity: 25 kHz;
time structure: DC; contamination with electrons: ∼
10%; contamination with π-mesons: < 10−3.
The measurements were performed at various temper-
atures with pure D2 gas (16 runs), with equilibrated
H2+2HD+D2 (4 runs) and non-equilibrated H2+D2(3
runs) gas mixtures, and also with HD gas (3 runs). Typ-
ical running times were 20 ÷ 30 hours per temperature
point. The strategy of the experiment was to select a
clean sample of muon stops in the IC fiducial volume, to
ensure close to 100% efficiency in detection of the dd fu-
= (40 ÷

Page 7

sion events, and to provide energy and time distributions
of the fusion events measured with the IC for physics
analysis.
4.1. Event triggers
The event triggers were generated according to the fol-
lowing criteria. The first-level trigger was made from the
signals of the beam telescope:
Tr1 = S1 · S2 · S1(−7µs + 9µs).
This trigger identified muons within the beam tele-
scope acceptance (2 cm2) with an additional require-
ment that there should be no other muons detected by
the S1 counter (area of 15 × 15 cm2) in the time inter-
val from 7µs before and 9µs after the coincidence sig-
nal. The second-level trigger used the trigger signals
from the B anodes. These channels had two-threshold
charge-integrating trigger circuits with the “Elowthresh-
old” (120 keV) and the ”Ehighthreshold” (800 keV). The
energy resolution was 30 keV (σ). The logic scheme pro-
vided three triggering modes:
(7)
• Elowtrigger. A low threshold signal during the first
2.0 µs (maximal drift time) on at least one B anode
signaling an incoming muon.
• Elow· Elowtrigger. Two separate low threshold sig-
nals (∆t ≥ 0.3 µs) within 10 µs on the same B
anode indicating a muon stop followed by a fusion
event separated in time.
• Ehightrigger. A signal on any B anode above the
Ehighthreshold indicating a (p+t) event or a3He
signal piling up with the muon signal.
Since the rate of muon stops was much higher than
the rate of fusion triggers and could not be processed
in globe, the Elow trigger was prescaled by a factor Ps
equal to 100, 50 or 20 depending on the fusion probability
per muon stop. Summarizing, the final trigger in this
experiment was
Tr2 = Tr1 · (Elow/Ps+ Elow· Elow+ Ehigh).
At typical run conditions, the Tr1 rate was ∼ 2 kHz,
while the rate of Tr2 was about 20 Hz.
4.2. Muon stop selection
The first step of the muon-stop selection procedure was
to find the B anode which detected the muon-stop. In
the case of the Elow·Elow trigger and the Ehigh trigger,
the muon-stop B anode is just the anode producing the
corresponding trigger signals. In the case of the Elow
trigger, the muon-stop anode is the most downstream
anode along the muon track which detected the muon
signal. All the other steps of the selection procedure are
identical for all three trigger modes. These steps were as
follows:
(8)
• The arrival time Tµ of the muon signal from the
muon-stop B anode was measured, and it was
required that there should be no signals on the
A1÷A3anodes within the time interval Tµ±50 ns.
This restricted penetration of the muon tracks into
the A anodes region to a small boarder region of
(0.3÷0.4) mm where stopping muons produce sig-
nals below the thresholds in the A anode channels.
The amount of such muon stops escaping the area
covered by the B anodes was about 1.5%, and this
amount was under control as explained below.
• Vertically, the fiducial volume was determined by
the cuts on the drift time of the ionization electrons
detected by the muon-stop B anode (Fig. 9) which
set the muon stop location at 1.5 mm from the
cathode and from the grid, such that the3He,3Heµ
and t-tracks would not touch the electrodes.
• Very useful for the muon stop selection proved to
be a combination of the energy deposit Eion the
muon-stop B anode and the energy deposit Ei−1on
the preceding B anode: S=Ei+ 2Ei−1. Figure 10
demonstrates the S distributions of the muon sig-
nals detected with the Elow trigger and with the
Elow·Elow trigger. One can see a sharp peak at
S= 1.15 MeV corresponding to undisturbed muon
signals with a small peak at S= 1.75 MeV due
to pile up of the3He signals with the muon sig-
nals. Piling up with the (t+p)-signals shifts the
muon signals to S> 2 MeV (not seen in Fig. 10).
Both S distributions shown in Fig. 10 are identi-
cal except the low energy noise contribution to the
Elow trigger events. In the analysis, the selected
S region was 0.83 MeV<S< 1.44 MeV. This cut
is loose enough to provide equal efficiencies within
better than ±0.1% for muon registration in both
trigger modes, Elow trigger and Elow·Elow trigger.
The identical muon selection in these two modes
is important. This allows to use the Elow trigger
events for controlling the detection efficiency in the
Elow·Elowmode and also for absolute calibration of
the measured fusion rates.
Our main event trigger (Elow· Elow) required muon
stop and fusion signals from the same B anode. This
caused a class of events escaping triggering, i.e. muons
which intruded so little into the anode region that the
created charge signal remained below the trigger thresh-
old. This event class was carefully reconstructed using
the Elow trigger events.The measured correction on
a level of (3 ÷ 7)% with systematic uncertainty ±0.7%
was applied to compensate for these losses. The absolute
calibration of the fusion rates was performed using the
number of the muon stops in the Elowtrigger events se-
lected by the S cuts as shown in Fig. 10. Note that this
number does not include the muon-fusion pile up events

