ARTICLE IN PRESS

Nuclear Instruments and Methods in Physics Research A 611 (2009) 235–238

Contents lists available at ScienceDirect

Nuclear Instruments and Methods inPhysics Research A

0168-90

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/nima

Precision measurement of the neutron-3He spin-dependent scattering lengthusing neutron interferometry

M.G. Huber a,�, F.E. Wietfeldt a, T.R. Gentile b, W.C. Chen b,c, M. Arif b, D.A. Hussey b, D.A. Pushin d,L. Yang e, T. Black f

a Tulane University, New Orleans, LA 70188, USAb National Institute of Standards and Technology, Gaithersburg, MD 20899, USAc Indiana University, Bloomington, IN 47408, USAd Massachusetts Institute of Technology, Cambridge, MA 02139, USAe Stanford Linear Accelerator, Menlo Park, CA 94025, USAf University of North Carolina-Wilmington, Wilmington, NC 28403, USA

a r t i c l e i n f o

Available online 6 August 2009

Keywords:

Neutron interferometry

Polarized 3He

Few-body systems

02/$ - see front matter Published by Elsevier

016/j.nima.2009.07.062

esponding author.

ail address: mhuber@tulane.edu (M.G. Huber)

a b s t r a c t

There is strong theoretical interest in the study of few-body nucleon systems. Experimental

measurements of neutron scattering lengths are able to perform precise tests of nucleon–nucleon

models. Neutron interferometry provides some of the most precise values of spin-independent neutron

scattering lengths, bc, including measurements for n-H, n-D, and n-3He to better than a percent relative

uncertainty. The spin-dependent neutron scattering length, bi , of n-3He has been measured once before

by Zimmer et al. using a polarized 3He target inside a spin echo apparatus. Their result along with

measurements of the spin-independent scattering length of n-3He disagree with current theoretical

models. An experiment to measure bi in n-3He to less than 1% relative uncertainty has been conducted

at the Neutron Interferometer and Optics Facility at the National Institute of Standards and Technology

using small, flat-windowed 3He cells. This experiment has different systematics than that of the spin

echo measurement, and is the first use of a polarized gas target in a neutron interferometer.

Published by Elsevier B.V.

1. Introduction

Recent interest in few-body nuclear systems is fueled byadvances in theoretical approaches which now predict fewnucleon properties such as scattering lengths to the 10�3 levelof precision. Experimental limits in measuring scattering lengthshave also recently achieved this precision. Neutron interferometry(NI) has provided precise values of spin-independent neutronscattering lengths, bc, to better than a percent for n-H, n-D [1], andn-3He [2]. The spin-dependent neutron scattering length, bi, isdifficult to measure because it requires a polarized beam andtarget. For 3He, bi has been measured only once before by Zimmeret al. [3] using pseudomagnetic spin precession in a spin echoapparatus. Their result bi ¼ ð�2:36570:020Þ fm combined withspin-independent data is inconsistent with theoretical models bymore than 4s, where s is the standard uncertainty.

Theoretical calculations using realistic nucleon–nucleon (NN)potentials such as Argonne AV18 [4], CD-Bonn [5], and Nijmegen[6] do not produce the experimentally determined bindingenergies of 3He and 3H by several hundred keV thus demonstrat-

B.V.

.

ing the need for the inclusion of additional three nucleoninteractions (3N). Popular three nucleon interactions such asUrbana UIX [7] or V�3 [8] added to NN models correct thediscrepancy in binding energies, but at the expense of otherobservables. Low-energy neutron scattering lengths providecrucial tests of various theoretical approaches to NNþ 3N models.

The n-3He scattering length is also important to effective fieldtheory (EFT) approaches. EFT methods are based on expandedQCD Lagrangians separated by the pion mass into two distinctenergy regions. In EFTs calculations can be explicitly carried out inthe lower energy region, but the higher energy region requires theinput of experimentally determined low-energy observables suchas scattering lengths to parameterize mean-field behavior.Effective field theories are attractive because they provide cleartheoretical uncertainties from estimates of the relative contribu-tion of higher order terms [9].

