Also at ]University of Maryland, College Park, MD 20742 Also at ]Institute for
Quantum Computing, Waterloo, ON N2L 3G1, Canada

# Neutron interferometric measurement of the scattering length
difference between the triplet and singlet states of n-3He

M. G. Huber<EMAIL_ADDRESS>M. Arif  W. C. Chen  [  T. R. Gentile  D.
S. Hussey  National Institute of Standards and Technology, Gaithersburg, MD
20899, USA  T. C. Black  University of North Carolina-Wilmington, Wilmington,
NC 28403, USA  D. A. Pushin  [  University of Waterloo, Waterloo, ON N2L 3G1,
Canada  C. B. Shahi  F. E. Wietfeldt  Tulane University, New Orleans, LA
70188, USA  L. Yang  University of Illinois at Urbana-Champaign, Urbana, IL
61801, USA

###### Abstract

We report a determination of the n-3He scattering length difference $\Delta
b^{\prime}=b_{1}^{\prime}-b_{0}^{\prime}=$ ($-5.411$ $\pm$ $0.031$
(statistical) $\pm$ $0.039$ (systematic) ) fm between the triplet and singlet
states using a neutron interferometer. This revises our previous result
$\Delta b^{\prime}=$ ($-5.610$ $\pm$ $0.027$ (statistical) $\pm$ $0.032$
(systematic) ) fm obtained using the same technique in 2008 [M. G. Huber,
Phys. Rev. Lett. 20, 102 (2009) & M. G. Huber, Phys. Rev. Lett. 17, 103
(2009)]. This revision is due to a re-analysis of the 2008 experiment that
includes a more robust treatment of the phase shift caused by magnetic field
gradients near the 3He cell. Furthermore, we more than doubled our original
data set from 2008 by acquiring six months of additional data in 2013. Both
the new data set and a re-analysis of the older data are in good agreement.
Scattering lengths of low Z isotopes are valued for use in few-body nuclear
effective field theories, provide important tests of modern nuclear potential
models and in the case of 3He aid in the interpretation of neutron scattering
from quantum liquids. The difference $\Delta b^{\prime}$ was determined by
measuring the relative phase shift between two incident neutron polarizations
caused by the spin-dependent interaction with a polarized 3He target. The
target 3He gas was sealed inside a small, flat windowed glass cell that was
placed in one beam path of the interferometer. The relaxation of 3He
polarization was monitored continuously with neutron transmission
measurements. The neutron polarization and spin flipper efficiency were
determined separately using 3He analyzers and two different polarimetry
analysis methods. A summary of the measured scattering lengths for n-3He with
a comparison to nucleon interaction models is given.

neutron interferometry, polarized 3He, few-body systems

###### pacs:

03.75.Dg, 28.20.Cz, 21.45.-v

##  I Introduction

Understanding the properties of nuclei from the point of view of a collection
of individual interacting nucleons is an important goal of nuclear physics
Lazauskas  _et al._ (2005); Viviani  _et al._ (2005). Unfortunately, direct
calculations of low-energy nuclear phenomena using Quantum Chromodynamics
(QCD) is currently impractical. Instead properties of few-body nuclear systems
are calculated using a variety of phenomenological models. The prevailing two
nucleon (NN) models, AV18 Wiringa  _et al._ (1995), CD-Bonn Machleidt (2001),
and Nijmegen Stoks  _et al._ (1994), incorporate a fit to _np_ and _pp_
scattering data Stoks  _et al._ (1993) for energies up to 350 MeV. Problems
arise with NN models when applying them to systems containing more than two
nucleons. This is most evident by their failure to accurately predict the
binding energy of the triton by 800 keV Pieper  _et al._ (2001). For this
reason three nucleon interactions (3N), which arise in lowest order in the
meson exchange model from the exchange of two pions between three separate
nucleons, are included with NN models to describe larger few-body systems.
Three nucleon interactions, including Urbana Pudliner  _et al._ (1997), Tucson
Melbourne Coon  _et al._ (1979), Brazil Robilotta and Coelho (1986) or
Illinois Pieper  _et al._ (2001) potentials, can correct for this and now
accurately predict many nuclear levels for atomic number up to $13$ Wiringa
_et al._ (2014); Wiringa and Pieper (2002); Pieper  _et al._ (2001). However,
the increase in the prediction accuracy of binding energies has not meant that
NN+3N models have accurately predicted low-energy scattering data in systems
involving more than two nucleons Viviani  _et al._ (2001); Black  _et al._
(2003).

Nuclear effective field theoriesEpelbaum  _et al._ (2002); Entem and Machleidt
(2003) have also been a successful approach to understand few-body nuclear
systems. Effective field theories separate nucleon interactions into two
distinct energy regions that are separated by the pion mass. Below the pion
mass threshold interaction diagrams are explicitly calculated. For higher-
energy processes, the interactions are described by using low-energy
observables such as scattering lengths to parameterize mean-field behavior.
For instance using the triton binding energy Kirscher _et al._ Kirscher  _et
al._ (2009) predicted a value of the scattering length for the singlet state
in n-3He to 8 % relative uncertainty. Although presently not as precise as
other approaches, effective field theories are attractive because they provide
clear theoretical uncertainties from estimates of the relative contribution of
higher order terms Bedaque  _et al._ (1999).

A final motivation for measuring the n-3He incoherent scattering length to
high precision is that $\Delta b$ also arises in the study of quantum liquids.
Inelastic neutron scattering in liquid 3He for a momentum $\bm{Q}$ and energy
transfer $E$ is described by a dynamic structure factor $S(\bm{Q},E)$ Glyde
(1995). The dynamic structure factor can be separated into coherent $S_{c}$
and incoherent $S_{i}$ terms as Sköld  _et al._ (1976); Glyde  _et al._ (2000)

$\displaystyle S(\bm{Q},E)=S_{c}(\bm{Q},E)+\frac{\left|\sqrt{3}\Delta
b^{\prime}\right|^{2}}{\left|b^{\prime}_{0}+3b^{\prime}_{1}\right|^{2}}S_{i}(\bm{Q},E).$
(1)

One can see that the incoherent density term is weighted by a ratio composed
of both the triplet $b^{\prime}_{1}$ and singlet $b^{\prime}_{0}$ scattering
lengths where $\Delta b^{\prime}=b_{1}^{\prime}-b_{0}^{\prime}$. (The real
part of the scattering length is denoted by ′). For the n-3He system the sum
$b^{\prime}_{0}+3b^{\prime}_{1}$ has been previously measured to $<$ 1 %
relative uncertainty. An accurate determination of $S_{c}(\bm{Q},E)$ and
$S_{i}(\bm{Q},E)$ from $S(\bm{Q},E)$ relies on determining $\Delta b$ to
similar precision.

Neutron scattering lengths can be determined very precisely using neutron
interferometry. In the case of silicon, neutron interferometry has been
utilized to measure the scattering length to 0.005 % relative uncertainty
Ioffe  _et al._ (1998). In the last few years, there have been several
precision measurements using neutron interferometry with light nuclei targets.
These include measurements of the spin-independent scattering length
$b^{\prime}$ of n-1H, n-2H Schoen  _et al._ (2003), and n-3He Huffman  _et
al._ (2004); Ketter  _et al._ (2006) to less than one percent relative
uncertainty.

Here we report a determination of the scattering length difference $\Delta
b^{\prime}=b_{1}^{\prime}-b_{0}^{\prime}$ of n-3He using a neutron
interferometer that is based on a reanalysis of Huber _et al._ Huber  _et al._
(2009a, b) and additional data. A review of the relevant neutron optical
theory is covered in Sec. II. The experimental setup and measured phase shift
caused by the 3He target sample is discussed in Sec. III. In Sec. IV we
describe neutron polarimetry measurements using 3He analyzers with two
different analysis methods. Systematically limiting this technique is the
uncertainty in the triplet absorption cross section of 3He. This limit in
determining $\Delta b^{\prime}$ and other non-statistical uncertainties are
discussed in Sec. V. Finally, we compare world averages of the current
experimental results of the coherent and incoherent scattering lengths to
various theoretical predictions in Sec. VII.

##  II Neutron Scattering

Neutron scattering from a single target atom can be described by the wave
function Goldberger and Seitz (1947)

$\displaystyle\Psi=e^{i\bm{k}\cdot\bm{r}}+\frac{e^{ikr}}{r}f(\theta).$ (2)

The first term in Eq. (2) describes the incident neutron where $\bm{k}$ is the
incident wave vector and $\bm{r}$ is the position of the neutron. The latter
term corresponds to the scattered wave in the first Born approximation with a
scattering amplitude $f(\theta)$ that can be expanded in terms of $k$ as Sears
(1989)

$\displaystyle f(\theta)=-a+ika^{2}+\mathcal{O}(k^{2})+\ldots\approx-a,$ (3)

where $a$ is called the free scattering length. The approximation in Eq. (3)
is valid because the magnitude of $a$ is of $\mathcal{O}($1 fm$)$ and for low
energy neutrons $k$ is of $\mathcal{O}($$10^{-4}$ fm-1$)$. In general, $a$ is
complex such that $a=a^{\prime}+ia^{\prime\prime}$ where
$a^{\prime},a^{\prime\prime}$ are both real numbers. As discussed later,
neutron absorption by the target atom is related to the imaginary term
$a^{\prime\prime}$. Most importantly $a$ represents a measurable quantity of
the interaction that is unique for each nuclear isotope. In neutron
interferometry, even when considering a gas target, the forward scattered
momentum transfer is zero. For this reason, it is more relevant to define
everything using the bound scattering length

$\displaystyle
b=\left(\frac{M_{\mathrm{N}}+m_{\mathrm{n}}}{M_{\mathrm{N}}}\right)a.$ (4)

Here $M_{\mathrm{N}}$ and $m_{\mathrm{n}}$ are the mass of the target atom and
neutron, respectively.

Figure 1: A simplified schematic of an interferometer. A beam splitter
separates a neutron wave function into two paths that are recombined in a
final analyzing beam splitter. before being detected. A sample placed in one
path causes a relative phase shift $\phi_{\mathrm{sam}}$ because of a change
in wavelength due to the index of refraction of the material.

To describe neutrons scattering from a homogeneous material one uses the time-
independent Schrödinger equation

$\displaystyle\left(\frac{\hbar^{2}K^{2}}{2m_{\mathrm{n}}}+V_{\mathrm{opt}}\right)\psi=\frac{\hbar^{2}k^{2}}{2m_{\mathrm{n}}}\psi\mbox{,}$
(5)

where $K$ is the magnitude of the neutron wave vector in the material. The
optical potential

$\displaystyle V_{\mathrm{opt}}=\frac{2\pi\hbar^{2}}{m_{\mathrm{n}}}Nb,$ (6)

is the effective potential of the material with an atom density $N$. The index
of refraction

$\displaystyle n=\frac{K}{k}=\sqrt{1-\frac{\lambda^{2}Nb}{\pi}},$ (7)

of the material can be derived using Eqs. (5) and (6). Here $\lambda=2\pi/k$
is the neutron wavelength in vacuum. The index of refraction for neutron
optics is conceptually the same as for light optics with two subtle
differences. For one the index of refraction for neutrons is only a small
deviation from unity; ie. $1-n=\mathcal{O}(10^{-6})$ and generally $n$ is less
than one.