Page 8

which may constitute (2÷5)% of the total number of the
muon stops. However, this correction to the number of
muon stops, could be precisely determined from the time
distribution of the intervals between the fusion events.
In conclusion, the clean muon stops were selected in the
fiducial volume well isolated from the chamber electrodes,
and the absolute number of the selected muon stops was
controlled on a better than 1% precision level. A typical
rate of the clean muon stops was ∼ 100 s−1at ∼ 2 kHz
rate of the Tr1 trigger. The number of the clean muon
stops collected in each run varied from 2·106to 45·106.
4.3. Energy distributions of dd fusion events
Figure 11 shows a typical energy spectrum measured in
pure D2 gas at 120.3 K (the gas density was 5% of liq-
uid hydrogen density). This spectrum contains the first
fusion signals arriving after the muon signals, being sep-
arated from them in time. The spectrum was integrated
over the (0.5÷7.5)µs time interval. The energy of the fu-
sion signals was obtained by summing up the integrated
charges of the fusion signals on the muon-stop B anode
with the integrated charges of the signals on the neigh-
bor B or A anodes closest to the muon-stop anode if the
arrival times of these signals coincided within ±50 ns.
One can distinguish in the shown energy spectrum the
following components:
•3He peak at 0.6 MeV corresponding to the fu-
sion channel dd→3He+n (initial energy E0(3He) =
0.82 MeV, track length R(3He) = 0.26 mm).
•3Heµ peak at 0.75 MeV corresponding to the fu-
sion channel dd→3Heµ+n (E0(3Heµ) = 0.80 MeV,
R(3Heµ) = 0.6 mm).
• Continuous distribution from 0.9 MeV to 4 MeV
corresponding to the fusion channel dd→t+p
(E0(t) = 1.01 MeV, E0(p) = 3.02 MeV, R(t) =
1 mm, R(p) = 16 mm).
The events below 3.6 MeV are due to proton tracks es-
caping the sensitive volume controlled by the two neigh-
bor anodes. The minimal energy in the (t+p) distribu-
tion is determined by the energy of the triton which never
escapes the controlled sensitive volume. One can see also
in Fig. 11 two small peaks originating from piling up of
the second and the first fusion signals: the (3He+3He)
pile up peak at 1.2 MeV and the (t + p)+3He peak at
4.2 MeV. Note that the shifts of the observed peak en-
ergies with respect to the initial energies are due to the
charge recombination effect.
The events below 0.45 MeV are due to two background
reactions: pd fusion and muon capture on gas impurities.
As concerns the pd fusion, it occurs due to the presence
of a hydrogen admixture in deuterium (in the D2 runs
the admixture of HD molecules was 0.6% according to
the chromotographic analysis). There are two pd fusion
channels:
pd →3He (0.2 MeV) +µ (5.3 MeV) and
pd →3Heµ + γ (5.5 MeV).
The 0.2 MeV3He particles from the first channel are
detected producing a peak just above the 120 keV regis-
tration threshold with a small tail towards higher energy
due to partial absorption of the 5.3 MeV muon energy.
This tail ends at 0.4 MeV well below the region where
the dd fusion products are detected. The second channel
of the pd fusion is not directly detected in our experi-
ment because the3Heµ energy is below the registration
threshold. So, the pd fusion does not produce any back-
ground interfering with the dd fusion events. From this
point of view, it could be ignored. However, it proved
to be useful to determine this background and to sub-
tract it from the low energy part (≤0.40 MeV) of the
spectrum. The remaining events in this energy region
are entirely contributed by the muon capture reactions.
This gives the possibility to use this energy region for ab-
solute calibration of the muon capture background. The
shape of the pd→3He+µ peak was measured in our HD
experiment (see Fig. 20 in section 5). Also, it was shown
that the registration efficiency of this pd fusion channel
was ωpd= 85%. Knowing λpdµ(T), Cp, and ωpd, it was
possible to evaluate the pd background for each D2run.
Moreover, in some D2 runs with exceptionally low Ni-
trogen concentration it was possible even to determine
the Cpconcentration in deuterium. The obtained results
(Cp = 0.2% ÷ 0.3%) were in agreement with the chro-
matographic data.
The energy distribution of the µ-capture events
was measured in a special run with the gas mixture
H2+N2(140 ppm) at 45 bar pressure at room temper-
ature (Fig. 12). Also, the registration efficiency of the
µ-capture events was found in those measurements: εr=
0.55. Figure 11 presents the energy distribution of the
µ-capture events normalized to the yield of these events
in the energy region 0.1 MeV÷0.45 MeV measured in the
D2run at 120.3 K.
In the data of H/D runs, where the low energy region
of the dd fusion spectrum is heavily contaminated by the
products of pd fusions, such normalization of the capture
events needs some additional check. This was done in the
data with pure HD gas, where the pile up probability of
two dd fusion signals was extremely low, so the normal-
ization of the capture events was possible using the high
energy (> 5 MeV) part of the spectrum (see Fig. 20 in
section 5).
Taking the determined number of the µ − capture
events Ncapand assuming the gas impurity is Nitrogen,
it was possible to find the muon transfer rate to Nitro-
gen ΛdZand the Nitrogen concentrations CZin each run
using an approximate relation:
ΛdZ= λdZ· CZ· ϕ = Ncap·
< Λddµ> +λ0
Nµ· εr· λc/(λ0+ λc), (9)
where λdZ= 1.45·1011s−1[30] is the muon transfer rate

