# Measurement of the Solar Neutrino Flux with an Array of Neutron Detectors in
the Sudbury Neutrino Observatory

B. Jamieson (for the SNO collaboration)  University of British Columbia,
Vancouver, BC V6T 1Z1, Canada

###### Abstract

The Sudbury Neutrino Observatory has measured the 8B solar neutrino flux using
an array of 3He proportional counters. Results obtained using a Markov-Chain
Monte-Carlo (MCMC) parameter estimation, integrating over a standard extended
likelihood, yield effective neutrino fluxes of:
$\phi_{nc}=5.54^{+0.33}_{-0.31}$(stat) ${}^{+0.36}_{-0.34}$(syst)$\times
10^{6}$ cm-2s-1, $\phi_{cc}=1.67^{+0.05}_{-0.04}$(stat)
${}^{+0.07}_{-0.08}$(syst)$\times 10^{6}$ cm-2s-1, and
$\phi_{es}=1.77^{+0.24}_{-0.21}$(stat) ${}^{+0.09}_{-0.10}$(syst)$\times
10^{6}$ cm-2s-1. These measurements are in agreement with previous solar
neutrino flux measurements, and with neutrino oscillation model results.
Including these flux measurements in a global analysis of solar and reactor
neutrino results yields an improved precision on the solar neutrino mixing
angle of $\theta~{}=~{}34.4^{+1.3}_{-1.2}$ degrees, and $\Delta
m^{2}=7.59^{+0.19}_{-0.21}eV^{2}$.

##  I THE SNO DETECTOR

The Sudbury Neutrino Observatory (SNO) snonim  is a low background neutrino
detector $\sim$2 km (6000 mwe overburden) underground in the Vale Inco
Creighton nickel mine in Sudbury, Canada. The detector consists of 1000 tonnes
of D2O in a 12m diameter acrylic vessel surrounded by an inner shield of 1700
tonnes of H2O. At the edge of the inner light water, a support structure
holding about 9500 photomultiplier tubes (PMTs) provides 54% coverage. An
additional outer shield of 5300 tonnes of light water surrounds the PMTs.

The SNO detector detects the neutrino reactions:
$\nu_{x}+e^{-}~{}\rightarrow~{}\nu_{x}+e^{-}$ (ES),
$\nu_{e}+d~{}\rightarrow~{}p+p+e^{-}$ (CC), and
$\nu_{x}+d~{}\rightarrow~{}p+n+\nu_{x}$ (NC). The SNO detector provides the
unique detection of the neutrons in the NC reaction by three different
methods, one for each phase of SNO running. The first phase detected gamma
rays from the triton production in the detector
($n+d~{}\rightarrow~{}t+\gamma+6.25~{}MeV$)snod2o . In SNO’s second phase NaCl
was added to the heavy water, and increased the neutron capture through
neutron capture on the Cl
($n+^{35}C~{}\rightarrow~{}^{36}Cl+\gamma+8.6~{}MeV$), where the 8.6 $MeV$ is
the sum of a cascade of gamma rayssnosalt .

For the final phase of SNO reported on here the neutron is detected when it is
captured in an array of 36 3He proportional counters (NCDs) via the reaction
$n+^{3}He~{}\rightarrow~{}p+t+0.76~{}MeV$ncdnim ncdprl . The NCD phase
measurement separates the NC and CC signal detection which significantly
reduces the CC spectrum contamination by the 6.25 $MeV$ neutron captures on
deuterium. The NCD phase is more complex however since $\sim$10% of the
Cherenkov light is blocked by the array, the radioactivity of the counters
adds a non-negligible background, and the signal rate of $\sim$1000
neutrons/year is fairly low.

##  II NCD DETECTOR ENERGY SPECTRUM CALIBRATION

The neutron energy spectrum is measured with calibration data from a 24NaCl
brine that produces neutrons by the gamma capture on deuterium
($\gamma+d~{}\rightarrow~{}p+n$). The spectrum is characterized by a peak at
0.76 $MeV$ with features at 0.57 $MeV$ and 0.19 $MeV$ where either only the
proton or triton are seen in the proportional counter.

The 24NaCl brine is the calibration source most like the neutrons produced
from solar neutrinos since it can be uniformly distributed in the D2O, and it
also provides a measurement of the neutron detection efficiency ($0.211\pm
0.007$). The mixing of the brine can be seen by looking at the light output
from different parts of the detector, and only data from after the brine was
uniformly mixed was used. In addition, the detection efficiency from the MCNP
MCNP  Monte Carlo code yielded an efficiency of 0.210(3). Finally a time-
series based analysis using neutron bursts from a 252Cf source confirmed these
neutron efficiency measurements.