2. Optical theory

In low-energy neutron scattering the amplitude of thescattered wave is described by its scattering length, b. For theinteraction between a neutron with spin rn and a nucleus with

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Fig. 1. The n-3He experiment (not to scale). A polarized neutron beam (red)

entering from the left Bragg diffracts within the first blade of the interferometer

coherently splitting the neutron into two separate paths. 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 due to 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. A third 3He detector labeled C4 monitors the

decay of the 3He polarization. (For interpretation of the references to color in this

figure legend, the reader is referred to the web version of this article.)

M.G. Huber et al. / Nuclear Instruments and Methods in Physics Research A 611 (2009) 235–238236

spin I the scattering length is

b ¼ bc þ2biffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

IðI þ 1Þp rn � I ð1Þ

where bc is called the coherent scattering length and has beenmeasured recently using unpolarized 3He gas as bc ¼

ð5:85370:007Þ fm [2] by the Neutron Interferometer and OpticsFacility (NIOF) at the National Institute of Standards andTechnology (NIST) and separately at the Institut Laue-Langevin(ILL) as bc ¼ ð6:01070:021Þ fm [10]. The discrepancy betweenthese two measurements has yet to be explained. In Eq. (1) bi isthe incoherent scattering length and corresponds to the spindependent part of the scattering length. Technically speaking b isthe bound scattering length which is related to the free scatteringlength a by a ¼ bMN=ðmn þMNÞ where mn is the neutron’s massand MN is the mass of the atom. In the case of n-3He a � 0:75 b[11].

It is often more convenient to work with scattering lengths fora given total spin, J ¼ I7rn. The neutron and 3He nucleus form atriplet (J ¼ 1) and singlet state (J ¼ 0) such that

b1 ¼ bc þ

ffiffiffiffiffig0

g1

rbi ðtriplet stateÞ ð2Þ

b0 ¼ bc �

ffiffiffiffiffig1

g0

rbi ðsinglet stateÞ ð3Þ

where g1 ¼34 and g0 ¼

14 are the statistical weight factors for 3He.

Alternatively in terms of bc and bi

bc ¼ g1b1 þ g0b0 ð4Þ

bi ¼ffiffiffiffiffiffiffiffiffiffig1g0p

ðb1 � b0Þ: ð5Þ

A more complete treatment of neutron optics can be found inRef. [12].

The phase shift of a neutron wavefunction can be directlydetermined using neutron interferometry allowing precise mea-surements of neutron scattering lengths. Conceptually, an NI isanalogous to a Mach–Zehnder interferometer in light optics. Aneutron interferometer is composed of a silicon ingot machined toproduce identical crystal blades on a common base. This insuresthat the lattice planes of the interferometer are aligned to withinthe Darwin width, typically a few mrad. A monochromatic,collimated neutron beam Bragg diffracts at an angle of yB ¼

arcsinðl=2dÞ in the first blade of the interferometer where l is theneutron wavelength and d is the lattice spacing of silicon. Thiscoherently splits the neutron’s wavefunction into two spatiallyseparate paths (Fig. 1). Bragg diffraction at a second bladeredirects the two paths so that they can interfere with oneanother at the final blade of the interferometer. A pair ofproportional detectors labeled O- and H-measure the neutronintensity which is now a coherent superposition of paths I and II.The count rate in the O-beam detector is

I0 ¼ c0 þ c1cosF ð6Þ

where c0 and c1 are parameters specific to the experiment and F isthe relative phase difference between paths I and II.

When a sample is placed in path I it causes an additional phaseshift of

fsam ¼ �NlbD ð7Þ

where N and D are the atomic density and thickness of thesample, respectively. Along with a target sample a control samplecalled a phase flag is placed in the beam paths such that

F ¼ fsam þ fflagðeÞ. Rotation of the phase flag by an anglee changes its effective thickness and varies the phase such thatall the parameters in Eq. (6), fflagðeÞ, and fsam can be determined.

The phase difference between the neutron states can be relatedto the triplet and singlet scattering lengths by

b1 � b0 ¼�2Df0

N3lD3P3: ð8Þ

Here Df0 is the difference in phase shift for when the flipper is onversus off after correcting for an incident neutron beam that hasnon-perfect polarization (i.e. Pna1). N3 and D3 are the atomicdensity and inner length of the 3He cell, respectively, and P3 is the3He polarization.

