Results From Indirect Measurements Of The 1S0 NeutronNeutron Scattering Length
C.R. Howell
Department of Physics, Duke University and the Triangle Universities Nuclear Laboratory, Durham, NC USA
Abstract. The ultra-high sensitivity of the 1S0 nucleon-nucleon (NN) scattering length to the strength of the nuclear
potential makes it a central parameter for quantifying charge symmetry breaking in the strong nuclear force. Of the
three 1S0 NN scattering lengths, the value of the neutron-neutron (nn) scattering length ann is the most uncertain. A
number of reactions that produce two neutrons with low relative momentum have been used to determine ann. However,
the two reactions that give ann with the least theoretical uncertainty are pion-deuteron capture (π-+d → n+n+γ) and
neutron-deuteron breakup (n+d → n+n+p). Discrepancies between the results of different nd breakup experiments raise
serious concerns about the use of this reaction to determine ann. In this paper the results obtained using these two
popular reactions will be summarized and discussed in the context of the experimental techniques and theory used to
determine ann in selected experiments.
to the differences in the masses of the u and d quarks
(QMD) and in the electromagnetic forces of the uu and
dd interactions[1]. In the meson exchange picture of
the NN potential, CSB is due to the isospin mixing of
the ρ and ω mesons. The mixing amplitude is directly
proportional to the QMD [1]. It has been shown also
that CSB can be explained as being due to dynamical
effects in the one-boson exchange NN potential that
are caused by the mass difference of the neutron and
proton [2]. Given that the mass difference of the
neutron and proton is mostly due to the QMD, the
fractional change in the strength of the NN potential
due to a QMD of 3 MeV is approximately
INTRODUCTION
Soon after the neutron was discovered by
Chadwick in 1932, Heisenberg invented the concept of
isospin space. Because of the small mass difference
between the neutron and proton, he suggested that they
should be treated as two states of the same particle, the
nucleon. As in the case of spin, Heisenberg felt that
the Hamiltonian for the strong interaction should be
invariant under arbitrary rotations in isospin space, i.e.,
the strengths of the neutron-neutron (nn), protonproton (pp) and neutron-proton (np) strong nuclear
forces should be identical; the symmetries of charge
independence and charge symmetry were born.
However, these are only approximate symmetries, i.e.,
there are no fundamental conservation rules that
prevent their violation, and it is well established that
both are violated. In the single-boson exchange model
of the nucleon-nucleon (NN) potential, charge
dependence (Vpp ≠ Vnp) is caused mostly by the
additional exchange of charged mesons in the case of
the np force. The cause for charge-symmetry breaking
(CSB), Vpp ≠ Vnn, is more interesting.
2(md − mu ) 2(3MeV / c 2 )
≈
= 0.006.
M
1000MeV / c 2
(1)
In the above equation md and mu are the bare masses
of the u and d quarks, respectively, and M is the
approximate mass of a nucleon. Therefore, an
observable that is sensitive to the strength of the NN
potential at the level of 0.5% must be used to observe
CSB effects. The 1S0 NN scattering length is such an
observable.
The sensitivity of the 1S0 NN scattering length to
the potential strength can be understood to first order
using effective range phenomenology. For a square
well potential, the fractional change in the scattering
length due to a small change in the potential depth is
given by
The charge symmetry operator rotates a neutron
into a proton and vice versa. At the quark level it
rotates an up (u) quark to a down (d) quark and vice
versa. So, by measuring the degree to which charge
symmetry is broken, information about the differences
in the responses of the u and d quarks to the strong
nuclear force is obtained. In fact, all observed chargesymmetry breaking can be accounted for as being due
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
© 2003 American Institute of Physics 0-7354-0149-7/03/$20.00
283
δann
ann
 a  δV
= 1.23 nn 
 rnn  V
.
breakup experiments. Five of the experiments
reanalyzed by Tornow et al. [7] using modern theory
(2)
Using typical values of ann = -18.8 fm and rnn = 2.8 fm,
Equation (2) implies that a 1% change in the depth of
the square well potential will result in about a 10%
shift in the value of ann. For most realistic NN
potential models, a 1% change in the potential strength
results in a 30% shift in the value of the calculated
scattering length. It is this high sensitivity of aNN to
the strength of the NN potential that makes it well
suited for CSB studies. The difference in the values of
the 1S0 nn and pp scattering lengths, ann and app,
respectively, gives a direct measure of CSB, and
consequently the QMD.