Page 9

at the Nitrogen atomic density of 4.25·1022nuclei/cm3
(liquid hydrogen density, LHD), λc= 0.057 µs−1is the
muon capture rate in Nitrogen, λ0 = 0.455 · 106s−1is
the muon decay rate, ϕ is the gas density normalized
to LHD, εr is detection efficiency of the muon capture
reaction, < Λddµ > is the integral
determined approximately by the ratio of the detected
fusion events Nf to the number of the selected muon
stops Nµ: < Λddµ>= λ0·Nf/(Nµ−Nf).
Such calculations for the considered D2 run at 120.3 K
gave ΛdZ= 0.0024· 106s−1and CZ= 0.33 ppm.
4.4. Time distributions of dd fusion events
To study the time distribution of the dd fusion events, we
have selected the3He and3Heµ events, that is the events
in the energy range from 0.4 MeV to 0.8 MeV (Fig. 11).
This choice has the following advantages:
ddµ formation rate
• Short track length of3He and3Heµ particles leads
to short duration of the signals (≤200 ns) and, as
a consequence, to a minimal dead time after muon
signals.
• Short duration of the3He and3Heµ signals makes
them similar in shape to the muon signals (note
that muon tracks are nearly parallel to the grid
surface). This facilitates precision measurement of
the time difference between the fusion and muon
signals.
• The pile up probability of the (3He+3Heµ) signals
with the next fusions being similar to that of the
muon signals, the overall pile up correction for nor-
malization of the (3He+3Heµ) time distributions is
strongly reduced if compared with the case of the
(t+p) time distributions.
• The3He+n fusion channel can be simultaneously
detected by the neutron detector. This gives a pos-
sibility to determine the actual initial point (time
zero) of the experimental
measured with the ionization chamber.
3He time distribution
The fusion time TICwas determined as the difference
between the mean time of the3He (3Heµ) signal and the
mean time of the muon signal, TIC =<tF > − <tµ>,
both signals detected on the muon-stop B anode. With
this measuring technique, slight asymmetries of the muon
entrance distribution may cause some systematic shifts
of the TIC=0 timing. However, this effect was carefully
evaluated by comparison with the fusion time TNof the
same event measured with the neutron detector: TN=
tn– t0
neutron counters correlated with the3He (3Heµ) signal
in the IC and t0
distribution. The value of t0
sion better than ±1 ns by the front edge (half-maximum)
of the time distribution of dd fusion neutrons in pure HD
n. Here tnis the time of a neutron hit in one of the
nis the ”time zero” in the neutron time
nwas determined with preci-
gas, where the dd fusions process starts immediately af-
ter the muon stop (see section 6.3). If compared with
the ”time zero” t0
µ−-decay (Fig. 13), the neutron ”time zero” t0
to be shifted by 12 ns, which reflects the mean time-of-
flight of the 2.45 MeV neutrons befor absorbtion in the
scintillators. The relatively slow rise of the neutron time
distribution shown in Fig. 13 is due to thermalization
time of dµ atoms in D2 gas. Figure 14 shows the time
correlation between the fusion signals detected by the IC
and by the neutron detector. The measured difference
∆T=TIC−TN is taken as a correction to the time mea-
sured by the IC. This correction for different runs varied
from -9 ns to +2 ns. Taking into account possible system-
atic errors, we concluded that the “time zero” in the time
distributions measured with the IC was known with pre-
cision ±2 ns limiting this error source in measurements
of λ3/2
notation TIC≡t.
Figure 15 presents the time distribution of the3He
(3Heµ) events measured with the IC at 120.3 K. The
drop in the IC spectrum at t ≤ 0.4 µs is due to piling up
of the fusion signals on the muon signals. Such pile up
events were eliminated on a level of the muon selection
as was explained in Section 4.2. On the other hand, the
neutron time distributions were measured in coincidence
with all3He (3Heµ) events including the pile up events.
This yielded the time spectrum starting from TN=0.
Both the IC and the neutron time spectra clearly show
the two-exponential shape. These spectra contain infor-
mation on the ddµ formation rates Λ3/2
on the spin flip rate Λ21. Roughly speaking, Λ3/2
Λ1/2
t=0 of the fast and slow exponents, respectively, while
Λ21is the slope of the fast exponent.
The region at t> 3µs is the so-called steady-state re-
gion with equilibrium population of the spin states of the
dµ atoms. The ddµ formation rate in this region is given
by the following expression:
efor time distribution of electrons from
nproved
ddµto ±0.4%. For simplicity, we shall use further the
ddµand Λ1/2
ddµand
ddµand
ddµare determined by the intercepts with the Y-axis at
Λddµ(T) = (Λ1/2
ddµ(T) + γΛ3/2
ddµ(T)) · (1 + γ)−1,
(10)
γ = 2·e−∆/kT, ∆ = 0.0485 eV, k= 8.62 · 10−5eV/K.
In our main analysis, we used the IC time distribu-
tions as they provided higher statistics, better precision
in absolute normalization, and low background. The only
background to be taken into consideration is the back-
ground due to muon capture on gas impurities (N2). Fol-
lowing the kinetics scheme (Fig. 2), the time distribution
of the µ-capture events was calculated using the expres-
sion:
Yc(t) = A·ea·t+B·eb·t+C·ec·t,
where all the constants are known functions of Λ3/2
Λ1/2
A+B+C = 0, (11)
ddµ,
ddµ, Λ21, ΛdZ, λc, and λ0.

