Can the Neutron-Neutron Scattering Length be
Determined from n-d Breakup Experiments?
W. von Witsch
Institut für Strahlen- und Kernphysik, Universität Bonn, D-53115 Bonn, Germany
Abstract. Two recent nd breakup experiments are discussed which were aimed at the determination of ann, the neutron-neutron
scattering length. In both cases, the data were analyzed by means of state-of-the-art Monte-Carlo simulations based on rigorous
three-nucleon calculations with realistic nucleon-nucleon potentials. Yet, the two results differ greatly, whereat one of them
agrees with the value obtained via the 2H(π¯,nn)γ reaction, while the second one reproduces the average result of all previous
nd breakup experiments. Moreover, some other recent findings regarding the nd breakup process raise the old question anew
whether the nn scattering length can be extracted at all from this reaction using present-day theories.
INTRODUCTION
EXPERIMENTAL DETAILS
The 1S0 nucleon-nucleon (NN) scattering length aNN is
a powerful magnifying glass to study the NN
interaction. Because the 1S0 state is almost bound, the
scattering length has a large negative value, and small
changes in the depth and width of a two-nucleon
potential cause large changes in aNN. The neutronneutron and proton-proton scattering lengths ann and app
are of special interest because, in principle, they allow a
sensitive test of charge symmetry in the strong
interaction. While app can be measured very accurately
via pp scattering, in the case of ann the measurement is
difficult because one must resort to multi-particle
breakup reactions with two neutrons in the exit channel.
Numerous attempts have been made to determine ann,
using mostly the 2H(π¯,nn)γ and 2H(n,nn)p reactions
and investigating the region of the nn final-state
interaction (FSI), where the two neutrons travel
together with small relative energy. While today the
2
H(π¯,nn)γ reaction seems to provide for ann a consistent
average value of –18.6±0.4 fm [1,2,3], the situation
regarding the nd reaction is confusing. Specifically, two
very thorough investigations based on this reaction
have recently been reported [4,5], with mutually
exclusive results. In both cases, the data were analyzed
by means of state-of-the-art Monte-Carlo simulations
based on rigorous Faddeev-type calculations with
realistic nucleon-nucleon potentials [6]. In this talk, I
will compare the two experiments, emphasizing their
respective strong and weak points, and finally address
the question of whether, perhaps, with present-day
theories the nn scattering length cannot be obtained
from this reaction after all.
The TUNL Experiment
The first experiment [4] was performed at the Triangle Universities Nuclear Laboratory (TUNL) at Durham, N.C. A collimated neutron beam of 13 MeV was
produced with the 2H(d,n)3He reaction by bombarding
an 8-bar gas target with a 10-MeV, 2-µA deuteron
beam from the tandem accelerator. The reaction target
consisted of a glass cylinder, 6 cm high and 4 cm in
diameter, filled with C6D12 liquid scintillator in which
pulses from breakup protons as well as recoil deuterons
were observed.
The nn interaction was investigated in the traditional
"final-state geometry", i.e. the two neutrons were detected in coincidence with two liquid-scintillator n
detectors placed behind each other at the same angle on
the same side of the beam. The front detector was ringshaped, and the coaxial back detector was positioned
such as to fill the solid angle of the opening in the ring
detector. Absolute cross sections were measured at four
angles between 20.5o and 43.0o simultaneously. The
integrated beam-target luminosity was obtained by
recording the yield for nd elastic scattering along with
the breakup data. The efficiency of the n detectors was
determined by putting them in neutron fields of known
intensities at TUNL and at the Physikalisch-Technische
Bundesanstalt (PTB) in Braunschweig, Germany. The
value of ann, deduced from the absolute cross sections,
was –18.7±0.60 fm.
In a parallel experiment, the well-known 1S0 np scattering length, which is –23.75±0.01 fm [7], was measured concurrently by placing several additional n detec-
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timing for the determination of the neutron energies.
However, it entailed strong multiple scattering effects
which had to be simulated accurately. The recoil geometry chosen for the Bonn experiment avoided this, and the
passive target permitted a much higher beam intensity
which, together with the 100% efficiency of the proton
detector and a larger cross section, nearly compensated
for the smaller target thickness. Elastic deuterons could
easily be distinguished from breakup protons by means of
their energy and TOF. In addition, there was no problem
of cross talk between closely spaced n detectors. Perhaps
the biggest advantage of this geometry, however, was
the simultaneous appearance in the spectrum of quasifree np scattering (QFS), where the cross section is
independent of ann and thus provides a built-in normalization for the FSI peak. In this way, all problems regarding the absolute calibration of the n beam and of
the n detector efficiency were essentially eliminated –
assuming that the theory describes the np QFS cross
section correctly. (A certain complication arose from
the fact that the neutron detector was positioned on the
recoil axis of the 2n system with low relative energy, so
that both neutrons might hit the detector, thereby increasing its effective efficiency. However, even though
this was a sizeable effect, it is governed simply by
three-body kinematics and thus could be taken into
account very accurately in the MC simulations.)
tors on the opposite side of the beam, resulting in anp =
–23.5±0.8 fm, in agreement with Ref. [7].
