# A Complete Approach to Determine the 3He neutron incoherent scattering
length $b_{i}$

H. Lu ID Indiana University/CEEM, 2401 Milo B. Sampson Lane, Bloomington, IN
47408, USA  O. Holderer ID Forschungszentrum Jülich GmbH, Jülich Centre for
Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), 85747 Garching,
Germany  A. Ioffe ID Forschungszentrum Jülich GmbH, Jülich Centre for Neutron
Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), 85747 Garching, Germany
S. Pasini ID Forschungszentrum Jülich GmbH, Jülich Centre for Neutron Science
(JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), 85747 Garching, Germany  P.
Pistel  Forschungzentrum Jülich GmbH, ZEA-1, 52425 Jülich Germany  Z. Salhi ID
Forschungszentrum Jülich GmbH, Jülich Centre for Neutron Science (JCNS) at
Heinz Maier-Leibnitz Zentrum (MLZ), 85747 Garching, Germany  B. M. Goodson ID
School of Chemical and Biomolecular Sciences, Southern Illinois University,
Carbondale, IL 62901, USA  W. M. Snow ID Indiana University/CEEM, 2401 Milo B.
Sampson Lane, Bloomington, IN 47408, USA  E. Babcock ID e.babcock@fz-
juelich.de Forschungszentrum Jülich GmbH, Jülich Centre for Neutron Science
(JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), 85747 Garching, Germany

(August 28, 2024)

###### Abstract

We report the first results from a new approach for measuring the 3He neutron
incoherent scattering length $b_{i}$. $b_{i}$ is directly proportional to the
difference $\Delta b=b_{+}-b_{-}$ in the two low-energy s-wave neutron-nucleus
scattering amplitudes $b_{+}$ and $b_{-}$, corresponding to the singlet $J=0$
and triplet $J=1$ states of the neutron-3He interaction, respectively. An
accurate measurement of $b_{i}$ can help distinguish among different models of
three-nucleon interactions by comparison to ab initio nuclear theory
calculations. The neutron birefringence caused by $\Delta b$ results in
neutron spin rotation around the nuclear polarization. We measured $\Delta b$
using polarized neutron spin rotation and the transmission of neutrons through
a 3He gas target polarized in situ by spin-exchange optical pumping. This
brief test measurement, conducted at the FZ-Jülich neutron spin echo
spectrometer at the Heinz Maier Leibnitz Zentrum (MLZ), yielded $\Delta
b=[-5.27\pm 0.05$ (stat.) $-0.05$ (syst.)] fm. We argue that this method can
be improved in precision to resolve the discrepancies between two prior
measurements of $b_{i}$ which are dependent on the polarized absorption cross
section $\sigma_{p}$. Further with absolute 3He polarization via NMR (in a
properly-shaped cell) concurrent with accurate neutron transmission
measurements, $\sigma_{p}$ can be measured to obtain independent values of
$b_{+}$ and $b_{-}$.

Precision measurements of the scattering amplitudes in the $n+^{3}$He system
provide important tests for ab initio theoretical calculations of the
properties of few-nucleon systems. Three-body (3N) interactions among nucleons
are now estimated to provide about 5% of the total binding energy of stable
nuclei Atkinson et al. (2020). The development of a global model of bound
nuclei that can both explain the binding energies of stable nuclei and can
also make reliable predictions out to the extremes of nuclear stability is a
major long-term goal for nuclear physics, with important scientific
applications for astrophysics and for our understanding of the process of
formation of the heavy elements Johnson et al. (2020). Although theoretical
models for the possible forms of nuclear three-body forces exist and give a
rough estimate for the relative sizes and spin/isospin dependence of three-
and higher-body effects compared to two-nucleon forces, more precise
experimental data on systems with few nucleons is needed to determine the
relative strengths of these forces. The binding energies of 3H, 3He, and 4He
are essential data for this purpose in the $A=4$ system and are measured with
high precision. Theoretical calculations of the binding energy of 4He using
the Green’s function Monte Carlo technique Kamada et al. (2001) including some
using phenomenological three-nucleon interactions Wiringa et al. (2000); Nogga
et al. (2002); Lazauskas and Carbonell (2004); Viviani et al. (2005) differ
from experiment by 1%. Theoretical analysis also shows that the information on
the nuclear three-body force from the binding energy of 4He is not independent
of that from three-body bound systems and is mainly sensitive to the spin-
independent component of the nuclear three-body force Pudliner et al. (1995);
Machleidt and Entem (2011); Carlson et al. (2015); Baroni et al. (2018). To
better constrain the spin-dependent parts of the nuclear three-body force,
data are required on the spin-dependent scattering of three- and four-body
systems with precision at the sub-percent level.