Page 10

The integral of this spectrum should be normalized to
the number of µ-capture events in the selected energy
range under the3He (3Heµ) peaks. This procedure was
checked in the D2run at 50.2 K in which the N2concen-
tration was very high (15 ppm).
In that run, the time distribution was obtained of the
µ-capture events selected in the range (0.1 ÷ 0.4) MeV.
It was shown that the above expression describes well
the measured distribution. Moreover, it was possible in
the fitting procedure to determine ΛdZ and λc.
measured λc = 0.057(6) proved to be in good agree-
ment with the corresponding value known for nitrogen
λc(N2) = 0.065(5) [31]. The background calculated in
this way is presented in Fig. 15. One can see that in this
particular case the µ-capture background is very small.
It became more important for higher levels of impurities
and for lower dd fusion rates. However, in all cases the
µ-capture background was under control and could not
deteriorate the analysis precision. Other sources of back-
ground (double muon events, etc) were negligible. This
was checked in a run with the µ+beam under the same
intensity and energy of muons.
5. Experimental data and analysis
5.1 Running conditions
Tables II to 5 demonstrate the experimental conditions
and statistics collected in our measurements. Some com-
ments to these tables.
According to the µCF theory, the ddµ formation rate
may be slightly different for the ortho- and para-states
of the D2 molecules. In our case, the gas filling of the
ionization chamber was performed with ”equilibrium gas
mixture” corresponding to T= 77 K (the temperature
of the Zeolyte columns in the gas purification system,
Section 3.4). It was checked experimentally that the or-
tho/para ratio at the exit of the purification system is
2.4 : 1 which is exactly the equilibrium ortho/para ratio
that should be at T= 77 K. During the gas filling the
ionization chamber was at room temperature. Then the
chamber was cooled down to the lowest temperature. Af-
ter that the temperature was increased step-by-step from
run 1 to run 3 in the D2− 94 experiment and from run
1 to run 12 in the D2− 95 experiment. The total run-
ning time was 6 days and 18 days in the D2− 94 and
D2− 95 experiments, respectively. It is anticipated that
the ortho-para ratio was unchanged during these running
periods as there were no materials inside the chamber
which could catalyze the ortho-para transitions. Note
that there was a gas refilling in the D2− 95 experiment
between run 4 and run 5, as it was noticed that there was
too much Nitrogen in deuterium after the first gas filling
caused by an accidental error in the gas filling procedure.
As described in Section 3.3, the gas temperature was
stabilized and measured in each run with absolute pre-
cision of ±0.15 K. The gas pressure was measured with
precision varied from ±0.5% at T= 30 K (4 bar pressure)
to ±0.2% at T= 300 K (45 bar pressure). These measure-
The
ments allowed to calculate the normalized to LHD deu-
terium gas density applying the standard formula with a
correction factor for the non-ideal gas:
ϕ = 0.341 · P · (1 − ξ · P) · T−1,
where P is the gas pressure in bar,
T is the temperature in Kelvin,
ξ(H2) = 5.9−4and ξ(D2) = 5.6−4.
Summing up the possible errors in T, P and in the cor-
rection for the non-ideal gas, we conclude that the deu-
terium density was determined with precision of ±0.8%
at T= 30 K and ±0.4% or better at T≥ 60 K. Note
that the gas densities presented in Tables II to IV show
some continuous decrease with increasing temperature.
This is explained by the redistribution of the gas amount
between the cooled IC volume and the gas filling tubes
which were at room temperature.
The statistics collected in each run is illustrated in Ta-
bles II to V by the number of clean muon stops Nµand by
the number NHeof the corresponding (3He+3Heµ) fusion
events. Nµis defined as Nµ= Nµ(Ps)·Pswhere Nµ(Ps) is
the number of the clean muon stops triggered by the Elow
trigger with prescaling factor Psand selected as was de-
scribed in Section 4.2. NHeis the number of the events in
the energy range from 0.4 MeV to 0.8 MeV (Fig. 11) after
subtraction the µ-capture background NBGwhich is also
given in Tables II to V. This background, as well as the
transfer rate to the impurity ΛdZ and the impurity con-
centration CZ were determined following the procedure
described in Section 4.3. The relative concentrations of
the H2, D2, and HD molecules in H2/D2−96 and HD−96
experiments were measured with a gas chromatograph.
The results are presented in Table VI.
In our analyses we used the measured energy and time
distributions of the dd fusion events. Some of these dis-
tributions from the D2, D2+H2, D2+2HD+H2, and HD
runs are displayed in Figs. 16 to 20.
(12)
TABLE II: Experimental conditions and statistics in experi-
ment D2− 94
Run T, K ϕ,% Nµ, 106NHe
151.0 5.05 12.9156223 2343 0.0056
271.0 5.04 6.12 100648 1623 0.0054
396.0 5.02 3.29 79231
NBG ΛdZ,106s−1CZ, ppm
0.76
0.74
0.78 9900.0057
5.2. Analysis of amplitude distributions of dd fusion
events
5.2.1. Muon sticking probability
The fusion process in the ddµ molecule may lead to for-
mation of the (3Heµ)* atom moving away from the point
of dd fusion with kinetic energy 0.8 MeV, the muon being
bound in one of the orbits (n,l). This process is called
the ”initial” muon sticking (i.e. sticking immediately af-
ter ddµ fusion) with probability denoted as ωo
d. There is a

Page 11

TABLE III: Experimental conditions and statistics in exper-
iment D2− 95
Run T, K ϕ,% Nµ,106NHe
150.25.13 5.8382917
232.25.14 3.7763488
336.2 5.14 5.4689116
440.35.14 6.18 95896
545.35.05 5.2480633
6 60.3 5.04 5.7297437
7 120.3 4.99 3.97177023 365
8 150.3 4.97 2.60 165676 349
9 200.2 4.94 1.70155290 256
10 250.1 4.89 2.54270154 879
11300.0 4.85 1.95214643 985
12350.0 4.81 2.04 192674 7902
1328.3 2.76 3.5463033
NBG
16574 0.116
446
7273
13717 0.063
71
142
ΛdZ,106s−1CZ, ppm
15.6
0.40
4.70
8.45
0.041
0.073
0.33
0.47
0.56
0.99
1.71
13.3
0.05
0.003
0.035
0.0003
0.00053
0.0024
0.0034
0.004
0.007
0.012
0.093
0.000219
TABLE IV: Experimental conditions and statistics in experi-
ment H2/D2− 96. NE – non-equilibrium gas state (H2+D2),
EQ – equilibrium gas state (H2+ 2HD+D2)
Run T, K ϕ,% Gas Nµ,106NHe
150.25.12 NE 19.5
2150.2 5.02 NE 11.0
3 300.2 4.97 NE 3.96
450.25.07 EQ 34.1
5100.2 5.00 EQ 15.6
6150.2 4.97 EQ 18.9
7 300.2 4.91 EQ 5.56
NBG ΛdZ,106s−1CZ
290931 2491 0.0053(6)
372245 1070 0.004(1)
205744 117 0.0040(10)
309715 2987 0.0032(6)
202395 1393 0.0035(10)
371105 1744 0.0036(6)
166115 4740.0043(9)
0.71
0.55
0.56
0.44
0.48
0.50
0.60
certain probability (defined by “reactivation coefficient”
R) that the muon could be shaken off (muon stripping)
during the slowing down of the (3Heµ)* atom in collisions
with nuclei of the traversed matter. The probability ωd
for “final” sticking, after the3Heµ atom came to rest, is
defined as ωd= ωo
The first measurement of the muon sticking probability
in the dµd fusion was realized in the Gatchina experiment
in 1983 [15]. The final sticking probability was found to
be ωd= 0.122(3) at ϕ = 0.102.
In the presented here experiment, we used the same
experimental method as in the Gatchina experiment to
measure the final sticking probability, however, with
d· (1−R).
TABLE V: Experimental conditions and statistics
in experiment HD−96
Run T, K ϕ,% Nµ,106NHe
1300.3 4.87 25.9
2 150.3 4.78 25.2
350.24.74 45.8
NBG ΛdZ,106s−1CZ, ppm
73023 2102 0.0032
51575 1672 0.0027
60220 2903 0.0043
0.45
0.39
0.63
TABLE VI: Chromatographic data on H2/HD/D2 concentra-
tions in experiments H2/D2− 96 and HD−96
Experiment Run CD2,% CH2,% CHD,%
H2/D2− 96 1 ÷ 3 47.6
H2/D2− 96 4 ÷ 7 25.7
HD−96
HD−96
HD−96
much higher statistics. This method exploits the differ-
ence in the recombination of the initial ionization pro-
duced in the ionization chamber by the singly charged
(3Heµ)+and doubly charged (3He)++particles (see Sec-
tion 3.2) resulted in separation of the3He and3Heµ peaks
in the measured amplitude distributions of the fusion sig-
nals. We measured ωdin pure D2 gas at room tem-
perature at two deuterium densities ϕ = 0.0485 and
ϕ = 0.0837. Figure 21 displays the low energy part of
the measured amplitude spectrum.
the3Heµ peak is separated from the3He peak on one
side and from the (3He+3He) pile up peak on the other
side.Yet, there is some background under the
peak from the neighbor peaks and also from the strip-
ping events, that is from the events corresponding to the
3Heµ particles which have lost the muon at the end of
their tracks. Fortunately, there is a way to exclude com-
pletely this background (the survived muon method). If
we select the fusion events that were followed by next
fusions, we then know that the muon could not remain
stuck on the first3He, or it could not have catalyzed
the next fusion. Therefore, thus selected spectrum of the
first fusion events will contain all the events as in the un-
selected spectrum except the final sticking events. Note
that we selected only the (3He+3Heµ) signals from the
next fusion events. This selection helped to eliminate
some small background under the3Heµ peak caused by
the3Heµ capture events which might simulate the next
fusion. Figure 21 shows such survived muon spectrum
normalized to the unselected spectrum in the region of
the3He peak. The difference between these two spectra
in the region of the3Heµ peak gives the number N(3Heµ)
with high precision. Then the final muon sticking prob-
ability could be calculated as
49.0
26.6
1.84
2.07
2.15
3.4
47.7
97.38
96.91
96.75
1
2
3
0.78
1.02
1.10
One can see that
3Heµ
ωd=
N(3Heµ)
N(3Heµ) +
N(3He)
(1−Wpileup)
.
(13)
Here N(3He) is the number of events in the3He peak
and (1−Wpileup) is the pile up correction that takes into
account some reduction in N(3He) caused by piling up of
the second fusion signals on the3He signals. Note that
there is no pile up correction for N(3Heµ) as there are no
subsequent fusions in this case. The pile up probability
Wpileup was calculated from the measured distribu-
tions of the time intervals between the first and the sec-

