A. Badertscher¹, M. Daum°, P.F.A. Goudsmit¹, M. Janousch¹, P.-R. Kettle°, J. Koglin²°, V.E. Markushin°, J. Schottmüller³° and Z. G. Zhao¹
° | PSI, Paul-Scherrer-Institut, CH-5232 Villigen-PSI, Switzerland |
¹ | IPP, Institut for Particle Physics, ETH Zürich, CH-8093 Zürich, Switzerland |
² | Physics Department, University of Virginia, Charlottesville, Virginia 22901, USA |
³ | Physik-Institut der Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland |
Abstract
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We have measured the Doppler broadening of neutron time-of-flight spectra from the reaction pi¯ p -> pi° + n in atomic states. From the data, we infer that the kinetic energy distribution of pi¯p-atoms in liquid and gaseous hydrogen contains discrete `high-energy' components with energies as high as 200 eV attributed to Coulomb de-excitation. In liquid hydrogen, evidence for Coulomb de-excitation transitions with delta n = 2 has been found.
Introduction
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Evidence for a substantial fraction of highly energetic (>>1 eV)
pi¯p-atoms in the charge exchange (CEX) reaction
pi¯ p -> pi° + n,
in liquid hydrogen, was seen in our previous experiments [1-3].
The effect is observed as a Doppler broadening of the neutron time-of-flight
(TOF) peak, which is related to the kinetic energy distribution
f(Tpi p) of
pi¯p-atoms at the instant of the CEX-reaction.
Later experiments gave
further evidence for such `high-energy' components in both liquid and
gaseous hydrogen [4,5].
A precise knowledge of the kinetic energy distribution of the
pionic hydrogen atoms is important
for the determination of the strong interaction width of the ground state
in pionic hydrogen from the measurement of pionic X-ray
transitions [6-9].
The observed Doppler broadening of the TOF-spectra [1-5] can
be attributed to Coulomb de-excitation [10]
(pi¯ p)n + p ->
(pi¯ p)n' + p ¹,
where the de-excitation energy associated with the transition is shared as
kinetic energy between the collision partners. Other cascade processes, such as
external Auger effect, are only able to cause a
moderate acceleration of the pionic atom (~ 1 eV) [14].
The kinetic energy of the pi¯p-atom after a Coulomb
de-excitation transition is given by
neglecting the initial kinetic energy of the pionic atom. Here delta Enn' is the energy difference of the two atomic states n and n'; Mpi p and mp are the masses of the pi¯p-atom and the proton, respectively. The corresponding idealized energy distribution f(Tpi p) is shown in Fig.1, where only transitions with delta n =1 are considered. For the sake of simplicity, the low energy part of f(Tpi p) is approximated by a uniform distribution between 0 and T1. This Tpi p-distribution (Fig.1a) leads to the step-like neutron TOF-distribution shown in Fig.1b, where tau is the difference between TOF (t) and the mean TOF (t0). The times tau1 and taunn' in Fig.1b are related to the kinetic energies T1 and Tnn' as follows [15]:
Here, l is the length of the flight path, and v0 = 0.894 cm/ns is the neutron velocity for pi¯p-atoms undergoing the CEX-reaction at rest. At this point we assume that the pi¯p-atoms are not significantly decelerated between Coulomb de-excitation and nuclear capture. This assumption is supported by cascade calculations [5] for atoms with a kinetic energy of T >= 50 eV. Therefore, a signature for Coulomb de-excitation would be a step-like structure visible in the TOF-spectra of neutrons from the CEX-reaction. For atoms with T <= 20 eV the deceleration is important, and the Coulomb peaks are expected to be smeared out.
¹ It is not excluded that Coulomb de-excitation is part of some new mechanism, e.g. the formation of a resonant state, as was suggested for n = 2 [11-13].
The new experiment
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The present experiment was performed at the
piE1-channel of PSI; the experimental setup is shown in
Fig.2.
The following improvements to the previous experiments [1-4] were made:
(i) a reduction of the
background by introducing several neutron collimators as well as using
specially selected low noise photomultiplier (PM) tubes for the neutron
counters; (ii) improvement of the time resolution of the
neutron counters by placing them in adjustable holders so as to point
radially at the target for the different neutron flight-paths;
(iii) increased counting statistics by enlarging the solid angle of the neutron
detector and using a new beamline setup.
Pions of 117 MeV/c passed the beam counter S1
(see Fig.2) and a carbon degrader,
the thickness of which was optimized for a maximal stop rate in the hydrogen
target. The liquid target (LH2) had a length
of 9.3 cm
in the direction of the pion beam and a thickness of 0.5 cm in the
direction of the neutron flight path perpendicular to the pion beam.