Existing data on the difference $\Delta b=b_{1}-b_{0}$ of the two scattering
lengths $b_{0}$ and $b_{1}$ for the two total angular momentum $J=0,1$ values
of n-3He are inconsistent. $\Delta b$ is proportional to the n-3He incoherent
scattering length $b_{i}$. The two best measurements of $b_{i}$ using neutron
spin echo Zimmer et al. (2002) and neutron interferometry Huber et al. (2014)
are inconsistent based on the quoted errors, as are the three measurements of
the n+3He coherent scattering length $b_{c}$ using neutron interferometry
Kaiser et al. (1979); Huffman et al. (2004); Ketter et al. (2006). Different
theoretical calculations of $b_{1}$ and $b_{0}$ available for comparison at
the time employing NN+3N interactions, such as the standard potential models
AV18 + UIX, AV18 +UIX + V3 Hofmann and Hale (2003, 2008), and AV18 + LL2
Kirscher et al. (2010), were also not in agreement. Improved measurements of
both $b_{i}$ and $b_{c}$ are needed to help resolve the inconsistencies shown
in previous work. Improved precision on both $b_{i}$ and $b_{c}$ can also help
distinguish among different models of few-nucleon interactions. The
description of the 4He continuum just above the $n+^{3}$He threshold is
challenging for existing theory to treat, and changes to existing 3N force
models are proposed as a possible solution to existing discrepancies.
Fortunately several new theoretical techniques have been developed to tackle
nuclear four and five body systems Nollett et al. (2007); Kievsky et al.
(2008); Navratil et al. (2016); Lynn et al. (2016); Lazauskas and Carbonell
(2020); Flores and Nollett (2022) including chiral effective theory Machleidt
and Entem (2011); Piarulli and Tews (2020). Different ab initio calculational
methods for nucleon scattering in $A=3$ systems deliver internally consistent
results Viviani et al. (2011, 2017), and the resonating group method (RGM) has
been applied in the past to $A=4$ systems Quaglioni and Navrátil (2008);
Navrátil et al. (2010). We judge that the prospects for improved theoretical
calculations in the $n+^{3}$He system are good.

In general the total free n-nucleus scattering length is given by
$a=a^{\prime}+ia^{\prime\prime}$ where $a^{\prime}$ and $a^{\prime\prime}$ are
real. The imaginary term $a^{\prime\prime}$ arises from absorption, which is
very large for the n-3He system. For the forward scattering amplitudes
measured in this work the bound scattering lengths are observed. The bound
scattering length is related to $a$ by $b=a(A+1)/A$ where $A$ is the nucleus
to neutron mass ratio. The two s-wave neutron-nucleus scattering amplitudes
$b_{+}$ and $b_{-}$, corresponding to the total nucleus plus neutron angular
momentum $J=I+1/2$ and $J=1-1/2$ scattering channels from a nucleus of spin
$I$ and neutron of spin $s$, can be expressed as

$b=b_{c}+{\frac{2b_{i}}{\sqrt{I(I+1)}}}{s}\cdot{I}.$ (1)

The coherent scattering length is thus

$b_{c}=\frac{(I+1)b_{+}+Ib_{-}}{(2I+1)}$ (2)

and the incoherent scattering length is

$b_{i}=\frac{I\sqrt{I+1}(b_{+}-b_{-})}{(2I+1)},$ (3)

which is directly proportional to $\Delta b=b_{+}-b_{-}$. The two values
$J=0,1$ of the total spin for $I=1/2$ imply $b_{+}\equiv b_{1}$ and
$b_{-}\equiv b_{0}$ for the triplet and singlet scattering lengths,
respectively.

$\Delta b$ can be measured by observing the precession of the neutron spin as
neutrons pass through a polarized nuclear target, named “pseudomagnetic
precession” Baryshevsky and Podgoretsky (1964) in the literature. Although
this phenomenon was initially described Baryshevsky and Podgoretsky (1964);
Abragam and Goldman (1982) in terms of a fictitious “pseudomagnetic field”
inside the medium, $\Delta b$ originates from neutron-nucleus scattering. The
optical theorem Sears (1989) relates the spin dependence of the neutron
optical potentials associated with the scattering amplitudes $b_{+}$ and
$b_{-}$ to a two-valued neutron index of refraction ($n_{+}$,$n_{-}$)
depending on the relative orientation of the neutron spin and the nuclear
polarization:

$\begin{split}n^{2}_{\pm}=1-{\frac{4\pi}{k^{2}}}N(b_{coh}+b_{\pm}),\\\ \\\
\Delta n=(n_{+}-n_{-})\approx-{\frac{2\pi}{k^{2}}}N(b_{+}-b_{-}),\end{split}$
(4)

where $N$ is the number of nuclei per unit volume, $k=2\pi/\lambda$ is the
neutron wave number, and the approximation in the second expression is valid
in our case as the neutron index of refraction is $\simeq 1$. $\Delta n$ makes
the medium optically birefringent for neutrons so that the two helicity
components of the neutron spin accumulate different phases, $kn_{\pm}d$, in
the forward direction as neutrons propagate a distance $d$ through the target.
Therefore neutron spins orthogonal to the nuclear polarization direction of
the target precess around the nuclear polarization by an angle
$\phi^{*}=k\Delta nd$.