A neutron interferometer splits the neutron’s wave function along two
spatially separate paths labeled I and II. When a sample of thickness $D$ is
introduced into beam path II there is a phase difference relative to the path
I of

$\displaystyle\phi_{\mathrm{sam}}=k(1-n)D=-\lambda NDb^{\prime}\mbox{.}$ (8)

This phase shift is due to the change of the neutron wavelength as it passes
through a material. A conceptual illustration of this is provided in Figure 1.
A neutron interferometer is extremely sensitive to phase differences between
paths and therefore can be used to measure $\phi_{\mathrm{sam}}$ to high
precision. Along with information of the quantities $\lambda$, $N$ and $D$,
often performed using individual auxiliary measurements, one can use Eq. (8)
to determine the scattering length of the material.

In the case of neutrons with spin $\bm{\sigma}_{\mathrm{n}}$ incident on a
target with nuclear spin $\bm{I}$ the scattering length can be written as

$\displaystyle b=b_{c}+\frac{\displaystyle
2b_{i}}{\displaystyle\sqrt{I(I+1)}}\bm{\sigma}_{\mathrm{n}}\cdot\bm{I}\mbox{,}$
(9)

where $b_{c}$ and $b_{i}$ are called the coherent and incoherent scattering
lengths, respectively. Despite its name the incoherent scattering length is
due to a coherent interaction and corresponds to the spin-dependent part of
the scattering length. The scattering lengths for a given spin channel,
$\bm{J}=\bm{I}\pm\bm{\sigma}_{\mathrm{n}}$, are defined as

$\displaystyle b_{+}=$ $\displaystyle
b_{c}+\displaystyle\sqrt{\frac{\displaystyle g_{-}}{\displaystyle
g_{+}}}b_{i}$ (10a) $\displaystyle b_{-}=$ $\displaystyle
b_{c}-\displaystyle\sqrt{\frac{\displaystyle g_{+}}{\displaystyle
g_{-}}}b_{i}.$ (10b)

Here $g_{+}=(I+1)/(2I+1)$ and $g_{-}=1-g_{+}=I/(2I+1)$ are statistical weight
factors. Eqs. (10a) and (10b) are for general systems; for n-3He there is a
triplet (J $=1$) and singlet (J $=0$) channel allowing us to write the triplet
scattering length as $b_{+}\equiv b_{1}$ and the singlet scattering length as
$b_{-}\equiv b_{0}$ (likewise, $g_{+}\rightarrow g_{1}$ and $g_{-}\rightarrow
g_{0}$). Inverting Eqs. (10a) and (10b), the 3He coherent and incoherent
scattering lengths become

$\displaystyle b_{c}$ $\displaystyle=$ $\displaystyle g_{1}b_{1}+g_{0}b_{0}$
(11a) $\displaystyle b_{i}$ $\displaystyle=$
$\displaystyle\sqrt{g_{1}g_{0}}(b_{1}-b_{0}).$ (11b)

The total cross section for the n-3He interaction is
$\sigma_{t}=\sigma_{s}+\sigma_{a}$. Here $\sigma_{s}$ is the scattering cross
section given by

$\displaystyle\sigma_{s}=\sigma_{s,c}+\sigma_{s,i}=4\pi\left|b_{c}\right|^{2}+4\pi\left|b_{i}\right|^{2}.$
(12)

The absorption cross section $\sigma_{a}$ is related to the imaginary part of
the scattering length $b^{\prime\prime}$ by the optical theorem Feenberg
(1932)

$\displaystyle\sigma_{a}=\frac{4\pi}{k}b^{\prime\prime}.$ (13)

The measured unpolarized neutron absorption cross section for 3He(_n_ ,_p_)3H
is $\sigma_{\mathrm{un}}=$(5333 $\pm$ 7) b Mughabghab (2006) at the reference
thermal neutron wavelength $\lambda_{\mathrm{th}}=$ 1.798 Å. The uncertainty
quoted for $\sigma_{\mathrm{un}}$, as well as all other uncertainties quoted
below, is a standard uncertainty with a confidence level of 68 %. For 3He(_n_
,$\gamma$)4He the absorption cross section is (54 $\pm$ 6) µb Wolfs  _et al._
(1989) at these energies and thus for our purposes can be ignored.

The absorption cross section for polarized neutrons incident on polarized 3He
nuclei is given by

$\displaystyle\sigma_{a}=\sigma_{\mathrm{un}}\mp P_{3}\sigma_{p},$ (14)

where $P_{3}$ is the 3He polarization. The $\mp$ sign in Eq. (14) represents
the cases where the neutron and 3He spins are aligned parallel $(-)$ or anti-
parallel $(+)$. Similar to Eqs. (11a) and (11b), $\sigma_{\mathrm{un}}$ and
$\sigma_{p}$ can be defined in terms of singlet and triplet contributions as

$\displaystyle\sigma_{\mathrm{un}}=g_{1}\sigma_{1}+g_{0}\sigma_{0}$ (15a)
$\displaystyle\sigma_{p}=g_{0}\left(\sigma_{0}-\sigma_{1}\right).$ (15b)

Since $\sigma_{a}$ is dominated by the singlet channel it is often assumed
that $\sigma_{1}=0$ so that $\sigma_{\mathrm{un}}=\sigma_{p}$. Although
$\sigma_{1}$ is small there is no theoretical justification for assuming
$\sigma_{1}$ to be precisely zero. Neutron transmission measurements are
consistent with $\sigma_{1}=0$ only at the level of a few percent. The
uncertainty in the triplet absorption cross section leads to the largest
systematic uncertainty in both neutron interferometric and pseudomagnetic spin
precision measurements of $\Delta b$. This is discussed in more detail in
Section V.1.

##  III Experimental Procedure

Figure 2: The n-3He experiment (Not to Scale). (a) A monochromatic neutron
beam entering from the left is polarized by a supermirror. The polarization
direction can be changed using a precession coil spin flipper. A 2 mm $\times$
6 mm slit provided collimation just before the interferometer. (b) Neutrons
Bragg diffract in the first blade of the interferometer coherently splitting
the neutron into two separate paths. The two paths are diffracted in separate
mirror blades so that they mix and interfere at the analyzer blade of the
interferometer. One neutron path contains the 3He target cell while the other
path contains 8 mm of boron-free glass to compensate for the phase shift
caused by the target cell windows. A quartz phase flag is rotated to vary the
intensity in the two 3He filled proportional counters labeled the O- and
H-beam detectors. The 3He polarization was monitored by a third 3He detector
labeled C4.

We used a neutron interferometer to measure the phase difference between two
polarized neutron states that are transmitted through a polarized 3He target
cell. Neutrons are polarized in the vertical direction and can be flipped by
$\pi$ radians using a precession coil spin flipper. The target 3He
polarization direction stays constant throughout a measurement, but its
magnitude decreases exponentially in time. For neutrons with spin state
$\uparrow$ and $\downarrow$ incident on a polarized 3He sample we can insert
the effective scattering length for 3He (Eq. (9))

$\displaystyle b^{\prime}=b_{c}^{\prime}\pm\frac{\displaystyle
b_{i}^{\prime}}{\displaystyle\sqrt{3}},$ (16)

into Eq. (8) and find the phase difference

$\displaystyle\Delta\phi_{0}$
$\displaystyle=\phi_{\mathrm{sam}}^{\uparrow}-\phi_{\mathrm{sam}}^{\downarrow}$
$\displaystyle=-\lambda
N_{3}D_{3}\bigg{[}\bigg{(}b_{c}^{\prime}+N_{+}\frac{\displaystyle
b_{i}^{\prime}}{\displaystyle\sqrt{3}}-N_{-}\frac{\displaystyle
b_{i}^{\prime}}{\displaystyle\sqrt{3}}\bigg{)}$
$\displaystyle\quad-\bigg{(}b_{c}^{\prime}-N_{+}\frac{\displaystyle
b_{i}^{\prime}}{\displaystyle\sqrt{3}}+N_{-}\frac{\displaystyle
b_{i}^{\prime}}{\displaystyle\sqrt{3}}\bigg{)}\bigg{]}$ (17)

where $N_{3}$ is the 3He number density and $D_{3}$ is the active length of
the target cell. Here $N_{\pm}=(1\pm P_{3})/2$ is the fraction of 3He nuclei
in each polarization state. Using Eqs. (11b) and (17) one finds that the phase
difference between opposite neutron spin states is related to the triplet and
singlet scattering lengths by

$\displaystyle
b_{1}^{\prime}-b_{0}^{\prime}=\frac{\displaystyle-2\Delta\phi_{0}}{N_{3}\lambda
D_{3}P_{3}},$ (18)

The factors in the denominator of Eq. (18) are determined simultaneously with
$\Delta\phi_{0}$ by measuring the spin-dependent transmission of neutrons
through the 3He cell. This is an advantage over typical interferometric
measurements of $b^{\prime}$ in that none of the factors in the denominator
need to be determined individually; the spin-dependent transmission asymmetry
determines their product directly.

###  III.1 Neutron Interferometer and Facility

This experiment was done at the National Institute of Standards and
Technology’s (NIST) Center for Neutron Research (NCNR) in Gaithersburg, MD. A
20 MW reactor produces a steady source of neutrons that are moderated by a
liquid hydrogen cold source Williams and Rowe (2002).

Figure 3: The skew symmetric interferometer made by Cliff Holmes and others at
the University of Missouri-Columbia machine shop (on loan from Dr. Samuel A.
Werner).

These moderated neutrons are transported from the cold source to several
neutron instruments by 58Ni coated glass guides. At the Neutron Interferometer
and Optics Facility (NIOF) a double monochromator assembly reflects 2.35 Å
neutrons into an environmentally isolated enclosure Arif  _et al._ (1994). The
first monochromator is a single pyrolytic graphite PG(002) crystal that
reflects neutrons out of the neutron guide and toward a second monochromator 3
m away. This second monochromator vertically focuses the beam using nine
individually adjustable 1 cm x 5 cm PG(002) crystals Pushin  _et al._ (2008).
Further details of the facility can be found in Huber  _et al._ (tion).

A schematic of the experiment inside the enclosure is shown in Figure 2. After
the double monochromator assembly a pyrolytic graphite filter was used to
remove $\lambda/2=$ 1.175 Å neutrons from the beam. Neutrons passing through
the filter were polarized by a transmission-mode supermirror polarizer Ioffe
_et al._ (1996). The polarizer consisted of two separate 0.64 m and 0.73 m
long supermirrors that were slightly offset so that no incoming neutrons had
clear line of sight to the interferometer. Neutrons in the incorrect spin
state were reflected from the supermirror and absorbed on cadmium shielding.

Immediately downstream of the supermirror polarizer was a precession coil spin
flipper made from two orthogonal aluminum wire coils Badurek  _et al._ (1976).
One coil provided a magnetic field opposite of the guide field that created,
in the absence of the other coil, a zero field region in the center of the
coils. A second coil created a magnetic field

$\displaystyle
B_{\mathrm{f}}=\frac{\pi^{2}\hbar^{2}}{\mu_{\mathrm{n}}m_{\mathrm{n}}\lambda}L^{-1}$
(19)

perpendicular to the neutron polarization direction. The field
$B_{\mathrm{f}}$ was tuned such that the neutron undergoes half a Larmor
precession cycle. Here $L$ is the active length inside the coils,
$m_{\mathrm{n}}$ is the neutron mass,
$\bm{\mu_{\mathrm{n}}}=\gamma\hbar\bm{\sigma}_{\mathrm{n}}$ is the neutron
magnetic dipole moment, and $\gamma$ is the gyromagnetic ratio. When energized
the precession coil spin flipper allowed the neutron spin state to be rotated
$\pi$ radians with nearly 100 % efficiency.