Page 12

ond fusions:
Wpileup= 0.075(1) at ϕ = 0.0837 and Wpileup= 0.050(1)
at ϕ = 0.0485.
With these correction factors, the following results were
obtained for the final sticking probabilities:
ωd = 0.1224(6) at ϕ = 0.0837 and ωd = 0.1234(7) at
ϕ = 0.0485.
These results are in agreement with the Gatchina data
cited above within the uncertainties of the previous mea-
surements.
5.2.2. Branching ratios of dd fusion channels
As it was already pointed out, the time distributions
of fusion events were measured in this experiment
selecting the (3He+3Heµ) events. In order to obtain the
ddµ formation rates Λ3/2
distributions, one should know the branching ratio β for
these fusion channels (Fig. 2). As it became known from
the Gatchina experiment [15], the value of β is different
for the resonant and the non-resonant formation of the
ddµ molecule.Our measurements of the ratio of the
yields of the3He+n and t+p channels help to determine
branching ratios β3/2(T) and β1/2(T) for transitions
from the two spin states of the dµ atom.
data of the D2 experiment, we have determined the
ratio R=Y(3He+3Heµ)/Y(t+p) using the amplitude
distributions of the fusion signals in the steady-state
region selected by the time window 3µs≤t≤ 7.5µs.
The steady-state region was chosen due to less steep
sloping of the time distributions in this region that helps
to reduce the smearing correction to the yield of the
(t+p) events. This correction arises due to long tracks
of the protons leading to some shift of the measured
time distribution of the (t+p) events towards larger
t-values and, as a consequence, to some increase of the
number of the (t+p) events detected in the selected
time window. An example of the steady-state amplitude
spectra is shown in Fig. 22. Taking into account several
corrections, the yields of the (3He+3Heµ) and (t+p)
events were determined as follows:
ddµ(T) and Λ1/2
ddµ(T) from these
From the
Y(3He +3Heµ) = [N(3He +3Heµ)−
−NBG(3He +3Heµ)] · εtr+ Npileup,
(14)
Y(t+p) = [N(t+p) − NBG(t+p)] · εsm− Npileup.
Here N(3He+3Heµ) is the number of the events under the
3He and3Heµ peaks in the energy range (0.4÷0.8) MeV,
N(t+p) is the number of the events in the energy range
(0.8÷8) MeV, NBG(3He+3Heµ) is the number of the µ-
capture background events in the energy range (0.4 ÷
0.8) MeV, NBG(t+p) is the number of the µ-capture
background events in the energy range (0.8 ÷ 8) MeV,
Npileup is the number of pile up events (second fusion
signals piling up with the first3He fusion signals), εtris
the correction for trigger losses of some3He fusion events
(15)
at the edges of the anodes, εsm is the proton smearing
correction.
As concerns the µ-capture background, it was calcu-
lated as described in Section 4.4 with additional selection
in the time window 3µs≤t≤ 7.5µs. Note that the num-
bers NBG(3He+3Heµ) and NBG(t+p) proved to be close
to each other, so the overall correction to the ratio R was
less than 1%.
The correction Npileup takes into account the
events that were transferred from the3He peak to the
t+p region because of pile up with the second fusion.
This correction was calculated in a similar way as ex-
plained in Section 5.2.1. Its value ranged from 5% to
10% depending on the dd fusion rate. Note that there
is no correction for pile up of the second fusion with the
t+p events as such pile up events did not escape the de-
tected energy range (0.8 ÷ 8) MeV. The correction for
trigger losses was discussed in Section 4.2. Its value in
the D2 runs was Wtr = (1 − εtr) = (6.5 ± 0.7)%. The
value of the proton smearing correction was measured to
be εsm= (98.7 ÷ 98.0)%.
All the above mentioned corrections are well deter-
mined from the same experimental data set and impose
additional systematic error to the measured ratio R on
the level 0.8%. In this way, the ratio R(T) was deter-
mined using the experimental data obtained in D2− 94
and D2− 95 experiments. The results are presented in
Table VII and in Fig. 23. Note, that three D2− 95 runs
with exceptionally high Nitrogen contamination were not
used for determination of R(T).
The procedure of extraction of the R(T) values from
the data of the HD experiment was different from the one
described above in two points. First, the one-exponential
shape of the time distributions in this case allowed to
extend the time window to smaller t-region , 1µs≤t≤
7.5µs, thus increasing the statistics without enlarging the
smearing correction. Second, the determined branching
ratios were corrected for presence of the D2 admixture
in the HD gas. The obtained results are presented in
Table VIII and in Fig. 23.
5.3. Analysis of time distributions of dd fusion events
5.3.1. Analysis of D2runs
We use the kinetics diagram presented in Fig. 2 to ex-
tract the ddµ formation rates Λ1/2
spin flip rate Λ21(T) from the experimental time distri-
butions of the (3He+3Heµ) fusion events. Note that the
density dependent rates in this scheme can be related to
the rates normalized to the liquid hydrogen density (LHD
= 4.25·1022nuclei/cm3) as follows
Λ3/2
3He
ddµ(T), Λ3/2
ddµ(T), and the
ddµ=?λ3/2
Λpdµ= λpdµ· ϕ · Cp,
where?λ3/2
ddµ·ϕ·Cd, Λ1/2
ddµ=?λ1/2
ddµ·ϕ·Cd, Λ21=?λ21·ϕ·Cd,
ΛdZ= λdZ· ϕ · CZ,
(16)
ddµ,?λ1/2
ddµ,?λ21, λpdµ, λdZare the rates normalized
to LHD (ϕ=1), the first three of them being the effective