We can write the neutron precession angle $\phi^{*}$ created by the incoherent
scattering length $b_{i}\propto{\Delta b}$ of the polarized 3He Zimmer et al.
(2002); Huber et al. (2014) as

$\phi^{*}=-{\frac{1}{2}}\lambda P_{3}Nd{\Delta b}=-\frac{2\lambda
P_{3}Nd}{\sqrt{3}}b_{i},$ (5)

where $P_{3}$ is the 3He polarization, $N$=[He] is the 3He density, and $d$ is
the neutron path length through the 3He. For nuclei such as 3He which possess
a very large spin-dependent component to the neutron cross section, one can
determine the constant of proportionality $\lambda P_{3}Nd$ using neutron
measurements and write the measured quantity $\Delta b$ as follows:

$\Delta b=\frac{2\phi^{*}}{\lambda
P_{3}Nd}=\frac{\sigma_{p}}{\lambda_{th.}}\frac{{2}\phi^{*}}{{\rm
cosh^{-1}}R}.$ (6)

Here $R$ is the ratio of unpolarized neutron transmission of polarized 3He,
$T(P_{3})$, to the transmission of unpolarized 3He, $T(0)$, $\sigma_{p}$ is
the polarized 3He spin dependent neutron absorption cross section, and
$\lambda_{th.}=1.798\leavevmode\nobreak\ $Å is the thermal neutron wavelength
chosen by convention as a reference point for neutron absorption cross
sections. The total n-3He absorption cross section
$\sigma_{a}=(4\pi/k)b^{\prime\prime}$ obtained from the imaginary part of $b$
by the optical theorem Sears (1989) can be written in terms of the
polarization-independent and polarization-dependent terms as:

$\sigma_{a}=\sigma_{un}\mp P_{3}\sigma_{p},$ (7)

where the sign convention $\mp$ is for $P_{3}$ parallel (-) or anti-parallel
(+) to the neutron spin. Here $\sigma_{un}=(5333\pm 7)$ barn is the total
unpolarized neutron absorption cross section, and $\sigma_{p}$ can be
expressed as $\sigma_{p}=(1-\sigma_{1}/\sigma_{un})\sigma_{un}$. Both
$\sigma_{a}$ and $\sigma_{un}$ are measured to be proportional to the neutron
wavelength $\lambda$ to high precision Als-Nielsen and Dietrich (1964);
Borkazov et al. (1982); Keith et al. (2004). For an unpolarized neutron beam
of $n$ neutrons with half spin up ($n^{+}$) and half spin down ($n^{-}$)
neutrons the corresponding transmission of Eq. 7 is

$T^{\pm}={\frac{n^{\pm}}{n}}={\frac{1}{2}}{\rm exp}\left(-(\sigma_{un}\mp
P_{3}\sigma_{p}){\frac{\lambda Nd}{\lambda_{th.}}}\right).$ (8)

Thus, the transmission of unpolarized neutrons through polarized 3He is:

$T(P_{3})={\rm exp}\left(-{\frac{\sigma_{un}}{\lambda_{th.}}}\lambda
Nd\right){\rm cosh}\left({\frac{\sigma_{p}}{\lambda_{th}}}\lambda
P_{3}Nd\right).$ (9)

Since the unpolarized transmission is simply:

$T(0)={\rm exp}\left(-{\frac{\sigma_{un}}{\lambda_{th.}}}\lambda Nd\right)$
(10)

with two neutron transmission measurements giving $R=T(P_{3})/T(0)$ one
directly experimentally obtains the product
$\frac{\sigma_{p}}{\lambda_{th.}}\lambda P_{3}Nd$ from ${\rm cosh}^{-1}R$,
leading to eq. 6.

Thus one still needs to determine $\sigma_{p}$. If the triplet absorption rate
$\sigma_{1}$ were zero, so that there would be only absorption in the singlet
state $\sigma_{0}$, then $\sigma_{p}=\sigma_{un}$, where $\sigma_{un}$ is
known to $\simeq 0.1\%$. However the upper bound on $\sigma_{p}$ from previous
experiments is several-percent Passell and Schermer (1966); Borzakov et al.
(1982) and limits the precision of this technique. There is no reason to
expect that $\sigma_{1}$ is zero, as theoretical calculations show Hofmann and
Hale (2003, 2008). The work in Ref. Zimmer et al. (2002) used an average of
the experimental determinations Passell and Schermer (1966); Borzakov et al.
(1982) to arrive at a value that can be reinterpreted as $\sigma_{1}=57$ barn.
Conversely the work in Ref. Huber et al. (2014) used a combination of
theoretical predictions and the measured thermal absorption cross section to
estimate $\sigma_{1}=24$ barn Huber . For lack of better knowledge of
$\sigma_{1}$, we will use the latter value in our analysis below but also
present the result independent of the value of $\sigma_{1}$ as was also done
in Huber et al. (2014) for comparison purposes.

3He SEOP cells can also enable a measurement of $\sigma_{p}$. An independent
0.1$\%$ measurement of $P_{3}$ combined with accurate measurements of
$R(\lambda)$ through an in-situ polarized 3He sample using the time-of-flight
(TOF) method (i.e. a wavelength scanned neutron beam) as in Ref. Chupp et al.
(2007) could provide a $\simeq 0.1\%$ accuracy for $\sigma_{p}$, allowing
determination of $\sigma_{1}$ to $\approx 5$ barn accuracy. Several atomic
physics methods have determined $P_{3}$ to high precision Romalis and Cates
(1998); Babcock et al. (2005); Nikolaou et al. (2014); Wilms et al. (1997) so
this approach should be feasible.