The 40 bar gas target was operated at room temperature and had a
length of 21.2 cm in the direction of the pion beam and a
diameter of 14 mm.
Neutrons from the hydrogen target were detected after a flight path of variable
length (3 - 11 m) in a detector array consisting of 36
scintillator disks coupled directly to PM-tubes.
For the measurements in
liquid hydrogen, we used PILOT-U scintillators with a thickness of 5 mm,
whereas for the measurements in gas, NE102A scintillators with a thickness
of 15 mm were used in order to partially compensate for the lower pion
stop density. The neutrons from the CEX-reaction in the
target were accepted only if coincident with a suitably delayed pi-stop
signal and a corresponding gamma-ray signal from pi°-decay
in a NaI-calorimeter [16].
Data analysis
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The data are in form of TOF (TDC) and
pulse-height distributions (ADC). In a first step of
the analysis, which is very similar to that of Ref. [4], a lower and
an upper ADC-cut
were introduced for each neutron counter separately, in order to
suppress noise events from the PMs (lower ADC-cut) and accidental
events triggered
by photons from the pi°-decays or bremsstrahlung from beam
electrons (upper
ADC-cut). Both ADC-cuts were optimized for a maximal signal-to-noise ratio for
each neutron counter separately.
Before summing the spectra of all 36 neutron counters, the centre of each
neutron peak was shifted by a few TDC-channels to the same channel number, in
order to correct for small transit time differences in the PMs and cables as
well as small differences in the individual neutron flight paths.
In a second step, cuts were applied to the summed photon energy
from the NaI-calorimeter, accepting only events between 60 and 110 MeV.
In this way
photons from bremsstrahlung and from the
pi¯ p -> gamma n
reaction were suppressed. These cuts were also optimized for a maximal
signal-to-noise ratio. The small remaining background in the TOF-spectra,
consisting of a flat component from noise events and accidental peaks from
bremsstrahlung and 130 MeV photons from the reaction
pi¯ p -> gamma n,
was determined and subtracted.
The resulting TOF-spectra, taken at 3.82 m, 8.39 m and 11.10 m
between the liquid hydrogen target and the neutron detector
and at 3.83 m and 8.40 m
in gaseous hydrogen, are displayed in Figs.3 and
4, respectively. In both cases, the expected
step in the TOF-distribution from the Coulomb de-excitation transition
3 -> 2
(Tpi p = 209 eV) is clearly
visible.
From the resulting TOF-spectra we have obtained the kinetic energy distribution
f(Tpi p) using three different methods.
In method A, based on the TOF-distribution model
(cf. Fig.1b),
the three spectra measured in liquid hydrogen were fitted simultaneously
by TOF-distributions, generated by a
detailed GEANT Monte Carlo programme [17] which accounted for the
stopping distribution in the target, geometric effects
(intrinsic time resolution) and neutron scattering.
Similarly, the two spectra measured in gaseous hydrogen were
fitted to their respective Monte Carlo distributions.
The fit to the data taken with liquid hydrogen was restricted to a region
from -15
to +1 ns at 3.82 m, from -25 to +2 ns at 8.39 m and
from -30 to +2 ns at
11.10 m, to minimize the contributions from scattered neutrons.
For the data
taken with gaseous hydrogen, the corresponding fit regions were
from -15 to +2 ns at 3.83 m and from -25 to +2 ns at 8.40 m.
The free parameters were (i) four yields corresponding to the Coulomb
de-excitation transitions with principal quantum numbers 6 -> 5,
5 -> 4, 4 -> 3, and 3 -> 2; (ii) a yield
for all Coulomb de-excitation transitions with n > 6; (iii) an upper
energy bound for the above transition; (iv) two yields for the
Coulomb de-excitation transitions 6 -> 4 and 5 -> 3
with delta n = 2 (not included for gaseous hydrogen);
(v) the energy parameter T1
(cf. Fig.1a);
(vi) a distance independent
Gaussian time-jitter corresponding to electronic contributions to the
time resolution of the detector system; (vii) two (H2
gas) or three (LH2)
normalization factors for the
ordinates, and (viii) two (H2 gas) or three
(LH2) shifts to match the
time scales of the experimental histograms.
The energy shift observed in
Ref.[4] can be accounted for and made to vanish in our present
analysis if we include the delta n = 2 Coulomb
de-excitation transitions.
Then the resultant energies Tnn'
[cf. Eq. (1.1)] for n <= 6
perfectly match the theoretical values derived for Coulomb de-excitation,
and do not have to be taken as free parameters.