Page 13

rates, that is the rates including the back-decay, Cdand
Cpare the deuterium and hydrogen relative atomic con-
centration, Cd+Cp= 1, CZ is the atomic concentration
of the gas impurity.
The scheme in Fig. 2 has the analytical solution de-
scribing the time distribution of the (3He+3Heµ) events:
N3He(t) = Nµ· χnorm· (E1· eK1·t+ E2· eK2·t) · ∆t. (17)
Here, N3He(t) is the number of the (3He+3Heµ) events
(first fusions) detected in the time interval ∆t (∆t= 50 ns
in our analysis);
Nµis the number of the muon stops selected according
to the procedure described in Section 4.2.
χnorm= (1 − εtr)/(1 − ωd·Wpileup) is the normaliza-
tion factor taking into account the trigger losses of the
(3He+3Heµ) events and the difference in pile up correc-
tions for Nµand for N(3He+3Heµ); εtr= (6.5±0.7)% in
the D2runs (see Section 4.2);
Wpileup is the pile up probability of the first fusion
signals with the muon signals.
pile up probability of the second fusion signals with
the first3He fusion signals. The difference comes only
from the3Heµ signals as there are no fusions after the
3Heµ events. Therefore, the total pile up correction is
ωd·Wpileup, where ωd= 0.12 is the muon sticking prob-
ability. The value of Wpileup is determined from the
time distributions of the intervals between the muon and
the first fusion signals: Wpileup = (2 ÷ 5)% depending
on the fusion rates. So the total pile up correction is
ωd·Wpileup= (0.2÷0.6)%, and it is well determined from
our experimental data.
The constants E1, E2, K1, K2in (17) have explicit ex-
pressions through the rates and branching ratios entering
the kinetics scheme:
It is identical to the
K1= 1/2·{−(Λ1+Λ2)+[(Λ1−Λ2)2+4·Λ21·Λ12]1/2},
K2= 1/2·{−(Λ1+Λ2)−[(Λ1−Λ2)2+4·Λ21·Λ12]1/2},
E1= β1/2· Λ1/2
ddµ· A1+ β3/2· Λ3/2
ddµ· A3,
E2= β1/2· Λ1/2
ddµ· (η1− A1) + β3/2· Λ3/2
ddµ· (η2− A3),
whereΛ1= λ0+ Λ1/2
ddµ+ Λ12+ Λpdµ+ ΛdZ,
Λ2= λ0+ Λ3/2
ddµ+ Λ21+ Λpdµ+ ΛdZ,
A1=
Λ21
K1+ Λ1·A3,A3=K1+ Λ1
K1− K2·(η2−η1
Λ21·(K2+Λ1)).
Here, η1 = 1/2 and η2 = 3/2 are the populations of
the 1/2 and 3/2 spin states of the dµ atoms, λ0= 0.445·
106s−1is the free muon decay rate.
The spin flip rates Λ12and Λ21are related by the bal-
ance equation (6). The rate Λpdµ= 5.6(2) · 106s−1[32]
is known and ΛdZ are determined from our experimen-
tal data (Section 4.3 and Tables II to V). The remaining
three rates entering the above expressions Λ3/2
Λ21 are to be extracted from the fits to the measured
time distributions of the (3He+3Heµ) events.
As concerns β1/2(T) and β3/2(T), we represent them
by the following expressions through the branching ra-
tios βresand βnrfor the resonant and non-resonant ddµ
formation, respectively:
ddµ, Λ1/2
ddµ,
βF(T) =
(?λF
ddµ(T) − λnr
ddµ(T)) · βres+ λnr
?λF
F = 1/2,3/2.
ddµ(T) · βnr
ddµ(T)
,
(18)
Here λnr
malized to LHD.
Our measurements of the ratio R(T) in the D2 and
HD experiments allow to determine βresand βnr. Also,
λnr
distributions of the fusion events in the HD experiment.
However, the analysis of the HD data requires knowledge
of the ddµ formation rates in D2molecules presented as
a small admixture in the HD gas. Therefore, we used the
iteration procedure.
As the first step of the iteration procedure, the anal-
ysis of the time distributions of the (3He+3Heµ) events
from the D2runs was performed with some approximate
(though quite realistic) values of βres, βnr, and λnr
particular, we have chosen βnr= 0.500, βres=0.588 and
λnr
analysis of the (3He+3Heµ) time distributions from the
D2 runs yielded the preliminary values of?λ3/2
Using the obtained values of?λ3/2
from the HD run was performed (see Section 5.3.2), and
the ddµ formation rate λHD
sults are presented in Table VIII. The measured λHD
shows a weak linear dependence with temperature:
ddµ(T) is the ddµ non-resonant formation rate nor-
ddµ(T) can be determined from the analysis of the time
ddµ. In
ddµ= 0.05·106s−1. With these inputs, the performed
ddµ(T) and
?λ1/2
analysis of the time distributions of (3He+3Heµ) events
ddµ(T).
ddµ(T) and?λ1/2
ddµ(T) was determined. The re-
ddµ(T), the
ddµ(T)
λddµ(HD) = (2.6 · 10−4· T + 0.039) · 106s−1.
This result is in qualitative agreement with theoretical
calculations of the non-resonant ddµ formation rate (see
section 6.3). For the further analysis, we considered
λnr
Then data on R(T) from the D2runs were fitted with
the following expression:
(19)
ddµ= λddµ(HD), the latter being given by eq.(19).
R(T) =
(?λddµ(T) − λnr
ddµ(T)) · βres+ λnr
ddµ(T))(1 − βres) + λnr
ddµ(T) · βnr
ddµ(T) · (1 − βnr),
(?λddµ(T) − λnr
(20)