In order to determine $P_{3}$ on a neutron beamline, we propose to use the
“self-magnetometry” of a polarized 3He sample in a defined shape such as in a
long tube parallel or perpendicular to the applied $B_{0}$ field. The 3He
magnetization $M_{3}=\mu_{3}P_{3}N$ will generate a magnetic field of

$B_{3}=\mu_{0}M_{3}\left(1-{\frac{2}{3}}\right)=\mu_{0}{\frac{M_{3}}{3}}$ (11)

when the tube’s axis is parallel to $B_{0}$ and

$B_{3}=\mu_{0}M_{3}\left({\frac{1}{2}}-{\frac{2}{3}}\right)=\mu_{0}{\frac{-M_{3}}{6}}$
(12)

when the tube’s axis is perpendicular to $B_{0}$ Vlassenbroek et al. (1996);
Romalis and Cates (1998). Here the first term is the magnetization minus the
demagnetization factor and the $-2/3$ term, which is the same as the field
from a spherical volume, arises from the scalar contact term, meaning the 3He
spins are non-overlapping and cannot “see” one another so the self field must
be subtracted Romalis et al. (2014); Limes et al. (2019). We note the magnetic
moment of 3He $\mu_{3}/h$=-16217050 Hz/T is known to the ppb level Schneider
et al. (2022), and the the geometric correction factor in the field parallel
case for a cell with a length to diameter ratio $\simeq 5$ is $\simeq 2\%$ and
very well known Chen et al. (2006); Joseph (1966). Given at one bar pressure
at 25 ∘C there are $2.43\times 10^{25}$ atoms m3 and the gyromagnetic ratio of
3He, $\gamma_{3}=3.24\times 10^{7}$ Hz/T, the product
$f_{3}=\mu_{0}\mu_{3}P_{3}N\gamma_{3}=10.6$ Hz at $P_{3}=1$ and $N=1$ bar.
Thus for the field-parallel case, upon a reversal of $P_{3}$, an NMR frequency
shift of

$\Delta
f_{3}=2B_{3}\gamma_{3}=2\mu_{0}\frac{M_{3}}{3}\gamma_{3}=\frac{2}{3}\mu_{0}\mu_{3}P_{3}N\gamma_{3}\simeq
5\leavevmode\nobreak\ {\rm Hz}$ (13)

will be observed for $P_{3}=0.70$ at 1 bar pressure. Since $\lambda Nd$ of eq.
9 can be calibrated by unpolarized $T(0,\lambda)$ measurements using the well-
known $\sigma_{un}$ and given that one can expect $<5$ Hz NMR linewidths, 0.1%
accuracy in $P_{3}$ and thus $\sigma_{p}$ should be attainable for normal
pressures by signal averaging. This method would not require new on-beamline
detection methods to be developed other than a specialized cell for the
purpose. In-situ polarization of the 3He employing adiabatic fast passage
(AFP) 3He flipping and a TOF neutron beamline would be preferred.

Since a recent measurement of $b_{c}$ in n$+^{4}$He using perfect crystal
neutron interferometry Haun et al. (2020) reached $10^{-3}$ precision using a
technique that can be directly applied to 3He, our ideas to improve $b_{i}$
are the key additional input needed to confront theory. Therefore we intend to
perform a higher precision measurement of $\Delta b/\sigma_{p}$ and a
measurement of $\sigma_{p}$ to obtain an absolute value for $b_{i}$, which
could then also approach a $10^{-3}$ precision.

We have tested an accurate method to determine the real part of $\Delta
b/\sigma_{p}$ on the J-NSE Phoenix instrument Pasini et al. (2019) as part of
a different experiment to measure the same quantity for n-129Xe and n-131Xe Lu
et al. (2022). Our approach builds on the pioneering work of Zimmer et al.
Zimmer et al. (2002) by taking advantage of technical improvements in neutron
spin echo spectroscopy to measure neutron birefringence and by exploiting the
improved time stability and improved performance of polarized 3He gas targets
created using spin-exchange optical pumping (SEOP) Walker and Happer (1997).

Figure 1: A schematic drawing of the J-NSE neutron spin echo spectrometer
showing the coil arrangement and the SEOP-polarized 3He cell. Not shown are
optical pumping lasers whose light is perpendicular to the neutron flight path
and directed along $B_{SEOP}$ via a 45∘ mirror above the cell, and the oven
used to regulate the cell temperature for SEOP.

Measurements of neutron birefringence in polarized nuclei were originally
performed using the Ramsey method of separated oscillatory fields Ramsey
(1956, 1990). This so-called pseudomagnetic precession method Baryshevsky and
Podgoretsky (1964) uses two oscillating fields before and after a solid-state
nuclear-polarized sample to measure phase precessions caused by the sample. We
use a variation that also uses orthogonally-precessing polarized neutrons
moving through a nuclear polarized 3He gas sample but in a neutron spin-echo
(NSE) spectrometer Mezei (1972) in order to quantify the resulting phase
shifts in the neutron precession. NSE is similar to spin echo in NMR Hahn
(1950) but the neutron spin is precession-encoded in space along the path of
the traveling beam as opposed to in time with static spins as in NMR spin
echo.