A simulation of the NCD detector was used to model the energy spectrum from
background alphas from U, Th, and Po in the nickel of the NCD walls. The model
included effects of the energy loss, multiple scattering, electron-ion pair
generation, electron drift and diffusion, electron multiple scattering, ion
mobility, electron avalanche, space charge, signal generation, and a detailed
propagation through the electronics. The Monte Carlo simulation was tuned for
the surface to bulk alpha ratio, energy scale, energy resolution, alpha depth,
and contributions from different parts of the NCD using the alphas above 2
$MeV$. The simulation was found to reproduce the pulse width and energy
spectrum very well, including the effects of alphas from the NCD anode wires.

Instrumental events in the NCD detector were easily separated from ionization
events using an amplitude versus energy cut. Six of the 36 NCD strings with
high instrumental rates were removed from the analysis. Two probability
distribution functions for the instrumental backgrounds were included in the
signal-extraction to fit for an unconstrained number of instrumental events.

##  III BLIND ANALYSIS

Three blindfolds were implemented on the NCD phase measurement. One month of
the data was open for analysts to tune cuts. A hidden fraction of neutrons
that follow muons were added to the data, and an unknown fraction of candidate
events were omitted. Detailed internal documentation was reviewed by topic
committees before the box was opened to reveal the true solar neutrino flux
measurement.

The box was opened on May 2, 2008, and the results are presented as found
after correcting two inconsistencies. The three separate signal-extraction
codes had to correct pilot errors on the inputs to the final fit, which
resulted in no change in the central values, and made the final uncertainties
reported agree. An incorrect algorithm in fitting the peak value of the ES
posterior distribution was replaced.

##  IV NEUTRON BACKGROUNDS

The neutron backgrounds were measured for the D2O radioactivity, atmospheric
neutrinos, 16N neutrons, NCD counter neutrons from the bulk of the counters,
from hot-spots on the counters, and from the NCD cables. In addition,
backgrounds from the acrylic-vessel, reactor neutrinos, and other sources were
included in the background estimates. The neutron backgrounds are summarized
in the following Table, and were included in the signal extraction broadening
the uncertainties in the measured solar neutrino fluxes.

Table 1: Table of neutron backgrounds in the PMT and NCD data. Source | PMT neutrons | NCD neutrons
---|---|---
D2O radioactivity | 7.6$\pm$1.2 | 28.7$\pm$4.7
Atmospheric $\nu$, 16N | 24.7$\pm$4.6 | 13.6$\pm$2.7
Other backgrounds | 0.7$\pm$0.1 | 2.3$\pm$0.3
NCD bulk PD, 17,18O($\alpha$,n) | 4.6${}^{+2.1}_{-1.6}$ | 27.6${}^{+12.9}_{-10.3}$
NCD hot-spots | 17.7$\pm$1.8 | 64.4$\pm$6.4
NCD cables | 1.1$\pm$1.0 | 8.0$\pm$5.2
External-source neutrons | 20.6$\pm$10.4 | 40.9$\pm$20.6
Total | 77${}^{+12}_{-10}$ | 185${}^{+25}_{-22}$

##  V SIGNAL EXTRACTION METHODS

Parameter estimation, and estimation of the uncertainties on all fit
parameters (both fluxes and nuisance parameters for systematics) is done with
a Metropolis algorithm Markov-Chain Monte Carlo (MCMC)mcmc .

In SNO’s previous signal extractions, the negative log-likelihood (NLL)
function was simply minimized with respect to all parameters to get the best-
fit value, and the curvature of -$\log(L)$ at the minimum was used to
determine the uncertainties. The floating systematics approach also uses a
minimization, although with additional nuisance parameters added to account
for systematic uncertainties.

Minimizing the NLL is very challenging for the 27 flux parameters
($\phi_{nc}$,$\phi_{cc1...13}$,$\phi_{es1...13}$) and 35 systematic parameters
in the fit. The systematic parameters include PMT reconstruction uncertainties
estimated from calibration data pmtcalib , both PMT and NCD efficiencies, NCD
Monte Carlo and NCD instrumental uncertainties. In addition because the
likelihood function can be a bit choppy near the minimum, traditional
minimizers such as MINUIT run into trouble and often will not converge in
reasonable periods of time.