Page 14

where?λddµis the steady-state ddµ formation rate deter-
from the HD runs were fitted with the constant value
mined by eq. (10). Simultaneously, the data on R(T)
Rnr=
βnr
1 − βnr.
(21)
The results of the fit are shown in Fig. 23. From this fit
the final values of Rres = 1.455(11) and Rnr = 1.01(1)
were obtained. The corresponding branching ratios are:
βres = 0.593 and βnr = 0.502. With these new values
of βres and βnr and using λnr
second fit of the D2data was performed, and the values
of?λ3/2
further iteration was needed. The obtained results are
presented in Table VII and in Figs. 24 and 25. Also shown
are the steady-state formation rates?λddµcalculated with
ddµ(T) given by (19), the
ddµ,?λ1/2
ddµ,?λ21 were obtained. These values proved
to be close to the values from the first fit. Therefore, no
expression (10) using the measured?λ1/2
TABLE VII: ddµ formation and spin flip rates (106s−1), and
branching ratios in pure D2 gas
T, K ϕ,%?λ1/2
32.2 5.14 0.051(1) 4.13(7)
36.25.14 0.049(2) 3.96(6)
40.35.14 0.050(fix) 3.88(6)
45.35.05 0.0515(8)3.92(6)
50.25.13 0.0544(fix) 3.90(6)
51.05.05 0.0537(7)3.79(4)
60.35.04 0.063(1)3.89(6)
71.05.04 0.088(1) 4.05(5)
96.05.02 0.246(3) 4.42(7)
120.3 4.99 0.528(4) 4.98(10) 0.609(4)
150.3 4.97 0.943(5) 5.07(15) 1.129(8)
200.2 4.94 1.65(2)4.59(15) 1.97(2)
250.1 4.89 2.202(27) 4.28(12) 2.56(3)
300.0 4.85 2.549(23)3.75(fix) -
350.0 4.81 2.70(5)3.28(fix) -
ddµand?λ3/2
ddµ.
ddµ
?λ3/2
ddµ
3.98(5)
?λddµ
0.051(1)
0.049(2)
-
0.0515(8)
-
0.0538(7)
0.0637(10) 36.04(25) 1.08(2)
0.091(1) 35.38(25) 1.24(1)
0.270(3) 34.8(4)
35.0(5)
35.9(9)
34.2(1.5) 1.40(1)
37.0(2.0) 1.42(1)
-
-
?λ21
37.1(3)
36.6(2)
36.6(2)
36.8(3)
35.9(3)
35.77(16) 1.05(1)
R
28.32.76 0.053(3) 0.053(3) 37.0(4)-
1.07(3)
-
-
1.03(3)
-
1.34(1)
1.40(1)
1.43(1)
1.44(1)
1.44(1)
5.3.2.
2HD+D2) runs
Theanalysis
equilibrium gas mixture) was performed following exactly
the procedure used in the analysis of the D2data. The
only difference was that, while calculating the normal-
ized rates?λ3/2
Cd= 0.995 in the D2runs. Note that the H2and D2con-
centrations were determined with high precision (0.2%)
by the gas pressure measurements during the gas filling,
and they were also controlled with a chromatograph with
a precision of 2%. The obtained results are presented in
Table VI.
Analysis of the (H2+ D2), HD and (H2+
of the(H2+D2) experiment(non-
ddµ(T),?λ1/2
ddµ(T), and?λ21(T), the atomic deu-
terium concentration Cd = 0.476 was taken instead of
The analysis of the HD and (H2+2HD+D2) runs
required some modification of expressions (16), (17),
(18) used in the fitting procedure.
rates Λ1/2
components:
In particular, the
ddµ, Λ3/2
ddµwere replaced by the sum of two
ΛF
ddµ→ ΛF
ddµ(D2) + ΛF
ddµ(HD),F = 1/2, 3/2,
where
= λF
Here CD2and CHD are the relative concentrations of
D2 and HD molecules, respectively. Also, the products
βFΛF
ΛF
ddµ(D2) =?λF
ddµ(D2)· ϕ · CD2; andΛF
ddµ(HD)
ddµ(HD) ·ϕ · 1/2·CHD.
ddµin expressions (17) were replaced by the sum:
βFΛF
ddµ→ βF(D2) · ΛF
ddµ(D2) + βF(HD) · ΛF
ddµ(HD),
where
βF(D2) =
(?λF
ddµ(D2) − λnr
ddµ) · βres+ λnr
λF
ddµ· βnr
ddµ(D2)
,
(22)
F = 1/2, 3/2,
β1/2(HD) = β3/2(HD) = βnr.
(23)
In the analysis, we assumed that the hyperfine com-
ponents of the non-resonant ddµ formation rates in
HD molecules are equal to each other:
Λ3/2
the time distribution of the fusion events should be one-
exponential, and the observed deviation from this shape
(see Fig. 20) we attributed to the D2 admixture in the
HD gas.
In the fitting procedure of the HD data, the rates
?λ1/2
λddµ(HD), λ21(HD), and CD2. In fact, the spin flip rate
λ21(HD) could be reliably determined only at T = 50 K:
λ21(HD) = 32.2 (1.7)·106s−1. Then this value was fixed
in the fits of the data at 150 K and 300 K. The obtained
results are presented in Table VIII and in Fig. 25.
Note the close similarity of deuterium concentrations
CD2found from the fit to those measured by the chro-
matograph - an evidence of correct interpretation of the
HD data.
In the analysis of the (H2+2HD+D2) runs, the rate
Λddµ(HD) was taken from the HD analysis, see expression
(19). The fitting parameters were?λ1/2
Measurements of time distributions of fusion neutrons
in the HD experiment revealed a prominent peak at t<
50 ns (Fig. 26). It is interpreted as contribution of ddµ
Λ1/2
ddµ(HD)=
ddµ(HD)= Λddµ(HD). That means that in pure HD gas
ddµ(D2) and?λ3/2
ddµ(D2) were fixed (taken from the
D2 data analysis), and the fitting parameters were
ddµ(D2),?λ3/2
ddµ(D2),
?λ21. The results of the analysis are presented in Table IX.