3. Experiment

3.1. Facility

This experiment was performed at NIST’s NIOF located inGaithersburg, MD. A steady source of neutrons are provided by a20 MW reactor and then moderated with a liquid hydrogen coldsource. A pair of pyrolytic graphite (0 0 2) monochromatorscapable of selecting a neutron wavelength between 2.0 and 4.7 Areflect neutrons into a vibration suppressed [13] and temperaturecontrolled [14] enclosure. The second monochromator is made upof nine tunable 1 cm� 5 cm pyrolytic graphite crystals, whichallows maximum vertical intensity at the interferometer. Detailsof the facility can be found in Ref. [15].

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Fig. 2. The neutron polarization (red circles) and spin flipper efficiency (green

squares) defined as ðsþ 1Þ=2 using the asymmetry method. Lines are fit to the data.

(For interpretation of the references to color in this figure legend, the reader is

referred to the web version of this article.)

M.G. Huber et al. / Nuclear Instruments and Methods in Physics Research A 611 (2009) 235–238 237

This experiment used 2.35 A neutrons which were polarizedusing a transmission mode supermirror [16]. A graphite filter [17]located upstream of the supermirror eliminated l=2 neutronsfrom the beam which would not have been polarized by thesupermirror but would have been accepted downstream by the(2 2 0) reflection of the interferometer. A precession spin flipperallowed the neutron spin state to be rotated by 1803 with nearly100% efficiency. Permanent magnets provided a magnetic guidefield of 1–10 mT which preserved the neutron’s spin to theinterferometer.

The NIST glass shop fabricated four boron-free target cells foruse in this experiment. Each cylindrical cell had outer dimensionsof 25 mm� 42 mm diameter and were sealed with between 1.7and 2 bar of 3He gas. The windows of the cell were 4 mm thick, flatglass. The cells were made optically thin to allow for someneutron transmission even when the neutron and 3He spins werealigned anti-parallel. The 3He gas was polarized to � 65% usingspin exchange optical pumping (SEOP) [18] at a separate facility.The cells were transported to the NIOF with typical transport lossin 3He polarization of only a few percent. This eliminated theadded complexity and heat loads to the interferometer setup yetprovided the experiment with viable target samples. Helmholtzcoils placed around the interferometer provided a uniformmagnetic field of 1.5 mT which limited the loss of heliumpolarization due to magnetic field gradients. Cell lifetimes at theinterferometer varied per cell with a maximum lifetime of 115 h.

3.2. Polarimetry

The neutron polarization was determined using two slightlydifferent methods in a series of separate measurements. Thesemeasurements were carried out at various breaks in collection ofphase data to verify that the neutron polarization was constantthroughout the experiment. Low neutron fluence rates in the H-beam path prevented any practical polarization analysis behindthe interferometer. Instead the interferometer was removed andreplaced with one of three polarized 3He cells. These cells wereoptically thicker than the interferometer cells providing a highanalyzing power, PA, which was insensitive to small changes in3He polarization. A detector was placed behind the analyzer inorder to measure the absolute transmission of neutrons throughthe polarized gas. Two advantages of a 3He analyzer are the abilityto flip the cell’s polarization and the ability to determine theanalyzing power with unpolarized neutron measurements. Thiseliminated the need for a second spin flipper to obtain Pn, s, andPA. Since the interferometer only selects a small portion of thedirect beam one might argue that there could be a difference inthe measured neutron polarization through the direct beam andthe polarization of the neutrons traveling through the inter-ferometer. However with our polarizer Pn depends very weakly onwavelength and the beam spectrum is sufficiently narrow Dl=l �1% that this effect is negligible.

3.2.1. Method: asymmetry

The asymmetry, A, in the count rates for the neutron spinstates, Im and Ik, is related to the neutron polarization and spinflipper efficiency by

A ¼Im � Ik

Im þ Ik¼ð1þ sÞPnPA

2þ ð1� sÞPnPAð9Þ

where s ¼ jPmn =Pk

n j is the ratio of the neutron polarization for thetwo spin states and is less than unity due to imperfect spin flipperefficiency. The analyzing power of the 3He cell is

PA ¼ tanhðN3spD3P3Þ ð10Þ

where sp is the polarized cross-section of 3He. PA was between86% and 99% depending on the cell’s optical thickness and theinitial polarization of the 3He gas. The hyperbolic tangent term inEq. (10) can be rewritten as a ratio of two unpolarized neutrontransmission measurements such that