These fits gave a chi²/DOF of 0.96 with 739 degrees of
freedom (DOF)
for the measurements in liquid hydrogen and a chi²/DOF of 0.93
with 432
degrees of freedom for the measurements in hydrogen gas; this corresponds to
confidence levels of 76.2 % and 84.1 %, respectively. In the
gaseous hydrogen fit, the components with delta n = 2
were not included
because the statistics were not sufficient to resolve these transitions.
The values for the relative yields
Ann' of the transitions are
listed in Table 1. For the low energy component of
f(Tpi p),
we found
T1 = (1.0 ± 0.2) eV
for liquid hydrogen and
T1 = (1.6 ± 0.3) eV
for gaseous hydrogen. The upper
energy bound for the sum of all Coulomb de-excitation transitions with
n > 6 is
Tn>6 = (7 ± 1) eV
for liquid hydrogen and
Tn>6 = (7 ± 2) eV
for gaseous hydrogen.
For comparison, a fit to the liquid hydrogen data was made without
the Coulomb de-excitation components with delta n = 2 (see
Table 1). This fit gave a chi²/DOF of 1.06
with 741 degrees of freedom, which corresponds to a confidence level
of 12.4 %.
The difference in the chi²/DOF between the two fits in liquid
hydrogen is not
very significant; however, there is a strong hint for components
with delta n = 2 from two other model independent methods
(B,C) used to extract the kinetic
energy distribution f(Tpi p) from the data.
In method B, no assumptions about the positions of the peaks were made. Here,
the data measured in liquid and gaseous hydrogen
were fitted using a kinetic energy distribution consisting of 16
energy bins (from Ti-1 to
Ti); this corresponds to 16 equidistant
time bins (from taui-1 to
taui). The kinetic energy distribution
f(Tpi
p) was assumed to be constant within
each bin.
The relationship between the times taui and the
energies Ti is given by as in
Eq. (1.2)
with Ti [eV] = i²
(i = 1,...,16).
The fit was restricted to the same regions in the
TOF-spectra as described above. The resulting kinetic energy distributions are
shown in Fig.5. The yields
Ai correspond to the height of the respective bins.
In both kinetic energy distributions, a sharp decrease after the transition
4 -> 3 and indications for discrete peaks due to
Coulomb de-excitation can be seen. Moreover, in the kinetic energy
distribution for
liquid hydrogen, a small peak in the region between 100 eV and
121 eV is
visible which could be assigned to the Coulomb de-excitation transition
5 -> 3 corresponding to an energy of 107 eV.
In the distribution for gaseous hydrogen, the
error bars are larger due to the lower statistics.
Method C is based on the direct reconstruction of the kinetic energy
distribution from the deconvoluted TOF-spectra.
Here, the
TOF-spectra are deconvoluted with Monte Carlo generated
TOF-distributions, taking into account neutron scattering and the intrinsic time
resolution, as well as a distance independent Gaussian time-jitter
corresponding to
electronic contributions to the time resolution. From these monotonously
decreasing, deconvoluted
TOF-distributions F(tau), we have calculated
the cumulative
energy distributions W(Tpi p),
which are given by
with
For the calculation of W(Tpi p), only the fast side of the neutron TOF-spectra was used to minimize contributions from scattered neutrons. Finally, the kinetic energy distributions f(Tpi p) can be calculated as follows:
The final results in Fig.6 were obtained by averaging the kinetic energy distributions for the three different distances in liquid and the two distances in gaseous hydrogen. The shapes of these distributions are consistent with those of Fig.5.
Conclusions
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We have confirmed the existence of strong `high-energy' components (Tpi p >>1 eV) in the kinetic energy distribution f(Tpi p) of pi¯p-atoms in both liquid and gaseous hydrogen at the instant of the CEX-reaction. The results obtained with three different methods of reconstruction of the energy distributions are self-consistent. These `high-energy' components contain about half of the pi¯p-atoms (see Table 1), and about four percent have kinetic energies as high as ~200 eV. The shapes of the energy distributions strongly support the Coulomb de-excitation like origin of the `high-energy' components and further give evidence for delta n = 2 Coulomb de-excitation transitions. The inclusion of the delta n = 2 Coulomb de-excitation components in the present analysis made the previously claimed energy shifts in Tnn' [4] vanish. Our results favour the recent calculations of the Coulomb de-excitation process [18,19], which predict relatively high rates and resolve a long standing discrepancy between past calculations [10,14,20-22].
Acknowledgments
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We thank Z. Hochman, L. Knecht and H. Obermeier for their very competent technical assistance. The support from the Hallendienst and many other PSI staff members is gratefully acknowledged. This experiment was supported by the Schweizerischer Nationalfonds zur Förderung der wissenschaftlichen Forschung.
References
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Pictures of the NTOF-Experiment
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