The app has been measured directly to high
precision using two-nucleon scattering. However, the
extraction of the pure nuclear part of app from the
measured value (app = -7.802 ± 0.004 fm) requires a
large correction for electromagnetic (em) effects. This
sizeable correction results in a relatively large
uncertainty of about ±0.4 fm in the value of app. The
situation is quite different for ann. For technical
reasons direct measurements of ann using free neutrons
have never been successfully executed. Up to now, all
determinations for ann have been based on studies of
reactions with at least three particles in the exit
channel [3-6]. In this paper, the results from
measurements of neutron-deuteron (nd) breakup and
pion-deuteron (π-d) capture are reviewed. Studies
using these reactions were chosen for review because
historically they have provided the most trusted
determinations of ann. After the review, a value for ann
is recommended.
FIGURE 1. Summary of indirect determinations of ann from
experiments reported between 1964 and the present. The
data points for each reaction type are plotted on the
horizontal axis in chronological order. Five of the
experiments reanalyzed by Tornow et al. [7] are shown as
open circles. The second average value in the left section is
for the values obtained in the reanalysis [7]. The average
values given at the tops of the sections for kinematically
complete nd breakup measurements and the π-d capture
experiments don't include the results reported since 1997.
are displayed as open circles. The middle section
contains results from kinematically-complete nd
breakup experiments. The results from π-d capture
measurements are in the right section. The solid line
represents the statistically weighted average of the data
points for each reaction type from 1964 through 1997.
The dashed line in the left section is the average of the
reanalyzed data [7]. The main features of cross-section
measurements of each of the three reactions, are
presented.
INDIRECT METHODS AND RESULTS
In all methods the value for ann is determined by
fitting the measured cross section for the nn final-state
interaction (FSI) with ann either explicitly or indirectly
as one of the fit parameters. In methods using the nd
breakup reaction, the fit can be either of the absolute
cross section or of the shape of the cross-section
distribution (as a function of relative energy between
the two neutrons or the energy of the proton). The fit
is always to the shape of the cross-section distribution
in methods using the π-d capture reaction. The results
obtained using indirect methods are summarized in
Figure 1. The figure is divided into three sections.
The left most section contains the results from crosssection measurements of kinematically-incomplete nd
Kinematically-Incomplete nd Breakup
In the kinematically-incomplete (KI) nd breakup
experiments, the proton is detected near 0º, and its
energy is measured using either a magnetic
spectrometer or a charged-particle counter telescope.
With either apparatus, the target is a deuterated
polyethylene foil that is sufficiently thin to allow the
most energetic protons from the breakup reaction to be
detected and the types of the detected charged particles
to be identified using either time-of-flight-vs-energy or
∆E-vs-E techniques. The breakup cross section is
measured as a function of the energy of the detected
protons, and a value for ann is determined by fitting the
284
mg/cm2) with 14.1-MeV neutrons, which were
produced using the 3H(d,n) reaction. The trigger for a
breakup event was the four-fold coincidence between
the three gas counters and the E detector. The angular
acceptance of their device about 0º was such that the
maximum angle of the detected protons was 7.9, and
the HWHM of the angular distribution of the detected
protons was ±2.2º. The energy distribution of the
incident neutrons was measured by detecting the recoil
protons from np elastic scattering. They accounted for
three backgrounds: sample uncorrelated, hydrogen
contamination in the CD2 foil, and nd breakup protons
from degraded neutrons in their beam. All together
these backgrounds contributed about 15% of the
counts in the region of interest in the proton-energy
spectrum. Their final cross-section data are shown in
upper end of the proton energy spectrum, which is
enhanced by the nn FSI. Until the mid 1970’s simple
theoretical treatments of the nn FSI in nd breakup, like
the Watson-Migdal and effective range theories and
the impulse approximation, were used to describe the
region of the spectrum where the relative energies
between the two neutrons were less than about 1 MeV.