The MCMC method gets around this problem by interpreting NLL as the negative
log of a joint probability distribution for all of the free parameters. We
then integrate over all nuisance parameters to determine the distributions for
the fluxes. The origins of this procedure go back to Bayesian probability
theory, and in fact our approach could be considered to be a Bayesian analysis
with uniform priors assumed for the fluxes.

The advantages of the MCMC method are twofold. First, it converges much faster
than a 50+ parameter MINUIT minimization. Rather than minimizing over
parameters, we in fact integrate over nuisance parameters. Second, since we
integrate over nuisance parameters with the MCMC instead of trying to find a
best-fit point, we are insensitive to and in fact average over choppiness in
the NLL that would interfere with finding a minimum. Both the speed of
convergence and the insensitivity to numerical noise in the NLL means that the
MCMC method is better suited to handling large numbers of nuisance parameters.

##  VI RESULTS

The final corrected solar neutrino fluxes above a 6 $MeV$ Kinetic Energy
threshold from the unblinded NCD data are:
$\phi_{nc}~{}=~{}5.54^{+0.48}_{-0.46}\times 10^{6}$ cm-2s-1,
$\phi_{cc}~{}=~{}1.67^{+0.08}_{-0.09}\times 10^{6}$ cm-2s-1, and
$\phi_{es}~{}=~{}1.77^{+0.26}_{-0.23}\times 10^{6}$ cm-2s-1. The correlation
between $\phi_{cc}$ and $\phi_{nc}$ was only -0.19 in the NCD phase fit
including all systematic uncertainties. The $\phi_{es}$ is a 2.2 sigma lower
than the Super Kamiokande measurement, but the full set of fluexes has a
probability of 32.8% of being consistent with the six other flux measurements
from all of the SNO phases.

The NCD energy fit, unconstrained PMT energy fit, PMT angle to the Sun, and
radial position fits are shown in FIG. 1. It can be seen in the PMT energy fit
that the number of neutrons in the PMT is considerably less than either the
SNO D2O or salt phases, and provides the better CC separation than in those
phases.

Figure 1: These figures show the NCD energy spectrum fit (top left), the PMT
energy unconstrained fit (top right), the PMT angle to the sun fit (bottom
left), and the PMT radial position fit (bottom right).

Including the NCD phase flux measurements in a global oscillation analysis
results in an improved precision on the solar neutrino mixing angle of
$\theta~{}=~{}34.4^{+1.3}_{-1.2}$ degrees, and
$\Delta~{}m^{2}=7.59^{+0.19}_{-0.21}eV^{2}$.

###### Acknowledgements.

SNO gratefully acknowledges NSERC, Industry Canada, NRC, Northern Ontario
Heritage Fund, Vale INCO, Atomic Energy of Canada, Ontario Power Generation,
High Performance Computing Virtual Laboratory, Canada Foundation for
Innovation, Canada Research Chairs, Westgrid, US Department of Energy, NERSC
PDSF, UK STFC, and Portugal FCT.

## References

  * (1) SNO Collaboration, Nucl. Instrum. and Methods Phys. Res., Sect. A 449, (2000) 172.
  * (2) SNO Collaboration, Phys. Rev. Lett. 87, (2001) 071301. SNO Collaboration, Phys. Rev. Lett. 89, (2002) 011301. SNO Collaboration, Phys. Rev. Lett. 89, (2002) 011302.
  * (3) SNO Collaboration, Phys. Rev. Lett. 92, (2004) 181301. Phys. Rev. C72, (2005) 055502. SNO Collaboration, Phys. Rev. C75, (2007) 045502.
  * (4) J.F. Amsbaugh _et al._ , Nucl. Instrum. and Methods Phys. Res., Sect. A 579, (2007) 1054.
  * (5) SNO Collaboration, Phys. Rev. Lett. 101, (2008) 111301.
  * (6) Refer to the MCNP website for details:

         http://mcnp-green.lanl.gov/publication/mcnp_publications.html

  * (7) Metropolis  _et al._ , Journ. Chem. Phys. 21,(1953) 1087. W.K. Hastings, Biometrika 57, (1970) 97.
  * (8) M.R. Dragowsky _et al._ , Nucl. Instrum. and Methods Phys. Res., Sect. A 481, (2002) 284. I. Blevis  _et al._ , Nucl. Instrum. and Methods Phys. Res., Sect. A 517, (2004) 139\.