Helmholtz coils 56 cm in diameter were centered on the target cell and
provided a field of 1.5 mT. To preserve the neutron polarization between the
supermirror polarizer and the Helmholtz coils a series of permanent magnets
provided a magnetic guide field with a minimum of 1 mT. The heat output from
the Helmholtz coils, which was not actively cooled, increased the temperature
variation for this experiment. The temperature around the interferometer was
controlled with heating tape and calibrated platinum resistance sensors in
closed loop, proportional-integral-derivative (PID) operation and typically
achieved a temperature stability of ± 5 mK Pushin  _et al._ (2008). For this
experiment the interferometer enclosure was stable to only ± 20 mK because of
the increased heat caused by the Helmholtz coils.

A neutron interferometer consists of a perfect silicon crystal machined so
that there are several parallel crystal blades on a common monolithic base.
The interferometer used here is shown in Fig. 3. Neutrons entering the
interferometer Bragg diffract in the first (splitter) blade of the
interferometer. This coherently separates the neutron into two spatially
separate paths labeled I and II (see Fig. 2b). Both neutron paths are Bragg
diffracted in a second (mirror) blade and interfere in the final (analyzer)
blade of the interferometer. Conceptually, this is analogous to a Mach-Zehnder
interferometer in light optics. The two beams exiting the interferometer are
historically labeled the O- and H-beams. Neutrons are detected with near 100 %
efficiency using 25.4 mm diameter cylindrical 3He filled proportional
counters.

The target was a sealed glass cell containing 3He gas (see Fig. 4) placed in
path I of the interferometer. A phase flag, which consisted of 2 mm thick
quartz, was placed in both paths of the interferometer. Rotating the phase
flag by an angle $\epsilon$ varied the relative phase shift between the two
neutron paths and thus modulated the intensity in the O- and H-beam detectors.
For the O- and H- beam detectors, the intensity can be written for a neutron
with spin state $\uparrow(\downarrow)$ as

$\displaystyle I_{0}(\epsilon)$ $\displaystyle=$ $\displaystyle
c_{0}^{\uparrow(\downarrow)}+c_{1}^{\uparrow(\downarrow)}\cos\left[\phi_{\mathrm{flag}}(\epsilon)+\phi^{\uparrow(\downarrow)}\right]$
(20) $\displaystyle I_{H}(\epsilon)$ $\displaystyle=$ $\displaystyle
c_{3}^{\uparrow(\downarrow)}-c_{1}^{\uparrow(\downarrow)}\cos\left[\phi_{\mathrm{flag}}(\epsilon)+\phi^{\uparrow(\downarrow)}\right]$
(21) $\displaystyle\mbox{where }I_{0}$ $\displaystyle+$ $\displaystyle
I_{H}=\mbox{constant.}$ (22)

The coefficients $c_{i}^{\uparrow(\downarrow)}$ for $i=0,1,\ldots,n$ are
treated as fit parameters. Here
$[\phi_{\mathrm{flag}}(\epsilon)+\phi^{\uparrow(\downarrow)}]$ is the relative
phase difference between the two paths. The phase
$\phi^{\uparrow(\downarrow)}$ includes both $\phi_{\mathrm{sam}}$ and any
initial relative phase difference between paths I and II. The term
$\phi_{\mathrm{flag}}(\epsilon)=c_{2}f(\epsilon)$ is the phase shift due to
the phase flag where

$\displaystyle
f(\epsilon)=\frac{\sin\left(\theta_{B}\right)\sin\left(\epsilon\right)}{\cos^{2}\left(\theta_{B}\right)-\sin^{2}\left(\epsilon\right)}$
(23)

is the difference in optical path length for path I and II. Here $\theta_{B}=$
37.73 ° is the Bragg angle for the interferometer. Since 3He has spin-
dependent absorption, $c_{i}^{\uparrow}\neq c_{i}^{\downarrow}$ for $i=0,1,3$.

The contrast or fringe visibility $C$ of the interferometer is the ratio of
the amplitude $c_{1}$ and mean $c_{0}$ in Eq. (20). In practice the contrast
for a typical neutron interferometer is less than unity due to a host of
reasons including small crystal imperfections and temperature gradients. Under
the best experimental conditions neutron interferometers have demonstrated at
most around 90 % contrast. In this experiment there are two losses of contrast
that, although not unique, are of particular interest.

The first case is due the interaction of the neutron as it passes through the
glass windows of the target cell. As a neutron passes through the glass
windows it experiences a large phase shift $\phi_{\mathrm{win}}$. Although
this phase shift $\phi_{\mathrm{win}}$ is spin-independent and is canceled
when subtracting the phase measured in both neutron spin states,
$\phi_{\mathrm{win}}$ does affect the measured contrast and hence overall
precision of the experiment. This is because the incident neutron beam
contains a small wavelength spread of $\sigma_{\lambda}/\lambda=$ 1 %.
Neutrons of slightly different wavelengths will experience slightly different
$\phi_{\mathrm{win}}$ which dephases the detectable neutron beam after the
interferometer. This doesn’t affect the measured phase determined by Eq. (20)
but dephasing does decrease the contrast. Assuming a Gaussian spectrum of
$\lambda$, the measured contrast becomes
$C=C_{0}\exp{[-(N_{\mathrm{win}}D_{\mathrm{win}}b_{\mathrm{win}}^{\prime}\sigma_{\lambda})^{2}/2]}$
where $C_{0}$ is the initial contrast of the interferometer. Here
$N_{\mathrm{win}}$, $D_{\mathrm{win}}$, and $b^{\prime}_{\mathrm{win}}$ are
the density, thickness, and effective scattering length of the window,
respectively. This effect is negligible for the 3He itself because the density
of the gas is much lower. A more complete description of coherence and
subsequent contrast loss can be found in a number of sources including
Clothier  _et al._ (1991); Kaiser  _et al._ (1983); Klein  _et al._ (1983);
Petrascheck (1988); Rauch  _et al._ (1996); Pushin  _et al._ (2008). To
minimize the loss of contrast from the cell windows, 8 mm of compensating
glass made from 2 target cell windows was placed in path I of the
interferometer.

Another mechanism of contrast loss in this experiment is due to the fact that
the 3He sample is a neutron absorber; therefore both $c_{0}$ and $c_{1}$ are
decreased from what they would otherwise be in an empty interferometer. For
absorption we have Sears (1989),

$\displaystyle c_{0^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{c_{0}}{2}(1+e^{-\sigma_{a}N_{3}D_{3}})$ (24)
$\displaystyle c_{1^{\prime}}$ $\displaystyle=$ $\displaystyle
c_{1}e^{-\sigma_{a}N_{3}D_{3}/2}$ (25) $\displaystyle
C=\frac{c_{1^{\prime}}}{c_{0^{\prime}}}$ $\displaystyle=$ $\displaystyle
C_{0}\operatorname{sech}(\sigma_{a}N_{3}D_{3}/2)$ (26)

Absolute contrast during this experiment was a function of the environment,
effectiveness of compensation glass, neutron spin-state and the 3He
polarization and varied between 30 % and 75 %.

###  III.2 Glass Target Cells

The NIST optical shop fabricated four geometrically identical flat-windowed
3He cells. Two of these cells, named Cashew and Pistachio (Pistachio is shown
in Fig. 4) were used in this experiment. Each cell was made from boron-free
aluminosilicate glass Gentile and Fuller (2007) and consisted of two flat 25
mm diameter, 4 mm thick windows fused onto a 34 mm long cylindrical base.
Before the cells were sealed they were filled with between 1.7 bar and 1.9 bar
of 3He gas. Nitrogen and rubidium were also added in order to polarize the 3He
using spin exchange optical pumping (SEOP) Walker and Happer (1997).
Properties of the cells can be found in Table 1.

Figure 4: Two 3He cells. The larger of the two cells, called Skylight, was
used in the polarimetry measurements. The smaller flat-window cell called
Pistachio was used as a target cell. Rubidium deposits can be seen as dark
spots along the walls of the cell.

The environmental constraints at the NIOF required that the cells be polarized
at a separate facility. In this facility SEOP was employed to polarize the 3He
gas over a period of 2 days to an initial 3He polarization between 65 % and 75
%. It was also possible at the SEOP facility to monitor and flip the 3He
polarization by nuclear magnetic resonance (NMR) techniques Bloch (1946). The
cells were transported to the NIOF using a portable battery powered solenoid.
Losses in 3He polarization from transporting the cell between the two
facilities were measured to be $<2$ %. Helmholtz coils placed around the
interferometer provided a uniform magnetic field which minimized the loss of
3He polarization due to magnetic field gradients.

Table 1: The 3He cells properties. Quoted spin relaxation times are for the interferometer facility which had magnetic field gradients. $N_{3}\sigma_{p}D_{3}$ is the opacity of the cell. The pressure at 26 $\degree$C was determined assuming that the transmission through the cell windows was 88 % Chupp  _et al._ (2007). The cell Cashew’s relaxation time was 135 h in 2008 and 150 h in the 2013 data set. Pistachio was used only in 2008.  | Relaxation | $N_{3}\sigma_{p}D_{3}$ | Pressure |
---|---|---|---|---
Cell Name | time [h] | at 2.35 Å | [bar] | Function
Cashew (’08) | 135 | 1.1 | 1.9 | Target
Cashew (’13) | 150 | 1.1 | 1.9 | Target
Pistachio | 35 | 1.0 | 1.7 | Target
Skylight | 110 | 3.1 | 3.1 | Polarimetry
Haystack | 80 | 3.0 | 2.94 | Polarimetry
Whiteface | 35 | 3.6 | 3.50 | Polarimetry

Cell relaxation times at the interferometer varied per cell with a maximum of
150 h.

###  III.3 Phase Data

The phase shift caused by the spin-dependent interaction with the target 3He
was measured by rotating the phase flag from an angle of $\epsilon=$ 0 mrad to
$\epsilon_{\mathrm{max}}$ and then from $\epsilon=\epsilon_{\mathrm{min}}$ to
0 mrad in 2.18 mrad steps. The angles $\epsilon_{\mathrm{max}}$ and
$\epsilon_{\mathrm{min}}$ varied slightly per run with
$\epsilon_{\mathrm{max}}-\epsilon_{\mathrm{min}}=$ 56.68 mrad. Each run lasted
4 h to 9 h with a statistical mode of 4.4 h. At each angle of the phase flag
the spin flipper was operated in a off-on-on-off sequence to reduce the effect
of linear drifts. Two simultaneous interferograms, one for each precession
coil spin flipper state, were constructed from the background subtracted off-
on-on-off data. A typical pair of interferograms are shown in Fig. 5. Figure 6
shows the measured phases $\phi_{1}^{\uparrow}$ and $\phi_{1}^{\downarrow}$
over a span of a month that includes five cell transfers.

Figure 5: Typical interferograms generated by a off-on-on-off spin flipper
sequence is shown where the spin flipper is off (blue squares) and on (red
circles). Each point was counted for 150 s. The lower intensity for the
“Flipper ON” interferogram is due to stronger absorption in that case. The
uncertainties shown are purely statistical. Lines are fits of the data using
Eq. (20).  Figure 6: The measured phases $\phi_{1}^{\uparrow}$ (blue squares)
and $\phi_{1}^{\downarrow}$ (red circles) for a subset of the data that
includes 5 cell transfers. The data taken with the cell Pistachio are marked.
Uncertainties are smaller than the points.

Comparing the two interferograms yields a measured phase shift
$\Delta\phi_{M}$. A correction must be applied to
$\Delta\phi_{M}=\phi_{1}^{\uparrow}-\phi_{1}^{\downarrow}$ in order to
determine $\Delta b^{\prime}$ using Eq. (18). This is because the incident
beam is an incoherent mixture of both spin-up and spin-down neutrons (the
neutron polarization $P_{n}\neq 1$). The measured interferogram is actually a
sum of two different interferograms.