The Bonn Experiment
The second experiment [5] was done at energies of
25.3 and 16.6 MeV at the cyclotron of the Institut für
Strahlen- und Kernphysik at Bonn University. In this
case, high-energy (HE) neutrons produced via the
²H(d,n)³He reaction were used together with lower-energy
(LE) ones from the upper part of the breakup continuum
from the ²H(d,n)pd reaction. The gas target was liquidnitrogen cooled at a pressure of 39 bar, resulting in a HE
beam width of 4.0 MeV. In contrast to the TUNL
measurement, a different geometry was used in Bonn,
where only one of the neutrons was detected together with
the recoiling proton on the other side of the beam ("recoil
geometry"). In an evacuated "proton arm", a 50-mg/cm2thick CD2 target was suspended in the intensity plateau of
the tightly collimated n beam. The breakup protons first
had to pass through a thin scintillation foil detector (SFD),
positioned near the target but outside of the n beam,
before being stopped in a 5-mm-thick scintillator. The
signals produced in the SFD served as start signals for all
TOF measurements, including the TOF of the incoming n
beam. For the absolute normalization of the neutron
fluence, a double proton recoil telescope (PRT) was
placed in the beam, detecting protons emitted from a CH2
target at angles of ±35o. The efficiency of the n detector
was determined in situ by means of np scattering, using
the same setup with the CD2 target replaced by a CH2 foil.
At 25.3 MeV, a value of ann = –16.3±0.4 fm was
deduced from the absolute cross section while the
relative cross sections at 25.3 and 16.6 MeV,
normalized in the region of quasi-free np scattering (see
below), yielded –16.1±0.4 fm and –16.2±0.3 fm,
respectively.
Also at Bonn the np scattering length was measured
as a test for the experimental procedure. In a first
measurement, done at 25.3 MeV in recoil geometry, the
proton arm was replaced by a second n detector, and a
4.4-cm-diameter scintillator target was used, leaving the
rest of the setup unchanged. Subsequently, a
measurement in FSI geometry was made in which a Si
surface barrier detector was placed in front of an n
detector at 32O [8]. The results of the two experiments
were anp = –23.9±1.0 and –24.3±1.1 fm, respectively,
which is also in agreement with the value known from
free np scattering.
Discussion
Differing by almost four standard deviations, the ann
results of the two experiments are obviously conflicting, while being internally consistent. The four angles
measured at TUNL gave results which agree among
themselves and also with the average value from the
2
H(π¯,nn)γ reaction. Similarly, all three results of the
Bonn experiment agree with each other, whereat two of
them, at 25.3 and 16.6 MeV, were deduced from the
relative cross sections, and one, at 25.3 MeV, from the
absolute count rate. The Bonn results are also in agreement with those of all previous kinematically complete,
n-induced breakup experiments [9]. Thus, while the
TUNL results seem to indicate that there is no angular
dependency, those from Bonn show independence with
respect to the bombarding energy. Of course, the different geometries of the two experiments should not matter because, at relative energies Enn ≈ 0, the breakup
amplitude does not depend on which two of the three
final-state particles are detected. Nevertheless, although
both results appear to be well-founded experimentally,
one of them is obviously wrong. In order to resolve this
disturbing situation, it is planned to repeat the two experiments at TUNL simultaneously [10]. To this end,
both experiments are to be placed in the same neutron
beam, one behind the other, thereby eliminating many
relative uncertainties.
Comparison
Comparing the two experiments, each one had its
specific advantages. Only the final-state geometry
employed at TUNL allowed the use of a thick, active
target, resulting in a higher count rate and facilitating the
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Since all results [4,5,8] for the neutron-proton
scattering length agree with each other, and also with
the value known from free np scattering, it appears that
the nd breakup reaction should, in principle, also be
suitable for the extraction of ann. However, here a
caveat might be in order. While modern 3N calculations
describe most of the nd data very well, both below and
above the breakup threshold, there are some striking
exceptions. Specifically, in a recent experiment at Bonn
[11], np and nn QFS have been measured in the nd
reaction, with very surprising results: while for np QFS
the calculations reproduce the measured cross section
perfectly [12], there is a difference of almost 20% for
nn QFS, corresponding to more than 5 standard
deviations! (The nn part of this experiment will be
repeated at the China Institute for Atomic Energy in
Beijing [13].) Similarly, in the so-called "space star"
geometry the measured nd breakup cross sections are
some 25% larger than predicted [14]. Another case is
the well known "Ay puzzle", i.e. the fact that the theory
is unable to correctly describe the vector analyzing
power in nd and pd scattering [15]. Finally, no threenucleon-force (3NF) effects have been observed in Ref.
[4] although they are predicted by the theory and should
have been seen in the experiment. Taken together, these
observations strongly suggest that there are still some
important parts missing in the theory, most likely in the
3NF, and that there might even be an as yet unknown
difference between the nn and the np interaction. The
nd breakup reaction is still the best tool for the
investigation of these fundamental questions.
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