PA ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

Iun

Ipol

� �2s

ð11Þ

where Ipol(Iun) is the transmission of unpolarized neutronsthrough a polarized (unpolarized) 3He cell. To obtain unpolarizedneutrons the supermirror was translated out of the beam. Theposition of the supermirror was encoded and reproducible towithin 1mm. For Iun the analyzer cell was depolarized byconnecting the Helmholtz coils to 30 V AC. Pn and s weredetermined (Fig. 2) by measuring A and PA at several differentvalues of 3He polarization. The polarization of 3He gas alternatedbetween positive and negative values in order to individuallydetermine Pn and s. This was done outside the NIOF using nuclearmagnetic resonance to flip the 3He polarization.

3.2.2. Method: normalized transmission

Using a 3He cell as an analyzer allows for an additional way tocalculate the neutron polarization from absolute transmissionmeasurements. It can be shown that

Pmn ¼ Pn ¼

Im

Ipol� 1

� �PA ð12Þ

jPkn j ¼ sPn ¼ 1�

Ik

Ipol

� �PA ð13Þ

where the intensities are the same as above. As in the asymmetrymethod the analyzing power is determined by Eq. (11). With thismethod there is an increase in precision when the neutron andn-3He polarizations are anti-parallel, hence sPn (Eq. (13)) wouldnormally be determined more precisely than Pn (Eq. (12)). The3He polarization was flipped so that similar precision wasobtained for both sPn and Pn. Fig. 3 shows the results for whentwo spins were aligned anti-parallel.

Both the asymmetry and normalized transmission methodswere used to measure neutron polarizations and spin flip

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Fig. 3. The neutron polarization with (blue squares) and without (red circles) the

spin flipper activated calculated using the normalized transmission method. Lines

are fit to the data. (For interpretation of the references to color in this figure

legend, the reader is referred to the web version of this article.)

M.G. Huber et al. / Nuclear Instruments and Methods in Physics Research A 611 (2009) 235–238238

efficiencies with relative standard uncertainties of less than 0.04%.Combining the two methods yields Pn ¼ ð0:9290870:00075Þ ands ¼ ð0:9951070:00034Þ . There is a 2s disagreement between thetwo methods for the measured neutron polarization, and theuncertainties in the combined result have been expanded toincorporate this.

3.3. Data collection

A full 12 weeks of phase data was taken in 4–9 h scans. Scansconsisted of rotation of the phase flag in De ¼ 2:18 mrad steps. Ateach position of the phase flag an off-on-on-off spin flippersequence was performed. The intensity measured by the O-beamdetector was recorded and fitted to Eq. (6) to determine the phase.

The target cell’s polarization was monitored throughout theexperiment. A third detector labeled C4 was placed after theinterferometer and directly behind the 3He cell. The C4 detectormeasured the transmission of neutrons through the cell for bothneutron spin states. The difference in the C4 count rate for bothneutron spin states along with Eqs. (9)–(10) and the polarimetrymeasurements gave the product N3spD3P3 as the heliumpolarization decayed.

4. Concluding remarks

The authors are currently finalizing the data analysis for thespin-dependent scattering length of n-3He. This experiment isstatistically limited to � 0:4% relative uncertainty and experi-mental systematic uncertainties are expected to be comparable to

or less than this value. The largest source of systematicuncertainty is due to the small but nonzero 3He triplet absorptioncross-section, sþ. Current experiential limits on sþ are at the fewpercent level [19,20] such that the authors plan to use theoreticalestimates of sþ [21] in their analysis. However these estimates ofsþ will still dominate the overall systematic uncertainty. Betterexperimental determination of sþ is needed and would greatlyimprove the current overall uncertainty in both this experimentand [3] which is also limited by sþ. Four nucleon interactions haveyet to be included into the theoretical models due to the difficultyin handling long-range coulomb forces, but constitute a tinycorrection to NNþ 3NI predictions. This measurement is part ofthe ongoing exploration into few-body systems by the NIOF.

Acknowledgments

Special thanks to John Fuller and Jeff Anderson for making theglass cells. The development and application of the polarized 3Hecells and methods used in this experiment was supported in partby the Department of Energy, Basic Energy Sciences. Also wewould like thank Sam Werner and Helmut Kaiser for their helpfuldiscussions. This work is supported by NIST and the NationalScience Foundation through Grant PHY-0555347.

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