$\displaystyle I_{0}(\mbox{off})$ $\displaystyle=$ $\displaystyle
c^{\uparrow}_{1}\cos\left(\phi_{\mathrm{flag}}+\phi_{1}^{\uparrow}\right)$
(27) $\displaystyle=$
$\displaystyle\cos\left(\phi_{\mathrm{flag}}+\phi_{2}\right)+\eta_{-}\cos\left(\phi_{\mathrm{flag}}+\phi_{3}\right)$

where

$\displaystyle\eta_{-}$ $\displaystyle=$
$\displaystyle\frac{(1-P_{n})}{(1+P_{n})}e^{-N_{3}\sigma_{p}D_{3}P_{3}}$ (28)

is the ratio of the number of minority-spin neutrons to the number of
majority-spin neutrons that exit the 3He target cell. In Eq. (27) the mean
intensity has been subtracted but this does not affect the overall result
below. When the precession coil spin flipper is energized

$\displaystyle I_{0}(\mbox{on})$ $\displaystyle=$ $\displaystyle
c^{\downarrow}_{1}\cos\left(\phi_{\mathrm{flag}}+\phi_{1}^{\downarrow}\right)$
(29) $\displaystyle=$
$\displaystyle\eta_{+}\cos\left(\phi_{\mathrm{flag}}+\phi_{2}\right)+\cos\left(\phi_{\mathrm{flag}}+\phi_{3}\right)$

where

$\displaystyle\eta_{+}$ $\displaystyle=$
$\displaystyle\frac{(1-sP_{n})}{(1+sP_{n})}e^{+N_{3}\sigma_{p}D_{3}P_{3}}$
(30)

is again the ratio of the number of minority-spin to majority-spin neutrons.
We can now write the measured phase shift $\Delta\phi_{M}$ in terms of
$\Delta\phi_{0}=\phi_{2}-\phi_{3}$ which is the phase shift if the neutron
polarization had been perfect (100 %) using

$\displaystyle\Delta\phi_{M}$ $\displaystyle=$
$\displaystyle\arctan{\left(\frac{\displaystyle\sin{\Delta\phi_{0}}}{\displaystyle\eta_{+}+\cos{\Delta\phi_{0}}}\right)}$
(31)
$\displaystyle-\arctan{\left(\frac{\displaystyle\eta_{-}\sin{\Delta\phi_{0}}}{\displaystyle
1+\eta_{-}\cos{\Delta\phi_{0}}}\right).}$

No correction is necessary for the fact that the helium polarization
$P_{3}\neq 1$ because each individual neutron interacts with multiple 3He
atoms.

We have collected two sets of n-3He phase shift data taken in separate years.
The first run of this experiment done in 2008 and has previously been reported
in Huber _et al._ Huber  _et al._ (2009a, b). A second data set consisting of
six months of additional phase measurements was taken in the spring and summer
of 2013.

###  III.4 Phase shift due to a non-uniform magnetic field

When a magnetic field $B$ in the direction of the neutron polarization is
present, the additional magnetic potential
$V_{\mathrm{M}}=-\bm{\mu_{\mathrm{n}}}\cdot\bm{B}$ experienced by the neutron
creates a phase shift

$\displaystyle\phi_{\mathrm{mag}}=\pm\frac{\mu_{\mathrm{n}}m_{\mathrm{n}}\lambda
D}{2\pi\hbar^{2}}B=\pm\kappa DB.$ (32)

where $\kappa=$ $-$545 mrad/(cm-mT). The distance $D$ is the path length
inside the interferometer. The $(+)$ and $(-)$ signs correspond to neutron
polarization parallel and anti-parallel to the magnetic field, respectively.
When calculating the phase difference between precession coil spin flipper
states off and on the $\pm$ sign in Eq. (32) reverses, hence in the absence of
polarized 3He gas this difference is $2\phi_{\mathrm{mag}}$ for each path of
the interferometer. Since the length of both interferometer paths are equal,
$\phi_{\mathrm{mag}}$ can only be nonzero if the magnetic field in the two
paths are different. In this case, the phase shift difference will be

$2\phi_{\mathrm{mag}}=2\kappa(B_{1}-B_{2})D$ (33)

where $B_{1}$ and $B_{2}$ are the magnetic field strengths averaged over paths
1 and 2, respectively. For this interferometer the longer, parallel part of
the beam paths, which contained both the cell and compensating glass, was 6.4
cm with a total path length of 8.6 cm. A magnetic field gradient will be
manifested as a non-zero phase shift in the absence of polarized gas and a
non-zero y-intercept for a fit of the variation of phase shift with 3He
polarization.

By direct measurements without the cell inside the interferometer Huber _et
al._ Huber  _et al._ (2009a, b) determined $2\phi_{\mathrm{mag}}=$(2 $\pm$ 10)
mrad, hence consistent with zero but with a relatively large uncertainty. The
fit of the phase shift data also yielded a y-intercept consistent with zero.
An estimate of the gradient from the observed 3He relaxation time in the cell
Cashew yielded $2\phi_{\mathrm{mag}}=$ 0.72 mrad and this value was used to
assign a systematic uncertainty due to magnetic field gradients. However,
applying the incoherent beam correction in Eq. (31) Huber  _et al._ (2009a)
shifted the y-intercept to (16 $\pm$ 4) mrad, suggesting an underestimate of
the magnetic field gradient. This led us to directly map the magnetic field,
revisit the estimate from 3He relaxation, and perform a better evaluation of
the y-intercept (see Sec. VI). The need to study the effect of magnetic field
gradients on the measured phase shift when $P_{3}=0$ was the leading
motivation to repeat parts of the n-3He experiment in 2013 and revisit parts
of our analysis in 2008.

Figure 7: The difference between the maximum magnetic field in mT and the
magnetic field $B_{z}$ in the region of the target cell’s center (0,0)
measured using a fluxgate magnetometer. Contour lines show a $7\times 10^{-4}$
cm-1 field gradient.

Figure 7 shows a map of the magnetic field obtained with the Helmholtz coils,
which revealed a fairly linear gradient $(1/B_{z})dB_{z}/dy\approx$ 7$\times
10^{-4}$ cm-1. The two paths of the interferometer are separated by 2.2 cm,
hence this gradient yields $B_{1}-B_{2}\approx$ $2.3\times 10^{-3}$ mT and
thus $2\phi_{\mathrm{mag}}\approx$ 16 mrad. Although magnetic parts were
avoided near the 3He cell, there was a rotation stage with magnetic components
below the coils. This stage was necessary so that the interferometer could be
aligned to satisfy the Bragg condition.

The observed relaxation time $T_{1}$ of the 3He gas results from contributions
from dipole-dipole relaxation Newbury  _et al._ (1993), wall relaxation
Fitzsimmons  _et al._ (1969), and magnetic field gradients. The first two
components yield the “intrinsic” relaxation time of the cell,
$T_{\mathrm{i}}$. The observed relaxation time in the interferometer in the
presence of a field gradient is

$\displaystyle\frac{1}{T_{1}}=\frac{1}{T_{\mathrm{i}}}+\frac{1}{T_{\mathrm{fg}}}$
(34)

where the gradient contribution is given byMcIver  _et al._ (2009)

$\displaystyle\frac{1}{T_{\mathrm{fg}}}$ $\displaystyle=$
$\displaystyle\frac{6700}{p}\left(\frac{\left|\bm{\nabla}B_{x}\right|^{2}}{B^{2}}+\frac{\left|\bm{\nabla}B_{y}\right|^{2}}{B^{2}}\right)$
(35) $\displaystyle=$ $\displaystyle\frac{6700\beta^{2}}{p}\mathrm{h}^{-1}$

Here $p$ is the pressure in bar and $B_{x,y}$ are the perpendicular components
of the magnetic field where the applied field is in the $z$ direction. For the
cell Cashew, $T_{1}=$ 135 h and $T_{i}=$ 330 h which yields $\beta=1.1\times
10^{-3}$ cm-1. Whereas $\beta$ includes several components, by using
$\nabla\times\bm{B}=0$ and by assuming the gradient is dominated by the linear
gradient observed in the field map, one obtains $(1/B_{z})dB_{z}/dy\approx
1.0\times 10^{-3}$ cm-1, $B_{1}-B_{2}\approx$ $3.6\times 10^{-3}$ mT and thus
$2\phi_{\mathrm{mag}}\approx$ 25 mrad. It is likely that this value is an
upper limit because several gradient components contribute to relaxation. As
discussed in Sec. VI, we obtained y-intercepts of (16 $\pm$ 4) mrad in the
2008 run and (21 $\pm$ 3) mrad in the 2013 run, consistent with the estimates
from the field map and 3He relaxation.

Between 2008 and 2013 other interferometry experiments were performed at the
NIOF. Changes to the NIOF included different shielding, changes to the
polarizer, wavelength changes, the use of different interferometers and
mounting, stages, and a change in the monochromator crystal Huber  _et al._
(tion). Despite these changes the experimental conditions were reasonably well
reproduced. Planning for additional phase data at $P_{3}=0$ was started almost
immediately after 2008 as the magnetic field gradient became more of a
concern. The Helmholtz coils and spin flipper were at identical positions in
2008 and 2013. Other components of the experiment like the electronics,
detectors, the interferometer, cell mounting, and other system components were
the same between the two runs as they were reserved for this work and not
otherwise used. Initial $P_{3}$ was 10 % higher in 2013 because of advances in
helium polarization techniques Chen  _et al._ (2014). Another difference was
that in 2008 the polarization direction of $P_{3}$ was keep fixed; whereas in
2013 $P_{3}$ was twice polarized in the reversed direction. Lastly, the
neutron polarization was 3 % lower (see Section IV.2) in 2013 due to changes
in the supermirror alignment.

###  III.5 Measuring cell relaxation

Target cell transmission was measured _in situ_ during each scan using the C4
detector (see Fig 8). For each run the asymmetry

$\displaystyle
A=\frac{I^{\uparrow}-I^{\downarrow}}{I^{\uparrow}+I^{\downarrow}}$ (36)

was calculated from the individual off-on spin flipper asymmetries. The
asymmetry is related to the neutron polarization $P_{n}$ and spin flipper
efficiency

$\displaystyle s=\left|\frac{P_{n}(\mbox{on})}{P_{n}(\mbox{off})}\right|,$
(37)

where on(off) refers to the state of the precession coil spin flipper Wildes
(1999), by

$\displaystyle A=\frac{(1+s)P_{n}P_{A}}{2+(1-s)P_{n}P_{A}}.$ (38)

The values of $s$ and $P_{n}$ are known from the polarimetry measurements. For
each interferogram an averaged asymmetry $\overline{A}$ was calculated. The
analyzing power $P_{A}$ of a 3He cell can be written Rich  _et al._ (2002)

$\displaystyle P_{A}=\tanh\left(\xi\right)$ (39)

where

$\displaystyle\xi=N_{3}\sigma_{p}D_{3}P_{3}$ (40)

is the product of the opacity of the cell $N_{3}\sigma_{p}D_{3}$ and 3He
polarization. One can use Eqs. (38) and (39) to write $\xi$ in terms of the
measured asymmetry $\overline{A}$ as

$\displaystyle\xi=\tanh^{-1}\left(\frac{2\overline{A}}{(s+1)P_{n}+(s-1)P_{n}\overline{A}}\right).$
(41)

Figure 8: The average cell transmissions $I^{\uparrow}$(blue squares) and
$I^{\downarrow}$ (red circles) as measured by the C4 detector for a subset of
the data that includes 5 cell transfers. The data taken with the cell
Pistachio are marked. Uncertainties are smaller than the points.