In a NSE spectrometer, polarized neutrons are first flipped by $\pi/2$ to
induce precession in the orthogonal plane, they then pass through a high-field
flight path with an over 1 T$\cdot$m field integral encoding a large number of
spin precessions; this step is followed by a $\pi$ flip reversal of the
neutron polarization in the middle of the instrument and then by a second
high-field flight path identical to the first to decode the spins. The sample
is typically near the middle either before or after the $\pi$ flipper and the
additional precession it creates can be quantified by matching it to the
precession in additional phase (compensation) coils placed around the neutron
flight path. Thus the NSE method is like the Ramsey technique, but the
addition of the central $\pi$ flipper allows the sample-induced phase shift to
be quantified by DC phase coils rather than phase matching an oscillating RF
field directly. NSE has the benefit that, because the phase coils have field
integrals accurate to nT$\cdot$m compared to total instrument field integrals
of 1 T$\cdot$m or more, it can encode the spins very precisely and measure
very small changes in the neutron precession Pasini et al. (2019).

A schematic diagram of a NSE spectrometer is shown in Fig. 1. The nuclear
spins of the 3He sample were polarized in-situ using spin-exchange optical
pumping Walker and Happer (1997) in the usual sample area of the NSE
spectrometer after the $\pi$ flipper. The $B_{SEOP}$ field oriented
perpendicular (vertical) to the neutron flight path and the fields $B_{1}$ or
$B_{2}$ of the NSE spectrometer itself, which are longitudinal (i.e.
horizontal; see Fig. 1).

Because of the work on Xe isotopes the gas target was polarized in-situ to
maintain a steady nuclear spin polarization. For Xe this was mandatory because
of their comparatively short $T_{1}$ polarization lifetimes compared to the
duration of a typical NSE scan ($\simeq$20 min), especially 131Xe where
$T_{1}<30$ s. For the 3He experiment this also turned out to be advantageous
by decoupling time-dependent instrumental drifts from changes in $P_{3}$.
Using a home-built NMR system for free-induction decay detection, we were able
to determine that fractional changes to $P_{3}$ were below $0.3\%$ during our
NSE measurements. The in-situ polarization equipment Salhi et al. (2014) also
enabled on-beam AFP flipping of $P_{3}$ during continuous pumping. The
continuous polarization allows any time-dependent phase drifts in the NSE
spectrometer to be fit as a time-dependent background, and the AFP flipping
eliminates systematics due to possible non-perfect neutron spin-flips or non-
adiabatic transport of the neutron polarization through our setup.

A 5 cm diameter cylindrical 3He SEOP cell made of GE180 glass with about 0.4
bar of 3He was used Salhi et al. (2014). This cell has somewhat rounded ends
with a path length of 4.8 cm through its center. Ref. Salhi et al. (2019)
describes the SEOP instrumentation used to polarize the 3He spins. In contrast
to that neutron polarizer device, here the vertical magnetic field for SEOP
was provided by a set of 70 cm diameter Helmoltz coils for added flexibility
and to satisfy space constraints on the J-NSE Phoenix instrument; all other
components such as the lasers, NMR devices, and controls were taken directly
from the device in Salhi et al. (2019). High-fidelity data was obtained for a
6 cm2 area in a half-circle in the neutron-illuminated central portion of the
cell where the path length is approximately uniform.

A typical NSE scan is made by measuring the amplitude of the neutron
polarization vector as the phase-coil is scanned in small steps around the
point where the two NSE precession regions are balanced in field integral.
This action produces a spin echo envelope that shifts in proportion to
precession angle $\phi^{*}$ of the sample. The J-NSE Phoenix spectrometer
employs a position-sensitive detector allowing independent determination of
$\Delta b$ for approximately each 0.5 cm $\times$ 0.5 cm region of the 3He
cell, which can then be averaged. This practice eliminates problems that might
otherwise arise from a non-uniform neutron path length though the 3He cell. A
pair of NSE scans for the two states of the 3He polarization from one such
pixel is shown in Figure 2.

The NSE signal $I(B_{1})$ detected transmitted intensity after the neutron
polarization analyzer as a function of the phase coil field $B_{1}$. In the
expression:

$I(B_{1})=I_{0}[1-p\int d\lambda f(\lambda)cos(\phi_{1}-\phi_{2})],$ (14)

$\phi_{1}$ and $\phi_{2}$ are the precession angles in the first and second
precession coils, respectively, $f(\lambda)$ is the neutron wavelength
distribution, and $p$ measures the loss of contrast of the interference
pattern from neutron polarization efficiencies. The instrument records the
neutron intensity in each detector pixel as a function of the current of the
phase coil. The neutron wavelength distribution transmitted by the velocity
selector on this beamline is well fit by a triangular function, so the
resulting NSE signal is this form convoluted with a cosine function which is
used to fit the data for analysis.

Figure 2: Spin echo signals from the 3He target versus the current in the
phase coil ($I_{trim}$ )for one pixel of the neutron detector. The two NSE
profiles correspond to $P_{3}$ parallel and antiparallel to $B_{SEOP}$.