Page 15

TABLE VIII: ddµ formation and spin-flip rates (106s−1), and
branching ratios in HD gas
ϕ % λHD
ddµ
λ21
300.3 4.87 0.119(6) 32.2 (fix) 1.00(2) 0.82(8) 0.78
150.3 4.78 0.080(3) 32.2 (fix) 1.01(2) 1.06(5) 1.02
50.24.74 0.056(8) 32.2 (1.7) 0.99(2) 1.16(3) 1.10
T KRCD2% CD2(chr) %
TABLE IX: ddµ formation and spin flip rates (106s−1) in
H2/D2 gas mixture
Gas state ϕ %?λ1/2
50.2NE5.12 0.051(4)
100.2 EQ5.00 0.279(8)
150.2 EQ4.97 0.95(1)
150.2 NE5.02 0.934(10) 4.94(6)
300.2 EQ 4.91 2.55(6)
300.2 NE4.97 2.51(3)
T K
ddµ
?λ3/2
3.74(3)
4.50(5)
4.88(5)
ddµ
3.66(3)
?λddµ
0.051(4)
0.309(8)
1.126(10) 35.6(7)
1.11(1)
?λ21
35.5(3)
33.9(5)
50.2 EQ 5.07 0.058(4)0.058(4) 34.0(4)
35.5(7)
52(10)
35(5)
3.62(18) 2.80(6)
3.70(7)2,79(3)
formation in HD molecules by the epithermal dµ atoms
with kinetic energies higher than 0.3 eV (see Section 6.3).
6. Discussion
6.1. Muon sticking
Soon after discovery of muon catalyzed fusion, J.D. Jack-
son recognized the importance of muon sticking in the
µCF reactions as a fundamental limitation in the number
of the fusion cycles catalyzed by each muon. J.D. Jackson
also was the first to consider the muon stripping while the
Heµ atom is slowing down, and he came to the conclusion
that the reactivation process does not decrease apprecia-
bly the muon loss caused by muon sticking. His estimates
for the dµd fusion were: ω0
[33]. These calculations were renewed by S. Gershtein et
al. in 1981 [34] with the following results for dµd fusion:
ω0
The first experimental result on final muon sticking in
dµd fusion came from the Gatchina experiment in 1983:
ωd= (12.2±0.3)% [15]. This result proved to be slightly
below the theoretical prediction of S. Gershtein et al.
Moreover, the disagreement becomes even larger if, fol-
lowing the same theoretical approach as in [34], one takes
into account the excited states of the3Heµ atom with
n≥ 4 (in ref. [34] only the excited states with n≤ 3 were
considered). As it was demonstrated in ref. [35], in this
case the initial sticking probability becomes ω0
The problem proved to be in calculation of the ddµ wave
function ψddµ(r,ρ) at small inter-nuclear distances ρ → 0.
To calculate this wave function, S. Gershtein et al., as
well as J.D. Jackson, used the Born-Oppenheimer ap-
proximation assuming that the muon follows the motion
d= 16% and R= 0.04 ÷ 0.13
d= 15.5% , R= 0.05, ωd= 14.7%.
d= 17.2%.
of the nuclei adiabatically. In this approximation, the
ddµ wave function is replaced by a wave function of the
µ−meson in a K-shell orbit of the united-nuclei atom,
4Heµ in our case. Since then, new calculations appeared
focused on finding the united-nuclei limit of the correct
non-adiabatic wave functions of the muonic molecules.
Different techniques were applied: the adiabatic repre-
sentation [35] and the variational methods [36, 37]. The
calculated values of the initial sticking probability proved
to be quite close to each other (Table X). Averaging
these results, we obtain as the best theoretical predic-
tion (ω0
The calculation of the final sticking probability ωdre-
quires knowledge of the muon reactivation processes. Ac-
cording to the theory, 30% of the3Heµ atoms are pro-
duced in different excited states (nl). In principle, the
higher the excitation, the larger the muonic atom ion-
ization cross section.However, the ionization should
compete with very fast radiative de-excitation of the ex-
cited atoms.Therefore, the excited states contribute
to the muon stripping only at high gas densities, while
at low gas densities the stripping occurs mostly from
the ground state with some contribution from the 2s
metastable state of the3Heµ atom. The accurate calcu-
lation of the muon reactivation process requires kinetics
treatment that takes into account collisional excitations
and de-excitations, ionization, muon transfer, and radi-
ation. The rates of these processes are known mainly
from the theoretical calculations. Table X presents the
results from some kinetics analyses of the muon reacti-
vation process in the dµd fusion. One can see that the
theory predicts some gas density dependence of the muon
reactivation coefficient. For example, at the gas densities
of our experiment, the change in ϕ by ∆ϕ = +0.04 should
change the value of R by ∆R = +0.01. Our experimental
data are in qualitative agreement with this prediction:
d)th= (13.4±0.3)%.
∆Rexp=ωd(ϕ = 0.0485) − ωd(ϕ = 0.0837)
ω0
d
= 0.008(8).
Taking into account the theoretical ϕ-dependence of the
muon reactivation and averaging the results from the lat-
est analyses [39, 40, 41], we obtain:
Rth(ϕ = 0.07) = 0.10 ± 0.01 and ωth
(ω0
result agrees with our experimental data: ωd= 0.1224(6)
at ϕ = 0.0837 and ωd= 0.1234(7) at ϕ = 0.0485. This
agreement can be considered as an evidence of mutual
consistency and correctness of the modern theory of the
initial muon sticking and the theory of the muon reac-
tivation in muon catalyzed dd fusion. Note, however,
that, as concerns the muon reactivation, our experiment
reflects only the situation at relatively low gas densities
where the muon stripping occurs mainly in the ioniza-
tion collisions of the unexcited3Heµ atoms in the gas
medium.
d(ϕ = 0.07) =
d)th·(1−Rth) = 0.120(4). Within the quoted error, this