##  IV Auxiliary Measurements

###  IV.1 Neutron Wavelength

Because the skew symmetric interferometer uses a (220) reflection in silicon,
it is necessary to eliminate higher order $n=2,3,\ldots$ reflections from the
incident beam. Neutrons with wavelengths of $\lambda<$ 2.35 Å are poorly
polarized by the supermirror and could potentially affect the phase and
polarimetry measurements. Neutrons with wavelengths corresponding to $n\geq 3$
are suppressed by the liquid hydrogen cold source. However a non-negligible
amount (5 %) of $\lambda/2=$ 1.175 Å neutrons are present in the incident
beam. To eliminate these neutrons a pyrolytic graphite filter Shirane and
Minkiewicz (1970) consisting of nine separate PG crystals of varying thickness
(50 mm overall) was placed upstream of the supermirror polarizer. Neutrons of
wavelength 1.175 Å are preferentially reflected by the (114) plane of the
graphite and are absorbed by a surrounding boron shield.

A measurement of the fraction of $\lambda/2$ neutrons was performed with the
interferometer removed and a nearly-perfect silicon analyzer crystal (NPC)
placed in the direct beam before the interferometer. This crystal analyzer is
denoted as “nearly-perfect” because it contains a small mosaic spread (a small
variation in lattice vector direction throughout the crystal) of $1.5\times
10^{-4}$ mrad. The mosaic of the crystal allows a greater fraction of incident
neutrons to satisfy the Bragg condition; thus more reflect from the crystal
and increase the overall reflected intensity. The relative intensity of
$(I_{\lambda/2})/I_{\lambda}$ was measured by rotating the “nearly-perfect”
crystal $\pm\theta_{B}$ and examining the reflected beams. In addition to the
NPC, a disk chopper made from a rotating, neutron absorbing cadmium disk with
a small slit was used as well. The disk chopper allows time-of-flight analysis
of the neutron spectrum. Both techniques placed an upper limit of
$(I_{\lambda/2})/I_{\lambda}<0.1\%$ that was determined by comparing the
relative intensities with and without the filter in place. This ratio is
mainly limited by the accuracy in determining the small background signal. The
presence of 1.175 Å neutrons at this level had a negligible effect on
$\Delta\phi_{M}$ and polarimetry measurements.

###  IV.2 Polarimetry

Several neutron polarimetry measurements were made throughout the experiment
to verify that the neutron polarization was stable over the duration of the
experiment. Each polarimetry measurement took place during pauses in
collection of the phase data. Common techniques to measure neutron
polarizations and spin flipper efficiencies along with their difficulties are
described in Ref. Yerozolimsky (1999). In this experiment the neutron
polarization $P_{n}$ was measured with 3He cells and by using two analysis
methods, which we refer to as the asymmetry and normalized transmission
methods. Polarimetry cells are physically larger than target cells and one of
them is shown in Fig. 4. Properties of the cells are listed in Table 1.

A 3He analyzer had two advantages over crystal or supermirror analyzers.
First, the analyzing power $P_{A}$ of the cell was determined from unpolarized
neutron transmission measurements. Second, we could flip the cell’s
polarization by $\pi$ radians using nuclear magnetic resonance (NMR) at the
SEOP facility. This NMR induced flip eliminated the need for a second spin
flipper to uniquely determine $P_{n}$, $P_{A}$, and spin flipper efficiency.
These cells had three times the opacity of the target cells and thus provided
high analyzing powers that were relatively insensitive to variations in 3He
polarization.

The setup for both methods was the same. Low neutron fluence rates in the
H-beam prevented any practical polarization analysis behind the
interferometer. Instead, the interferometer was removed from its cradle and
replaced with one of the analyzing cells. Because the neutron polarization
produced by the supermirror polarizer should depend very weakly on wavelength
and the beam spectrum was sufficiently narrow ($\sigma_{\lambda}/\lambda=$ 1
%), the difference between the measured $P_{n}$ of the direct beam and the
neutron polarization of paths I and II of the interferometer is believed to be
negligible. The neutron transmission through the cell was measured using a 3He
detector located directly behind the analyzer.

For both methods the analyzing power of the cell was determined by the
transmission of unpolarized neutrons. The analyzing power of a 3He cell is
given by Eq. (39). For the polarimetry cells the range of initial $P_{A}$ was
between 86 % and 99 % but was typically around 97 % depending on the cell and
its initial 3He polarization. Eq. (39) can be rewritten as a ratio of two
unpolarized neutron transmission measurements as

$\displaystyle
P_{A}=\sqrt{1-\bigg{(}\frac{T_{\mathrm{un}}}{T_{\mathrm{pol}}}\bigg{)}^{2}}$
(42)

where $T_{\mathrm{pol}}$($T_{\mathrm{un}}$) is the transmission of unpolarized
neutrons through a polarized (unpolarized) 3He cell. These transmissions are
discussed later in Sec. IV.2.2. Unpolarized neutrons were obtained by
translating the supermirror out of the beam using an encoded linear stage. The
position of the supermirror was reproducible to within 1 µm. To measure
$T_{\mathrm{un}}$ the analyzer cell was depolarized by temporarily connecting
the Helmholtz coils to an alternating current voltage supply.

####  IV.2.1 Asymmetry Method

The asymmetry method used the difference in count rates for the two neutron
spin states, $I^{\uparrow}$ and $I^{\downarrow}$, to determine the neutron
polarization and spin flipper efficiency. Here $I^{\uparrow(\downarrow)}$ is
the intensity when the neutron and 3He polarization are aligned parallel
(anti-parallel). The asymmetry $A$ is related to the neutron polarization and
spin flipper efficiency by

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{I^{\uparrow}-I^{\downarrow}}{I^{\uparrow}+I^{\downarrow}}=\frac{(1+s)P_{n}P_{A}}{2+(1-s)P_{n}P_{A}}.$
(43)

To uniquely determine $P_{n}$ and $s$ using this method it is necessary to
have two separate asymmetries $A$ and $A^{\ast}$ where one reverses the
direction of the 3He polarization. Similar to Eq. (43) we have

$\displaystyle A^{\ast}$ $\displaystyle=$
$\displaystyle\frac{(1+s)P_{n}P_{A^{\ast}}}{2-(1-s)P_{n}P_{A^{\ast}}}$ (44)

The analyzing powers $P_{A}$ and $P_{A^{\ast}}$ were not the same because of a
few percent loss in $P_{3}$ caused by performing an NMR induced spin flip and
transporting the cell to and from the SEOP facility. Using Eqs. (43) and (44)
the spin flipper efficiency is

$\displaystyle
s=\displaystyle\frac{A\left[1+A^{\ast}\right]P_{A^{\ast}}+P_{A}\left[A-1\right]A^{\ast}}{A\left[A^{\ast}-1\right]P_{A^{\ast}}+P_{A}\left[1+A\right]A^{\ast}}.$
(45)

With knowledge of $s$, $P_{n}$ can be determined via

$\displaystyle P_{n}$ $\displaystyle=$
$\displaystyle\frac{2A}{P_{A}\left[(s-1)A+(1+s)\right]}$ (46) $\displaystyle=$
$\displaystyle\frac{2A^{\ast}}{P_{A^{\ast}}\left[(1-s)A^{\ast}+(1+s)\right]}.$

Measurements of $P_{n}$ and $s$ that were obtained using the asymmetry method
are shown in Fig. 9.

The intensities $I^{\uparrow}$ and $I^{\downarrow}$ for both this method and
normalized transmission method discussed in Section IV.2.2 were performed
symmetrically around measurements of $T_{\mathrm{pol}}$ and hence $P_{A}$ and
$P_{A^{\ast}}$. Specifically, $I^{\uparrow}$ (and likewise $I^{\downarrow}$)
was measured twice, once before and once after a measurement of
$T_{\mathrm{pol}}$. The averages of $I^{\uparrow}$ and $I^{\downarrow}$ were
then used in Eqs. (43) and (44). This was done to compensate for the decay of
$P_{3}$ while the measurements were being performed to a level where no
correction for $P_{3}$ decay was needed.

####  IV.2.2 Normalized Transmission Method

When $P_{n}=1$ the transmission of neutrons through a polarized 3He cell is

$\displaystyle T_{\mathrm{off(on)}}$ $\displaystyle=$ $\displaystyle
T_{\mathrm{g}}\exp{\left(-N_{3}\sigma_{a}D_{3}\right)}$ (47) $\displaystyle=$
$\displaystyle
T_{\mathrm{g}}\exp{\left(-N_{3}\sigma_{\mathrm{un}}D_{3}\right)}\exp{\left({\pm\xi}\right)},$

where Eqs. (14) and (40) have been used to relate the absorption cross section
$\sigma_{a}$ to $\xi$. The signs $(+)$ and $(-)$ is for ‘off’ and ‘on’ states
of the precession coil spin flipper, respectively. Here we have taken that the
initial neutron polarization and 3He polarization is in the same direction.
$T_{\mathrm{g}}$ is the transmission of neutrons through the cell windows.

For a neutron beam with $P_{n}\leq 1$ the transmission $T_{\mathrm{off}}$
becomes

$\displaystyle T_{\mathrm{off}}$ $\displaystyle=$
$\displaystyle\left(\frac{T_{\mathrm{g}}}{2}\right)e^{-N_{3}\sigma_{\mathrm{un}}D_{3}}$
(48) $\displaystyle\times$
$\displaystyle\bigg{[}\left(1+P_{n}\right)e^{\xi}+\left(1-P_{n}\right)e^{-\xi}\bigg{]}.$

Eq. (48) can be expressed more compactly as

$\displaystyle T_{\mathrm{off}}$ $\displaystyle=$ $\displaystyle
T_{\mathrm{g}}e^{-N_{3}\sigma_{\mathrm{un}}D_{3}}\left[\cosh\left({\xi}\right)+P_{n}\sinh\left({\xi}\right)\right].$
(49)

The transmission of unpolarized neutrons through a polarized 3He cell is given
by

$\displaystyle T_{\mathrm{pol}}=T_{\mathrm{un}}\cosh\left({\xi}\right),$ (50)

where

$\displaystyle
T_{\mathrm{un}}=T_{\mathrm{g}}e^{-N_{3}\sigma_{\mathrm{un}}D_{3}}$ (51)

is the transmission of unpolarized neutrons through an unpolarized 3He cell.
Dividing Eq. (49) by Eq. (50) yields

$\displaystyle\frac{T_{\mathrm{off}}}{T_{\mathrm{pol}}}=1+P_{n}\tanh\left(\xi\right).$
(52)

It follows from Eqs. (39), (42), and (52) that

$\displaystyle P_{n}=\frac{\frac{\displaystyle T_{\mathrm{off}}}{\displaystyle
T_{\mathrm{pol}}}-1}{\sqrt{\displaystyle{1-\bigg{(}\frac{T_{\mathrm{un}}}{T_{\mathrm{pol}}}\bigg{)}^{2}}}}.$
(53)

When one energizes the spin flipper one has the anti-parallel case where,

$\displaystyle sP_{n}=\frac{\displaystyle
1-\frac{T_{\mathrm{on}}}{T_{\mathrm{un}}}}{\displaystyle\sqrt{1-\bigg{(}\frac{\displaystyle
T_{\mathrm{un}}}{\displaystyle T_{\mathrm{pol}}}\bigg{)}^{2}}}.$ (54)

Figure 9: The neutron polarization (circles) and spin flipper efficiency
(diamonds) measured in 2008 (solid) and 2013 (open) determined using the
asymmetry method. The uncertainties shown are purely statistical. Fits of the
data are shown as solid lines for 2008 and dotted lines for 2013. Points with
larger uncertainties were taken when the cell had lower polarization. Figure
10: The neutron polarization (circles) and neutron polarization when
energizing the precession coil spin flipper (squares) measured in 2008 (solid)
and 2013 (open) determined using the normalized transmission method. The
uncertainties shown are purely statistical. All points shown were taken in the
anti-parallel state of n-3He (see text). Fits of the data are shown as solid
lines for 2008 and dotted lines for 2013. Points with larger uncertainties
were taken when the cell had lower polarization.