The 3He $b_{i}$ data reported here was obtained to verify the method for the
Xe measurement Lu et al. (2022), and thus time limited. 3He was measured 6
hours each for polarized and unpolarized 3He. The data was taken in the
following pattern: 2 NSE scans with $P_{3}$ positive; two scans with $P_{3}$
flipped to the negative state; two in the positive state; and 6 hours of scans
with $P_{3}$=0. Although not needed for the determination of $b_{i}$,
$P_{3}=70.6\pm 1.6\%$ was determined using $R$ and a value of [He]=$0.3556\pm
0.001$ bar from a separate transmission measurement using neutron TOF on the
FIGARO instrument Campbell et al. (2011). A graph of the phase versus time for
one of the pixels is shown in Fig. 3. This data was then fit for to a step
function with a linear time-dependent background to determine the measured
shift $\Delta\phi=2\phi^{*}$ for $+P_{3}$ to $-P_{3}$ from each pixel.

The $R$ measurement was performed by taking the weighted average of the mean
intensity value of the NSE signal during the NSE scans for the $+P_{3}$ and
$-P_{3}$ states to determine $T(P)$, and the mean intensity of the unpolarized
3He NSE scans to determine $T(0)$. This process was also performed for each
pixel of the NSE scan over our region of interest to account for any
variations in $d$ resulting from the cell’s shape and alignment in the neutron
beam. Using the mean value compensates for the small difference in
transmission of the positive versus negative $P_{3}$ due to a small residual
neutron polarization along $B_{SEOP}$.

Figure 3: Phase data from one pixel of the NSE detector over the entire
experiment. $b_{i}$ values from 36 pixels representing the center of the 3He
cell were averaged for the final result.

The measured $\Delta\phi$ value could have a small correction due to the
magnetic dipole field of the polarized gas, causing an extra precession signal
in the neutron phase that is also proportional to $P_{3}$. This effect was
discussed in Ref. Zimmer et al. (2002), however that work did not account for
the scalar-contact term that one must include for real particles Romalis et
al. (2014); Limes et al. (2019), which is different than one would expect from
the simple classical result where it is assumed that one can have non-
interacting and overlapping point-like particles. For n+3He the contact
interaction should be 0 for any reasonable first order approximations.
Therefore along their flight path neutrons do not sample the classical field
of the individual polarized 3He nuclei inside the cell. The field experienced
will consist of the long range dipole fields caused by a non-spherical
geometry inside the polarized cell and, since the neutron precession also
integrates the field along their entire flight path, the classical field of a
polarized (magnetized) volume outside of the cell. For the geometry of our
experiment this would lead to a correction $<0.07\%$ and is not relevant here.
However the general arguments of the fields experienced by nonzero spin
particle beams through polarized volumes is of interest for precision
measurements so we include a discussion of this effect in a supplement sup .

For neutron velocity selector instruments there is a small correction for the
triangular wavelength distribution caused by the wavelength-dependent
attenuation through the 3He target. Using the arguments described in the O.
Zimmer work Zimmer et al. (2002) this leads to a negligible correction factor
of $1.0003$ for our case because of the relatively small $Nd$ of the cell
used. This correction would not be needed for an instrument that uses TOF to
determine the transmitted spectrum. Our global detector count rates were $1\%$
or less of the total detector deadtime of 400 ns (e.g maximum count rate of
2.5 MHz) so detector deadtime corrections are also negligible.

Using this analysis we arrive at the final value of $\Delta b=[-5.27\pm 0.05$
(stat.) $-0.05$ (syst.)] fm using the values of $\sigma_{un}=5333(7)$ barn and
an estimated $\sigma_{p}=\sigma_{un}-\sigma_{1}=5309$ barn. From eq. 5 $\Delta
b=4b_{i}/\sqrt{3}$ gives the 3He neutron incoherent scattering length.
Rewriting the result to be independent of $\sigma_{p}$ as in Huber et al.
(2014); Huber  we obtain:

$\frac{\Delta b}{\sigma_{p}}=(-9.93\pm 0.09({\rm stat.})-0.09({\rm
syst.}))\times 10^{-4}\leavevmode\nobreak\ {\rm\frac{fm}{b}}.$ (15)

This value compares to $\Delta b/\sigma_{p}=(-10.1929\pm-.0760)\times 10^{-4}$
fm/b for the work of Ref. Huber et al. (2014) and $\Delta
b/\sigma_{p}=(-10.3628\pm-.0180)\times 10^{-4}$ fm/b for the work of Ref.
Zimmer et al. (2002). This preliminary data shows we can readily obtain our
target of $10^{-3}$ precision for $\Delta b/\sigma_{p}$ with a dedicated
measurement. The value using this basic method is free of systemics to our
knowledge. The variation of the preliminary result above with respect to
previous measurements could be attributed to slow experimental drifts and that
we only reversed the $P_{3}$ one time. The cell used had non-uniform neutron
flight path and a minor shift in the cell position could lead to a one-sided
error, estimations from shifting the center position of the data result in the
given systematic error here.