Starting around 1973, calculations for nd elastic
scattering and the breakup reaction became available
in which the three-nucleon continuum problem was
solved rigorously using separable NN potential models
in the Faddeev equations [8]. The results from the
last three experiments that are plotted in the left-most
section of Figure 1 were obtained using Faddeev
calculations. In all cases only the S-wave part of the
NN potential was included in the calculations.
Tornow et al. [7] have re-analyzed the proton-energy
spectrum from the most recent measurements using
rigorous nd calculations that use modern single-boson
exchange NN potentials with partial-wave forces for
j≤3. The average value obtained in the re-analysis is
3.3 fm more positive than the original value.
FIGURE 2. A cross-sectional view of the counter telescope
used by Shirato et al. [9]. The components are: (0) CD2
foil; (1)-(3) Gas proportional counters; (4) and (5) 0.2 mm
thick dE/dx and 2 mm thick E Si(Li) surface-barrier
detectors; (6) 0.015-cm diameter stainless steel counter wire;
(7) Coaxial cable of the dE/dx detector; (8) and (11) BNC
connectors; (9) Cooling pipe; (10) Gas filling tube; and (12)
Target wheel.
FIGURE 3. The nd breakup proton energy spectrum of
Shirato et al. [9]. All backgrounds have been subtracted
from the data.
The KI nd breakup experiment reporting the most
accurate value for ann was performed by Shirato et al.
[9]. Their value is the fourth most recent result plotted
in Figure 1. The experimental setup is shown in
Figure 2. Because of the statistical importance of their
result in computing the average, their experimental
technique is chosen as representative of the KI
experiments. They bombarded a CD2 foil (5.09
Figure 3 along with the fit for three different values of
ann. The fits to the spectrum were optimized by
searching on the values of three parameters: F, C and
ann. The systematic uncertainty in their cross-section
data is ±3.3%. This uncertainty is dominated by
uncertainties in normalization factors for the absolute
285
were measured and a combined analyzed was
performed. They obtained ann = -16.3 ± 1.0 fm.
cross section and has no impact on the uncertainty in
their determination of ann because the theoretical cross
sections are normalized to the data in the fitting
procedure, i.e., the value for ann is obtained by fitting
the shape of the proton-energy spectrum.
The other type of KC nd breakup experiment used
to determine ann will be referred to as using the protonrecoil geometry. In this setup the momentum of the
recoil proton and the momentum of one of the emitted
neutrons are measured. The deuterium target is a thin
foil in an evacuated scattering chamber that is attached
to a charged-particle arm. The main differences in the
systematic effects of this technique and the neutroncoincidence experiments are that the proton-recoil
measurements require accounting for the efficiency of
only one neutron detector and no corrections are
needed for neutron multiple scattering and attenuation
in the deuterium target. However, there is an effect in
the proton-recoil technique that does not exist in the
neutron-coincidence method. The detection correlation
of the two neutrons when both are emitted within the
angular acceptance of the neutron detector must be
taken into account in the proton-recoil measurements
when comparing the data to theory. The first highaccuracy proton-recoil KC nd breakup measurement
used to determine ann was reported by von Witsch et
al. [12,13]. Their experimental setup is shown in
Figure 5. Their neutron beam was produced using the
2
H(d,n) reaction with a gas deuterium target. Their
measurements averaged over incident neutron energies
from 17 to 27 MeV. They obtained ann = –16.9 ±
0.6 fm. This result is consistent with earlier nd
breakup measurements.
Kinematically-Complete nd Breakup
There are two general types of kinematicallycomplete (KC) nd breakup experiments used to
determine ann. In both types the kinematics are overdetermined experimentally, thereby enabling some
background reduction through imposing kinematical
constraints.