It should be noted that the uncertainty associated with determining the
polarimetry from Eq. (53) is different than that for Eq. (54). Propagating the
uncertainty $\sigma$ of the polarization product $Z_{\mathrm{off}}=P_{A}P_{n}$
and $Z_{\mathrm{on}}=P_{A}sP_{n}$ we find

$\displaystyle\sigma^{2}_{Z_{\mathrm{off}}}=\left(\frac{1}{T_{\mathrm{un}}}\right)^{2}\sigma^{2}_{T_{\mathrm{off}}}+\left(\frac{T_{\mathrm{off}}}{T_{\mathrm{un}}^{{}^{2}}}\right)^{2}\sigma^{2}_{T_{\mathrm{un}}}$
(55)
$\displaystyle\sigma^{2}_{Z_{\mathrm{on}}}=\left(\frac{1}{T_{\mathrm{un}}}\right)^{2}\sigma^{2}_{T_{\mathrm{on}}}+\left(\frac{T_{\mathrm{on}}}{T_{\mathrm{un}}^{{}^{2}}}\right)^{2}\sigma^{2}_{T_{\mathrm{un}}}.$
(56)

For this experiment $Z_{\mathrm{off}}\approx Z_{\mathrm{on}}\approx 0.9$ and
$T_{\mathrm{on}}\approx 0.05T_{\mathrm{off}}$. Using Eqs. (55) and (56) we
find that

$\displaystyle\sigma_{Z_{\mathrm{on}}}\approx 0.05\sigma_{Z_{\mathrm{off}}}.$
(57)

So despite the relative uncertainties of $T_{\mathrm{off}}$ and
$T_{\mathrm{on}}$ being comparable the overall uncertainty in determining
$P_{n}$ versus $sP_{n}$ differs by a factor of twenty. The contribution to the
overall uncertainty of $P_{n}$ from the uncertainty in $P_{A}$ is small
because of higher statistics without the supermirror in the neutron beam. By
reversing the 3He spin using NMR we can invert Eq. (57) so that

$\displaystyle\sigma_{Z_{\mathrm{off}}}\approx 0.05\sigma_{Z_{\mathrm{on}}}$
(58)

when the 3He polarization has been flipped. It follows that using the anti-
parallel state to determine $P_{n}$ or $sP_{n}$ is advantageous despite a much
smaller $T_{\mathrm{off(on)}}$ being measured because the overall uncertainty
is better. Thus we have chosen to use only the anti-parallel measurements for
the normalized transmission method (shown in Figure 10).

####  IV.2.3 Polarimetry Result

Both the asymmetry and normalized transmission methods yield neutron
polarizations and spin flipper efficiencies to less than 0.1 % relative
standard uncertainty. These results are shown in Table 2. For the 2008 data
set there was a $2\sigma$ disagreement in measured neutron polarization
between the two methods. To handle this discrepancy, the uncertainties for the
2008 polarimetry results are determined by adding the largest uncertainty of
the two methods in quadrature with the difference between the methods, for
example $\sigma_{P_{n}}=\sqrt{\sigma^{2}_{\mathrm{largest}}+(\Delta
P_{n})^{2}}=\sqrt{(0.0033)^{2}+(0.92874-0.92941)^{2}}$. In 2013 the
polarimetry (see Figs. 9 and 10) data were more consistent and this expansion
of their uncertainty was not done. An equal weighted average of the asymmetry
and normalized transmission methods yields

$\displaystyle P_{n}$ $\displaystyle=0.92908\pm 0.00075\quad\text{in 2008}$
(59) $\displaystyle=0.90227\pm 0.00055\quad\text{in 2013}$

and

$\displaystyle s$ $\displaystyle=0.99510\pm 0.00034\quad\text{in 2008}$ (60)
$\displaystyle=0.99475\pm 0.00089\quad\text{in 2013}$

The differences between the 2008 and 2013 neutron polarization is believed to
be due to non-reproducible changes in the angular separation between the two
mirror surfaces of the supermirror polarizer that is often varied between
experiments. In both 2008 and 2013 the neutron precession coil spin flipper
was the same device, was located in the same place, and showed much better
agreement.

Table 2: Results of the polarimetry for the various methods used. $\sigma_{R}$ is the relative standard uncertainty.  |  | 2008 | 2013
---|---|---|---
Var. | Method | Value ($\sigma$) |  $\sigma_{R}$, [%] | Value ($\sigma$) |  $\sigma_{R}$, [%]
$P_{n}$ | Asy. | 0.92874(33) | 0.04 | 0.90260(36) | 0.04
N.T. | 0.92941(17) | 0.02 | 0.90184(41) | 0.05
Asy. + N.T. | 0.92908(75) | 0.09 | 0.90227(55) | 0.06
$s$ | Asy. | 0.99502(31) | 0.03 | 0.99444(63) | 0.06
N.T. | 0.99516(23) | 0.02 | 0.99506(63) | 0.06
Asy. + N.T. | 0.99510(34) | 0.03 | 0.99475(89) | 0.09

##  V Systematic Effects

###  V.1 Absorption Cross Section

The quantity $\xi$ is a function of $\lambda$ and can be written as

$\displaystyle\xi=N_{3}\sigma_{p}D_{3}P_{3}=N_{3}\left[\frac{1}{4}\left(\sigma_{0}-\sigma_{1}\right)\right]\frac{\lambda}{\lambda_{\mathrm{th}}}D_{3}P_{3},$
(61)

where $\lambda_{\mathrm{th}}=$ 1.798 Å is the reference thermal neutron
wavelength. To extract $N_{3}\lambda D_{3}P_{3}$ from $\xi$, one needs the
singlet and triplet absorption cross sections $\sigma_{0}$ and $\sigma_{1}$.
The experimental value of $\sigma_{\mathrm{un}}\approx\sigma_{0}/4$ is well
known from transmission measurements as (5333 $\pm$ 7) b at
$\lambda_{\mathrm{th}}$ Mughabghab (2006). However, the triplet absorption
cross section is poorly known experimentally. Passell and Schermer Passell and
Schermer (1966) measured neutron transmission through 3He and determined the
ratio of singlet to total absorption cross section to be
$g_{0}\sigma_{0}/\sigma_{\mathrm{un}}=$ (1.010$\pm$ 0.032) . An indirect
measurement of the same quantity was made by Borzakov _et al._ Borzakov  _et
al._ (1982) where they determined $g_{0}\sigma_{0}/\sigma_{\mathrm{un}}=$
(0.998 $\pm$ 0.010) by examining deviations from a purely “$1/v$” absorption
law for neutron energies up to 150 keV. Both of these experiments support
$\sigma_{1}\approx 0$ but only at the 1 % level. Due to the lack of precision
measurements of $\sigma_{1}$, we used a theoretical prediction of the
imaginary part of the scattering length to estimate $\sigma_{1}$.

Calculations performed by Hofmann and Hale Hofmann and Hale (2003, 2008) of
the imaginary free scattering length $a_{1}^{\prime\prime}$ using R-matrix and
AV18+3N interactions give a range of values $a_{1}^{\prime\prime}$ of between
0.0012 fm and 0.0051 fm. However as noted in the same paper, AV18+3N models
under-predict the experimentally measured $a_{0}^{\prime\prime}$ by up to 30
%. To be conservative we used $a_{1}^{\prime\prime}=$ (0.005 $\pm$ 0.005 ) fm.
This allowed for the possibility that theoretical calculations are low by as
much as a factor of two. With $a_{1}^{\prime\prime}$ and the measured thermal
absorption cross section for unpolarized 3He we have

$\displaystyle\sigma_{0}-\sigma_{1}=\mbox{(21236 $\pm$ 100)\,b}.$ (62)

With Eqs. (41) and (62) one may extract $N_{3}\lambda D_{3}P_{3}$ from the
asymmetry measurements of $\xi$.

###  V.2 Polarimetry Effects

The effect of uncertainties in $P_{n}$ and $s$ on calculating $\Delta\phi_{0}$
is complicated by the fact that they affect both $\eta_{-}$ and $\eta_{+}$
directly and also indirectly through $\xi$. To determine the systematic
uncertainty in $\Delta\phi_{0}$ contributed by the uncertainties
$\sigma_{P_{n}}$ and $\sigma_{s}$ we studied a simulated set of
$[\Delta\phi_{M}]_{\mathrm{sim}}$ and $[\xi]_{\mathrm{sim}}$ data. This
simulated data was generated using a fixed value $\Delta b^{\prime}=$$-$5.400
fm and a randomly distributed set of $[\xi]_{\mathrm{sim}}$ to generate a
$[\Delta\phi_{M}]_{\mathrm{sim}}$. $\Delta\phi_{0}$ was then calculated using
the simulated $[\Delta\phi_{M}]_{\mathrm{sim}}$ and $[\xi]_{\mathrm{sim}}$
while varying $P_{n}$ and $s$ by their respective uncertainties. The variance
in $\Delta\phi_{0}$, and hence $\Delta b^{\prime}$, resulting from the
uncertainties $\sigma_{P_{n}}$ and $\sigma_{s}$ was taken as the systematic
uncertainty due to the polarimetry measurements.

##  VI Results

Figure 11 shows the measured $\Delta\phi_{0}$ versus $\xi$ for the 2008 data
set which was collected over several reactor cycles for a total of 12 weeks.
From Eq. (18) the value of $\Delta b^{\prime}$ can be determined by the slope
of $\Delta\phi_{0}$ in Figure 11. There are two significant changes in
determining $\Delta b^{\prime}$ from what was done previously in Huber _et
al._ Huber  _et al._ (2009a, b). The first and most significant is that in
Huber _et al._ the slope of $\Delta\phi_{0}$ versus $\xi$ was determined using
a one parameter fit. This fixed the y-intercept of the fit to be precisely
zero corresponding to $\Delta\phi_{0}=0$ at $P_{3}=0$. In the presence of a
magnetic field gradient, this approach is no longer valid. Instead we now
perform a two parameter fit of $\Delta\phi_{0}$ versus $\xi$. The fitted
y-intercept of the data shown in Fig. 11 yields $2\phi_{\mathrm{mag}}=$ (16
$\pm$ 4) mrad.

The other change we have made has been in the manner in which we cut
individual data points. In Huber _et al._ we cut the data based on the reduced
chi squared $\chi_{\mathrm{red}}^{2}$ of the interferogram fit. All fits with
$\chi_{\mathrm{red}}^{2}\geq 1.5$ were discarded and not included in our 2008
results. This was done to account for phase instabilities especially those
seen immediately following a cell transfer which introduced temporary
temperature and mechanical instabilities lasting 12 h or more. However,
discarding interferograms based on $\chi_{\mathrm{red}}^{2}$ values included
eliminating points taken in the middle of runs where the phase was more
stable. As discussed below, a systematic uncertainty of 0.012 fm attributed to
phase instabilities was also applied to the result in Huber _et al._ Since we
already incorporate an uncertainty due to phase instabilities, for this result
we make no cut based on the $\chi_{\mathrm{red}}^{2}$ in either the 2008 or
2013 data set. Phase instabilities were greater in 2013 as the temperature
stability that we had in 2008 was not reproduced. This is contrary to 2008
where temperature drifts were highly correlated to opening the facility doors
to perform a cell transfer (since $P_{3}\approx 0$ in most of 2013, transfers
were infrequent in that data run). The inclusion of data points with
$\chi_{\mathrm{red}}^{2}\geq 1.5$ does not affect the values determined by a
fit of $\Delta\phi_{0}$ versus $\xi$ but does decrease the statistical
uncertainty.