We implemented improvements to the measurement technique pioneered by Zimmer
et al. (2002) such as in-situ polarization of the 3He gas and the ability to
reverse the 3He polarization using AFP. The in-situ polarization approach
decouples the measured $\phi^{*}$ from time-dependent drifts that could
falsely correlate with $P_{3}$, and prevents possible inconsistencies induced
by removal and replacement of the 3He cell. The position-sensitive
determination of the cross section reduces possible path length errors, which
could be further reduced by using a flat-windowed 3He SEOP cell to eliminate
variations in path-length across the beam over time, and the AFP flipping
cancels errors from small residual longitudinal neutron polarization. Use of a
TOF NSE instrument such as the SNS-NSE Ohl et al. (2012) will eliminate the
neutron velocity selector correction. By increasing the measurement time to 1
week and using a cell with an optimized $Nd$ to minimize error propogation
from the cosh${}^{-1}(R)$ term we estimate one could reach a statistical
accuracy of $<0.1\%$ (or 0.005 fm). Previous work Musgrave et al. (2018) shows
that the transmission measurements needed to measure the proportionality
factor between $\phi^{*}$ and $\Delta b/\sigma_{p}$ can indeed be conducted
with the required precision. With the additional measurement of $\sigma_{p}$
to a comparable precision, a total $10^{-3}$ precision on $b_{i}$ for 3He can
be attained.

###### Acknowledgements.

H. Lu and W. M. Snow acknowledge support from US National Science Foundation
(NSF) grants PHY-1913789 and PHY-2209481 and the Indiana University Center for
Spacetime Symmetries. H. Lu received a Short-Term Grant, 2019 no. 57442045
from DAAD the German Academic Exchange Service. B.M. Goodson acknowledges
support from the NSF (CHE-1905341), DoD (W81XWH-15-1-0272, W81XWH2010578), and
a Cottrell Scholar SEED Award from Research Corporation for Science
Advancement. P. Guthfreund (ILL) and K Zhernenkov performed a calibration
measurement of $N$ (i.e [He]) on FIGARO Campbell et al. (2011) aiding this
work. We acknowledge G.M. Schrank for discussions and M. Huber for detailed
discussions of NIST work on $b_{i}^{3}$ and estimates of $\sigma_{1}$ for 3He.

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##  I Supplement

As described in the main paper, the measured $\Delta\phi*$ value could have a
small correction due to the neutron path integrated magnetic dipole field of
the polarized gas, causing an extra precession signal in the neutron phase
that is also proportional to $P_{3}$. This effect was discussed in Ref. Zimmer
et al. (2002), however that work did not account for the scalar-contact term
that one must include for real particles Romalis et al. (2014); Limes et al.
(2019), which is different than one would expect from the simple classical
result where it is assumed that one can have non-interacting and overlapping
point-like particles. For n-3He the contact interaction should be 0 for any
reasonable first order approximations, so the neutrons do not experience the
classical field of the individual polarized 3He nuclei inside the cell, but
only the integral of the long range dipole fields caused by a non-spherical
geometry inside the polarized cell Limes et al. (2019) and, since the neutron
precession also integrates the field along their entire flight path, the
classical field of a polarized (magnetized) volume outside of the cell.

The scalar contact term is $2\mu_{0}\delta^{3}({\bf r}){\bf m}/3$ for non-
interacting point dipoles Vlassenbroek et al. (1996) leading to a field of
$2/3\mu_{o}M$ where we replace $M_{3}$ for 3He specifc arguments by the
magnetization $M$. For this reason a parameter called $\kappa$ is introduced
to represent the actual magnitude of the contact interaction between
dissimilar particles as a ratio to that of the “classical field”. Thus,
$\kappa>0$ is often referred to as an enhancement factor, such as the case for
alkali-metal electrons with a noble gas (e.g. during SEOP) that experience
electron paramagnetic resonance shifts much larger than “expected” from the
classical field due to the contact interaction Romalis and Cates (1998). For
nuclei (or nucleons), the direct contact interaction is 0, but indirect $J$
couplings can still cause a (small) finite effect Limes et al. (2019).

The field external to the polarized gas volume retains the classical result
for a magnetized object; however, as described in Ref. Limes et al. (2019) the
apparent field experienced by spins inside the volume due to the dipole fields
and the contact interactions can be parameterized as

$B^{\prime}_{\rm inside}=\mu_{0}\left[M-\eta
M+{\frac{2}{3}}(\kappa-1)M\right],$ (16)

where here we need not worry about a field average as the neutron will sample
a line integral through the cell. Here $M$ is the magnetization of the noble
gas nuclei, $\kappa$ discussed above describes the spin-spin interaction, and
$\eta$ is referred to as the geometric demagnetization factor, i.e. this term
parametrizes the modification of the field inside a magnetized volume due to
its shape. The first term in Eq. 16 is the magnetization of the sample, the
second term is the $H$ from classical arguments, and the last term is the
quantum mechanics-required correction of the scalar contact interaction. One
can see here that for $\kappa=1$, i.e. the case with perfectly non-interacting
particles, one obtains the classical result. For our work, given that any
possible 3He-neutron coupling will lead to $\kappa=0$ in the first order
(which will be sufficient for this correction factor), assuming the classical
limit would result in an over-correction.