FIGURE 4. The neutron-coincidence experimental setup of
the ann measurements by Zeitnitz et al. [10,11].
In the most common type of measurement the
deuterium target is an active deuterated scintillator
(DS). In these experiments the momentum vectors of
both emitted neutrons and the energy of the emitted
proton are measured. This type will be referred to as
being done with the neutron-coincidence setup.
Sizeable experimental effects must be taken into
account when comparing theory to data taken in
particle-coincidence geometry. The most important
effects are: neutron attenuation in the DS, neutron
multiple scattering in the DS, the energy dependence
of the efficiency of the two neutron detectors and
angle and energy averaging over the experimental
acceptance. Some early high-precision neutroncoincidence experiments were done by B. Zeitnitz et
al. [10,11]. The features of their experiment are
general to many neutron-coincidence measurements.
The experimental setup used in their measurements is
shown in Figure 4. A beam of 18.4-MeV neutrons
bombarded a deuterated benzene scintillator (D0).
The neutrons were produced by the 3H(d,n) reaction.
Relative cross sections for both the nn and np FSIs
FIGURE 5. The proton-recoil experimental setup of the ann
measurements by von Witsch et al. [12,13].
The results for ann from the two most recent KC nd
breakup experiments of González Trotter et al. [5],
286
which was of the neutron-coincidence type, and of
Huhn et al. [6], which was performed with a protonrecoil setup, disagree by more than three standard
deviations of the reported experimental uncertainties.
This situation is a bit puzzling given that the data
taken in both experiments were analyzed with the
same theory. This situation is the focus on another
contribution in the same session of this conference.
The next substantial step in this problem would be
a direct measurement of ann. The high thermal neutron
flux at the YAGUAR pulsed reactor in Russia creates
opportunities for such measurements. The DIANNA
collaboration is planning the first direct measurement
of ann that uses neutrons from a reactor. Some details
of their proposed experiment are given in a paper in
this session of the conference.
π-d Capture
ACKNOWLEDGMENTS
-
The values obtained for ann from the π d capture
measurements are summarized in Figure 1. In these
experiments a degraded pion beam is stopped in a
liquid deuterium target. As in the case of the nd
breakup, the π-d capture experiments can be classified
as KI or KC measurements. In KI measurements
[14,15] only the gamma-ray energy (Eγ) is measured.
The end of the measured Eγ spectrum is fit with theory
to determine a value of ann. The first and third points
from the left in the π-d capture section of Figure 1
were obtained from KI measurements. In the KC
experiments, 2H(π-,nγ), the Eγ, the time of flight (TOF)
of one of the emitted neutrons and the angle between
the momenta vectors of the gamma-ray and the
detected neutron are measured. Typically, the TOF
spectrum of the detected neutron is fit with theory to
determine ann. The main experimental effects that must
be taken into account when fitting the neutron TOF
spectrum are the energy dependence of the neutron
detection efficiency, the energy dependence of the
attenuation and scattering of neutrons in the liquid
deuterium target and the surrounding materials. The
results from the two most recent KC experiments
[4,16] (the last two data points in the p-d section of
Figure 1) are in agreement within the reported
uncertainties, which include 0.3 fm due to theoretical
uncertainties.
This work was supported in part by the U.S.
Department of Energy, Office of High Energy and
Nuclear Physics, under Grant No. DE-FG02-97ER41033.
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SUMMARY AND OUTLOOK
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The most popularly accepted value for ann comes
from the π-d capture measurements. The general belief
is that these results are more reliable than those
obtained from nd breakup or from other reactions in
which there are three or more hadrons in the exit
channel.
This supposition is supported by the
observation that the ann values obtained in the most
recent high-precision π-d experiments are in agreement
while nd breakup experiments give discrepant results.
Therefore, the recommended value for ann obtained
using indirect methods is -18.6 ± 0.3 (experimental) ±
0.3 (theory) fm.
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