A two parameter fit of Fig. 11 gives $\Delta b^{\prime}=$ ($-$5.381 $\pm$
0.053) fm with $\chi^{2}/d.o.f.=530/(435-2)=1.2$. This $\chi^{2}$ represents a
low probability of fit ($<$ 1 %) and is due to random phase instabilities that
were most likely caused by small temperature fluctuations. To estimate the
systematic uncertainty due to this effect the uncertainty of $\Delta\phi_{0}$
was inflated by 0.016 rad in quadrature with the statistical uncertainty for
each point so that the $\chi^{2}/d.o.f.=1$. The average statistical
uncertainty for $\Delta\phi_{0}$ was $\approx$ 0.033 rad but varied strongly
with $P_{3}$. A histogram of the residual of the fit with a reduced
$\chi^{2}=1$ is shown in the inset of Fig. 11. The distribution of points in
the figure closely follows a Gaussian function centered at zero.

Figure 11 shows $\Delta\phi_{0}$ versus $\xi$ for the 2013 data set. In 2013
we polarized the 3He gas only four times focusing instead on measuring
$\Delta\phi_{0}$ at low $P_{3}$. Twice we polarized the 3He in the opposite
direction with respect to the neutron polarization defined by the supermirror
polarizer. In this case there is more neutron absorption when the precession
coil spin flipper is off. This reversed-polarized data is shown in the lower
left quadrant of Fig. 11. Again applying a two parameter fit of Fig. 11 gives
$\Delta b^{\prime}=$ ($-$5.439 $\pm$ 0.038) fm with a
$\chi^{2}/d.o.f.=1120/(507-2)=2.2$. To fix $\chi^{2}/d.o.f.=1$ the uncertainty
of $\Delta\phi_{0}$ was inflated in quadrature by 0.043 rad. For 2013 we find
that $2\phi_{\mathrm{mag}}=$ (21 $\pm$ 3) mrad which is consistent with both
the 2008 data and the field gradient measurement.

The weighted average of both data sets gives

$\displaystyle\Delta b^{\prime}$ $\displaystyle=$ ($-5.411$ $\pm$ $0.031$
(Stat.) $\pm$ $0.039$ (Syst.) ) fm.

The uncertainty budget for $\Delta b^{\prime}$ is given in Table 3. The
weighted average is performed by weighting both the statistical and systematic
uncertainties unrelated to neutron absorption on 3He in quadrature. The
systematic uncertainty related to 3He absorption was added to the total
systematic uncertainty in Eq. (VI).

Figure 11: $\Delta\phi_{0}$ vs $\xi$ values for (a) 2008 and (b) 2013. The solid line is a weighted average with a $\chi^{2}/d.o.f.=1$ (see text). Insets: Histograms of the residual distribution with Gaussian fits (solid lines). The residual is defined by $R=y_{i}-y$ where $y$ is the fit function and $y_{i}$ is the $i^{th}$ data point. (a) The Gaussian fit is centered at $-0.001$ rad with a full width half maximum (FWHM) of 0.091 rad. (b) The Gaussian fit is centered at $+0.002$ rad with FWHM of 0.117 rad. Table 3: The uncertainty budget for $\Delta b^{\prime}$. Uncertainties related to 3He absorption cross section that are identical for both data sets are summed in quadrature.  2008 |  | 2013
---|---|---
$\sigma$, [fm] | Parameter |  $\sigma$, [fm]
0.053 |  $\Delta\phi_{0}/\xi$ Fit (Statistical) | 0.038
0.028 | Triplet absorption cross section $\sigma_{1}$ | 0.028
0.007 | Total absorption cross section $\sigma_{\mathrm{un}}$ | 0.007
0.029 | Total systematic from cross sections | 0.029
0.025 | Phase instabilities | 0.040
0.005 | Neutron polarization $P_{n}$ | 0.004
0.002 | Spin flipper efficiency $s$ | 0.004
0.026 | Total non-cross section systematic | 0.040
0.053 | Total statistical | 0.038
0.039 | Total systematic | 0.049

##  VII Conclusions and Discussion

We have performed a precision measurement of the difference $\Delta
b^{\prime}=$ ($-5.411$ $\pm$ $0.031$ (Stat.) $\pm$ $0.039$ (Syst.) ) fm
between the triplet and singlet scattering lengths of n-3He using neutron
interferometry to 0.9 % relative standard uncertainty. The ultimate precision
of this technique is systematically limited by the triplet absorption cross
section corresponding to a relative uncertainty of 0.5 %. This result is in
good agreement with the only previous direct measurement of $\Delta
b^{\prime}=$ (-5.462 $\pm$ 0.046) fm performed by Zimmer _et al._ at the
Institut Laue-Langevin (ILL) Zimmer  _et al._ (2002). Ref. Zimmer  _et al._
(2002) used a spin echo apparatus to measure the relative difference in the
pseudomagnetic spin-precession Baryshevsky and Podgoretsky (1965); Abragam
_et al._ (1973) between a neutron passing though a polarized 3He cell compared
to an empty reference beam. That technique is fundamentally different than the
technique applied here. One can state the results independent of the triplet
absorption cross section and total absorption cross section from our results
and that of Ref. Zimmer  _et al._ (2002). This is done for two reasons: (i)
both groups estimated $\sigma_{1}$ differently and (ii) in the event of future
more accurate measurements of the absorption cross sections, one can
immediately update the spin-dependent n-3He scattering length. Zimmer _et al._
determined $\sigma_{1}$ from a average of the experimental results of Refs.
Passell and Schermer (1966) and Borzakov  _et al._ (1982) with the limitation
that $\sigma_{1}\geq 0$. Whereas, as described in Section V.1, we used a
theoretically predicted $\sigma_{1}$ but with an inflated uncertainty. Our
result stated independent of the triplet absorption cross section is

$\displaystyle\Delta b^{\prime}(\mbox{this work})=\quad\quad$
$\displaystyle\bigg{[}(-10.1929$ $\displaystyle\pm 0.0760)\times 10^{-4}\mbox{
fm/b}\bigg{]}$ $\displaystyle\times\bigg{(}1-$
$\displaystyle\frac{\sigma_{1}}{\sigma_{\mathrm{un}}}\bigg{)}\sigma_{\mathrm{un}}$
(64)

This is in disagreement with the result of Zimmer _et al._ of

$\displaystyle\Delta b^{\prime}(\mbox{Ref.~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zimmer_2002_EPJ}{\@@citephrase{(}}{\@@citephrase{)}}}})=\quad\quad$
$\displaystyle\bigg{[}(-10.3628$ $\displaystyle\pm 0.0180)\times 10^{-4}\mbox{
fm/b}\bigg{]}$ $\displaystyle\times\bigg{(}1-$
$\displaystyle\frac{\sigma_{1}}{\sigma_{\mathrm{un}}}\bigg{)}\sigma_{\mathrm{un}}$
(65)

by $2\sigma$ when factoring out the absorption cross sections.

Figure 12: An ideogram of the coherent scattering length measurements for
n-3He taken from Refs. Kitchens  _et al._ (1974); Kaiser  _et al._ (1977,
1979); Alfimenkov  _et al._ (1981); Huffman  _et al._ (2004); Ketter  _et al._
(2006). The blue band represents the weighted average $\pm\sigma$ of the three
experiments with smallest quoted uncertainties. Techniques used were neutron
interferometry (NI), total cross section (CS), and reflectivity (RE). Figure
13: Current experimental data on the n-3He system from this work, ILL 2006
Ketter  _et al._ (2006), NIST 2004 Huffman  _et al._ (2004), ILL 2002 Zimmer
_et al._ (2002), ILL 1979 Kaiser  _et al._ (1979) compared to theoretical
predictions Hofmann and Hale (2003, 2008). Bands represent the experimentally
determined values $\pm 1\sigma$.

There have been a number of experiments measuring the coherent scattering
length of n-3He defined by Eq. (11a) using techniques such as measuring
neutron reflectivity, relative phase shifts, and neutron transmissions. The
three most precise measurements of $b_{c}^{\prime}$ were done with neutron
interferometry; Kaiser _et al._ Kaiser  _et al._ (1977), Huffman _et al._
Huffman  _et al._ (2004), and Ketter _et al._ Ketter  _et al._ (2006).
However, the two most recent results differ by more than $7\sigma$. Figure 12
shows an ideogram of the coherent scattering length measurements. Each
measurement is represented by a Gaussian centered about their result with a
normalized area equal to 1/$\sigma$ Amsler  _et al._ (2008). The uncertainty
of the weighted average has been inflated in the manner described in Ref.
Amsler  _et al._ (2008).

Calculations employing models AV18+UIX, AV18+UIX+V${}^{*}_{3}$ Hofmann and
Hale (2003, 2008), and AV18+LL2 Kirscher  _et al._ (2009) have all predicted
similar values for the triplet and singlet scattering lengths. For example
$\Delta b^{\prime}(AV18+UIX)=$ (-5.753 $\pm$ 0.002) fm. Neither this work nor
the work of Zimmer _et al._ agrees with NN+3N calculations. Figure 13 shows a
selection of measured values of $b_{1}$ and $b_{0}$ beside some theoretical
predictions. Four nucleon interactions have yet to be included into the
theoretical models due to the difficulty in handling long-range Coulomb
forces, but should constitute only a tiny correction to NN + 3N predictions. A
calculation of pion-less effective field theory to Next to Leading Order (NLO)
shows promise Kirscher  _et al._ (2009), but the uncertainty of the predicted
value is still too large to compare to high-precision measurements. A recent
measurement of the total scattering cross section Guckelsberger  _et al._
(2000) that suggests a much larger scattering cross section and would lie
outside of Fig. 13 is omitted for space.

The recent work on the n-3He interaction can lead to further understanding of
low energy nucleon systems. Although there are several discrepant
measurements, scattering length measurements do not match theoretical models.
Taken alone, the coherent scattering length by Huffman  _et al._ (2004) agrees
with AV18+UIX, but doesn’t intersect a measurement of the spin-dependent
difference in triplet and singlet states. This work and Zimmer  _et al._
(2002) agrees with the R-matrix prediction. More work needs to be done to
resolve the discrepancy between different n-3He coherent scattering length
measurements. The uncertainty in the triplet absorption cross section needs to
be experimentally determined to better precision, if other measurements of the
spin-dependent quantity $\Delta b^{\prime}$ are to be made. The authors hope
that this work along with the previous scattering length measurements can
improve future NN+3N models, and is part of the ongoing exploration into few-
body systems at the NIOF.

###### Acknowledgements.

We wish to thank John Fuller and Jeff Anderson at NIST for making the glass
target cells. The development and application of the polarized 3He cells and
methods used in this experiment was supported in part by the Department of
Energy, Basic Energy Sciences. Also we would like thank Sam Werner and Helmut
Kaiser for their helpful discussions. This work is supported by NIST and the
National Science Foundation through grants PHY-0555347, PHY-0855445, and
PHY-1205342.