The values of $\eta$ have been calculated for normal geometries used in
polarization studies such as cylinders Chen et al. (2006); these values can be
solved for analytically in many cases or numerically. The most recognizable
“demagnetization” factor from classical electrodynamics is $\eta=1/3$ for a
sphere, which allows one to immediately obtain the “classical field” inside a
sphere, $B=2/3\mu_{0}M$ for example. Here as an additional note, for the case
of like particles where $\kappa=0$, by definition we see that $B^{\prime}_{\rm
inside}=0$; this is the reason why a magnetized sphere of hyperpolarized noble
gas nuclei do not experience NMR frequency shifts due to their own
magnetization, whereas the alkali-metal electrons notably not only experience
EPR frequency shifts due to the polarized noble gas, but a much larger shift
increased by $\kappa$ due to the enhanced probability of the alkali metal
electron being located at the noble gas nucleus in these systems Romalis and
Cates (1998); Babcock et al. (2005).

If the neutrons only experienced the field inside the cell, we would need to
calculate the “demagnetization” factor (i.e. edge effects) or $H$ for the
particular cell geometry in question; however, the spin precession of the
neutrons samples the cell’s field over the entire neutron flight path,
including outside the cell. It can be shown that for a (classical spherical)
dipole and thus an assembly of arbitrary dipoles that:

$\int_{\infty}^{\infty}H_{dipole}(z)dz=\left({\frac{1}{3}}-{\frac{1}{3}}\right)M=0,$
(17)

where $z$ is the axis of the dipole, i.e the direction of magnetization, and
also the direction of the neutron flight path in works such as Ref. Zimmer et
al. (2002). Here $1/3\mu_{0}M$ is the integral outside the sphere and
$-1/3\mu_{0}M$ is the well-known value of $H$ inside a spherical dipole. Thus
for work such as Zimmer et al. Zimmer et al. (2002), where the “self” field of
the cell’s dipole is aligned with the direction of neutron propagation, for a
spherical reference volume one quickly realizes that since outside the volume
$B=\mu_{0}H$ and inside the volume the effective field is described by Eq. 16
with $\kappa=0$, the additional precession experienced by the neutrons for the
case of will be

$\phi^{M_{\parallel}}={\frac{1}{3}}\mu_{0}Ml\frac{\gamma_{n}}{v},$ (18)

where $\gamma_{n}=2.98\times 10^{7}$ Hz/T is the gyromagnetic ratio of the
neutron, $v$ is the neutron velocity and $l$ the neutron path length through
the cell. Since we integrate over the whole neutron flight path, this result
will also hold for an assembly of dipoles in the shape of the 3He cell.
Therefore the data in that work was over-corrected by
$2/3\mu_{0}Ml\gamma_{n}$, the amount of the reduction in the apparent field,
$B^{\prime}$, from the scalar contact term. This would raise their result by
$\approx 0.15\%$. It is interesting to note that this result is analogous to
the result for the NMR frequency of an infinitely long cylinder parallel to
the $B_{0}$ field given in eq. 11 in the main paper where $\eta=0$
Vlassenbroek et al. (1996). Here we recover the same result as that for an
infinitely long sample, i.e. the solenoid-like field of a uniformly magnetized
rod minus the scalar contact term, because the full integrated field now also
includes the field outside of the cell as sampled by the neutrons.

For our geometry the case is different: we have a magnetized cylinder where
the cylinder axis, parallel to the neutron flight path, is perpendicular to
the holding field $B_{0}$ and the gas magnetization. For consistency we will
leave “z” as the direction of the neutron flight path and use “y” as the
direction of the noble gas polarization. Here the integral of $H$ from
$-\infty\rightarrow+\infty$ becomes:

$\int_{\infty}^{\infty}H_{dipole}(y)dz=-\left({\frac{1}{6}}+{\frac{1}{3}}\right)M=-\frac{1}{2}M.$
(19)

From the usual expression of a dipole field one sees that on the equatorial
plane, the field direction external to a dipole is reversed and half the
magnitude for any distance. Thus the integral along the orthogonal path to the
dipole is $-1/6\mu_{0}M$ and the additional precession experienced by the
neutrons is

$\phi^{M_{\perp}}=-{\frac{1}{6}}\mu_{0}Ml{\frac{\gamma_{n}}{v}},$ (20)

again assuming that $\kappa_{0}\simeq 0$. Like the prior result, for the case
of cell magnetization perpendicular to the $B_{0}$ field we recover the
relation for an infinitely long sample as in eq. 12 of the main paper for
which $\eta=1/2$ Vlassenbroek et al. (1996), again because the neutrons also
sample the field outside the cell (the - is defined by convention in eq. 16
for the demagnetization). Substituting in values for the $P_{3}$ and $N_{3}$
of the work here this correction would be +0.068% and is over an order of
magnitude below our statistical errors and would only be relevant for 0.1% or
better measurements of $b_{i}$. For the work of Ref. Zimmer et al. (2002) this
interpretation would raise their cited value by $0.15\%$ as the reported value
was over-corrected by $2/3\mu_{0}M_{3}$, the amount of the scalar contact
term.

For the work in the main paper this correction would be $<0.1\%$. However the
effect is clearly of importance to precision measurements with nonzero spin
particle beams through polarized targets and should not be de-facto neglected.