## References

  * Lazauskas  _et al._ (2005) R. Lazauskas, J. Carbonell, A. C. Fonseca, M. Viviani, A. Kievsky, and S. Rosati, Phys. Rev. C71, 034004 (2005).
  * Viviani  _et al._ (2005) M. Viviani, A. Kievsky, L. Marcucci, and S. Rosati, Nucl. Phys. A751, 226 (2005).
  * Wiringa  _et al._ (1995) R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C51, 38 (1995).
  * Machleidt (2001) R. Machleidt, Phys. Rev. C63, 024001 (2001).
  * Stoks  _et al._ (1994) V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen, and J. J. de Swart, Phys. Rev. C49, 2950 (1994).
  * Stoks  _et al._ (1993) V. G. J. Stoks, R. A. M. Klomp, M. C. M. Rentmeester, and J. J. de Swart, Phys. Rev. C48, 792 (1993).
  * Pieper  _et al._ (2001) S. C. Pieper, V. R. Pandharipande, R. B. Wiringa, and J. Carlson, Phys. Rev. C64, 014001 (2001).
  * Pudliner  _et al._ (1997) B. S. Pudliner, V. R. Pandharipande, J. Carlson, S. C. Pieper, and R. B. Wiringa, Phys. Rev. C56, 1720 (1997).
  * Coon  _et al._ (1979) S. A. Coon, M. D. Scadron, P. C. McNamee, B. R. Barrett, D. W. E. Blatt, and B. H. J. McKellar, Nucl. Phys. A317, 242 (1979).
  * Robilotta and Coelho (1986) M. R. Robilotta and H. T. Coelho, Nucl. Phys. A460, 645 (1986).
  * Wiringa  _et al._ (2014) R. B. Wiringa, R. Schiavilla, S. C. Pieper, and J. Carlson, Phys. Rev. C 89, 024305 (2014).
  * Wiringa and Pieper (2002) R. B. Wiringa and S. C. Pieper, Phys. Rev. Lett. 89, 182501 (2002).
  * Viviani  _et al._ (2001) M. Viviani, A. Kievsky, S. Rosati, E. A. George, and L. D. Knutson, Phys. Rev. Lett. 86, 3739 (2001).
  * Black  _et al._ (2003) T. C. Black, P. R. Huffman, D. L. Jacobson, W. M. Snow, K. Schoen, M. Arif, H. Kaiser, S. K. Lamoreaux, and S. A. Werner, Phys. Rev. Lett. 90, 192502 (2003).
  * Epelbaum  _et al._ (2002) E. Epelbaum, A. Nogga, W. Glöckle, H. Kamada, U.-G. Meißner, and H. Witała, Phys. Rev. C66, 064001 (2002).
  * Entem and Machleidt (2003) D. R. Entem and R. Machleidt, Phys. Rev. C68, 041001 (2003).
  * Kirscher  _et al._ (2009) J. Kirscher, H. W. Griesshammer, D. Shukla, and H. M. Hofmann, “Universal Correlations in Pion-less EFT with the Resonating Group Model: Three, and Four Nucleons,” (2009), arXiv:0909.5606 [nucl-th] .
  * Bedaque  _et al._ (1999) P. F. Bedaque, H. W. Hammer, and U. van Kolck, Nucl. Phys. A646, 444 (1999).
  * Glyde (1995) H. R. Glyde,  _Excitations in Liquid and Solid Helium_ (Oxford University Press, 1995).
  * Sköld  _et al._ (1976) K. Sköld, C. A. Pelizzari, R. Kleb, and G. E. Ostrowski, Phys. Rev. Lett. 37, 842 (1976).
  * Glyde  _et al._ (2000) H. R. Glyde, B. Fåk, N. H. v. Dijk, H. Godfrin, K. Guckelsberger, and R. Scherm, Phys. Rev. B61, 1421 (2000).
  * Ioffe  _et al._ (1998) A. Ioffe, D. L. Jacobson, M. Arif, M. Vrana, S. A. Werner, P. Fischer, G. L. Greene, and F. Mezei, Phys. Rev. A58, 1475 (1998).
  * Schoen  _et al._ (2003) K. Schoen, D. L. Jacobson, M. Arif, P. R. Huffman, T. C. Black, W. M. Snow, S. K. Lamoreaux, H. Kaiser, and S. A. Werner, Phys. Rev. C67, 044005 (2003).
  * Huffman  _et al._ (2004) P. R. Huffman, D. L. Jacobson, K. Schoen, M. Arif, T. C. Black, W. M. Snow, and S. A. Werner, Phys. Rev. C70, 014004 (2004).
  * Ketter  _et al._ (2006) W. Ketter, W. Heil, G. Badurek, M. Baron, E. Jericha, R. Loidl, and H. Rauch, Euro. Phys. J. A27, 243 (2006).
  * Huber  _et al._ (2009a) M. G. Huber, M. Arif, T. C. Black, W. C. Chen, T. R. Gentile, D. S. Hussey, D. A. Pushin, F. E. Wietfeldt, and L. Yang, Physical Review Letters 103, 179903 (2009a).
  * Huber  _et al._ (2009b) M. G. Huber, M. Arif, T. C. Black, W. C. Chen, T. R. Gentile, D. S. Hussey, D. A. Pushin, F. E. Wietfeldt, and L. Yang, Physical Review Letters 102, 200401 (2009b).
  * Goldberger and Seitz (1947) M. L. Goldberger and F. Seitz, Phys. Rev. 71, 294 (1947).
  * Sears (1989) V. F. Sears,  _Neutron Optics_ (Oxford University Press, 1989).
  * Feenberg (1932) E. Feenberg, Phys. Rev. 40, 40 (1932).
  * Mughabghab (2006) S. F. Mughabghab,  _Atlas of Neutron Resonances: Resonance Parameters and Thermal Cross Sections Z=1-100_ (Elsevier, 2006).
  * Wolfs  _et al._ (1989) F. L. H. Wolfs, S. J. Freedman, J. E. Nelson, M. S. Dewey, and G. L. Greene, Phys. Rev. Lett. 63, 2721 (1989).
  * Williams and Rowe (2002) R. E. Williams and J. M. Rowe, Physica B311, 117 (2002).
  * Arif  _et al._ (1994) M. Arif, D. E. Brown, G. L. Greene, R. Clothier, and K. Littrell, Vibr. Monit. Cont. 2264, 20 (1994).
  * Pushin  _et al._ (2008) D. A. Pushin, M. Arif, M. G. Huber, and D. G. Cory, Phys. Rev. Lett. 100, 250404 (2008).
  * Huber  _et al._ (tion) M. G. Huber, D. Pushin, and C. B. Shahi, Nuclear Instruments & Methods (2014 In preparation).
  * Ioffe  _et al._ (1996) A. Ioffe, P. Fischer, T. Krist, and F. Mezei, J. Phys. Soc. Jap. Suppl. A65, 80 (1996).
  * Badurek  _et al._ (1976) G. Badurek, H. Rauch, A. Zeilinger, W. Bauspiess, and U. Bonse, Phys. Lett. A56, 244 (1976).
  * Clothier  _et al._ (1991) R. Clothier, H. Kaiser, S. A. Werner, H. Rauch, and H. Wölwitsch, Phys. Rev. A44, 5357 (1991).
  * Kaiser  _et al._ (1983) H. Kaiser, S. A. Werner, and E. A. George, Phys. Rev. Lett. 50, 560 (1983).
  * Klein  _et al._ (1983) A. G. Klein, G. I. Opat, and W. A. Hamilton, Phys. Rev. Lett. 50, 563 (1983).
  * Petrascheck (1988) D. Petrascheck, Physica B151, 171 (1988).
  * Rauch  _et al._ (1996) H. Rauch, H. Wölwitsch, H. Kaiser, R. Clothier, and S. A. Werner, Phys. Rev. A53, 902 (1996).
  * Gentile and Fuller (2007) T. R. Gentile and J. Fuller, J. Amer. Glass. Soc. , 21 (2007).
  * Walker and Happer (1997) T. G. Walker and W. Happer, Rev. Mod. Phys. 69, 629 (1997).
  * Bloch (1946) F. Bloch, Phys. Rev. 70, 460 (1946).
  * Chupp  _et al._ (2007) T. Chupp _et al._ , Nuclear Instruments and Methods A574, 500 (2007).
  * Newbury  _et al._ (1993) N. R. Newbury, A. S. Barton, G. D. Cates, W. Happer, and H. Middleton, Phys. Rev. A 48, 4411 (1993).
  * Fitzsimmons  _et al._ (1969) W. A. Fitzsimmons, L. L. Tankersley, and G. K. Walters, Phys. Rev. 179, 156 (1969).
  * McIver  _et al._ (2009) J. W. McIver, R. Erwin, W. C. Chen, and T. R. Gentile, Review of Scientific Instruments 80, 063905 (2009).
  * Chen  _et al._ (2014) W. C. Chen, T. R. Gentile, Q. Ye, T. G. Walker, and E. Babcock, Journal of Applied Physics 116, 014903 (2014).
  * Wildes (1999) A. R. Wildes, Rev. Sci. Inst. 70, 4241 (1999).
  * Rich  _et al._ (2002) D. R. Rich _et al._ , Nucl. Inst. Meth. A481, 431 (2002).
  * Shirane and Minkiewicz (1970) G. Shirane and V. J. Minkiewicz, Nucl. Inst. Meth. 89, 109 (1970).
  * Yerozolimsky (1999) B. G. Yerozolimsky, Nucl. Inst. Met. A420, 232 (1999).
  * Passell and Schermer (1966) L. Passell and R. I. Schermer, Phys. Rev. 150, 146 (1966).
  * Borzakov  _et al._ (1982) S. Borzakov, H. Malecki, L. Pikel ner, M. Stempinski, and E. Sharapov, Sov. J. Nucl. Phys. 35, 307 (1982).
  * Hofmann and Hale (2003) H. M. Hofmann and G. M. Hale, Phys. Rev. C68, 021002 (2003).
  * Hofmann and Hale (2008) H. M. Hofmann and G. M. Hale, Phys. Rev. C77, 044002 (2008).
  * Zimmer  _et al._ (2002) O. Zimmer, G. Ehlers, B. Farago, H. Humblot, W. Ketter, and R. Scherm, Euro. Phy. J. 4, 1 (2002).
  * Baryshevsky and Podgoretsky (1965) V. B. Baryshevsky and M. I. Podgoretsky, JETP 20, 704 (1965).
  * Abragam  _et al._ (1973) A. Abragam, G. L. Bacchella, H. Glätti, P. Meriel, M. Pinot, and J. Piesvaux, Phys. Rev. Lett. 31, 776 (1973).
  * Kitchens  _et al._ (1974) T. A. Kitchens, T. Oversluizen, L. Passell, and R. I. Schermer, Phys. Rev. Lett. 32, 791 (1974).
  * Kaiser  _et al._ (1977) H. Kaiser, H. Rauch, W. Bauspiess, and U. Bonse, Phys. Lett. 71, 321 (1977).
  * Kaiser  _et al._ (1979) H. Kaiser, H. Rauch, G. Badurek, W. Bauspiess, and U. Bonse, Physik A291, 231 (1979).
  * Alfimenkov  _et al._ (1981) V. Alfimenkov, S. B. Borzakov, V. V. Tkhuan, A. Govorov, L. Lason, L. Pikel′ner, and E. Sharapov, Sov. J. Nucl. Phys. 33, 467 (1981).
  * Amsler  _et al._ (2008) C. Amsler _et al._ , Phys. lett. B667 (2008).
  * Guckelsberger  _et al._ (2000) K. Guckelsberger, W. Nistler, R. Scherm, and M. Weyrauch, Physica B276-B278, 975 (2000).