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Nuclear Instruments and Methods in Physics Research A 526 (2004) 91–95

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doi:10.1016

The spin-dependent nd scattering length—a proposedhigh-accuracy measurement

B. van den Brandta, H. Gl.attlib, H. GrieXhammerc, P. Hautlea, J. Kohlbrechera,J.A. Kontera, O. Zimmerc,*

aPaul Scherrer Institute, CH-5232 Villigen PSI, SwitzerlandbCommissariat "a l’Energie Atomique, CE Saclay/SPEC LLB, F-91191 Gif-sur-Yvette, France

cPhysik-Department, Technische Universit .at M .unchen, James-Franck-Strasse, D-85748 Garching, Germany

Abstract

The understanding of few-nucleon systems at low energies is essential, e.g. for accurate predictions of element

abundances in big-bang and stellar fusion. Novel effective field theories, taking only nucleons, or nucleons and pions as

explicit degrees of freedom, provide a systematic approach, permitting an estimate of theoretical uncertainties. Basic

constants parameterising the short-range physics are derived from only a handful of experimental values. The doublet

neutron scattering length a2 of the deuteron is particularly sensitive to a three-nucleon contact interaction, but

experimentally known with only 6% accuracy. It can be deduced from the two experimentally accessible parameters of

the nd scattering length. We plan to measure the poorly known ‘‘incoherent’’ nd scattering length ai;d with 10�3

accuracy, using a Ramsey apparatus for pseudomagnetic precession with a cold polarised neutron beam at PSI. A

polarised target containing both deuterons and protons will permit a measurement relative to the incoherent nd

scattering length, which is known experimentally with an accuracy of 2:4� 10�4:r 2004 Elsevier B.V. All rights reserved.

PACS: 21.45.+v; 25.40.Dn; 28.20.Cz; 25.10.+s

Keywords: Neutron physics; Neutron scattering; Few-body physics

1. Introduction

In the past few years, a new strategy has beendeveloped to describe nuclear forces at low energy.Chiral perturbation theory ðwPTÞ is an effectivefield theory, which describes interactions of pionsand between pions and nucleons (N). It leads to asystematic expansion of the scattering amplitudein powers of ratios of small momenta and low-

onding author.

ddress: oliver.zimmer@ph.tum.de (O. Zimmer).

- see front matter r 2004 Elsevier B.V. All rights reserve

/j.nima.2004.03.156

energy input parameters like the pion mass overthe breakdown scale of the theory. For the firsttime the accuracy of calculations can be estimatedin a model-independent theory of nuclear interac-tions, providing reliable predictions of manyimportant low-energy quantities. These areground-state properties of bound systems andprocesses involving external and exchange cur-rents, as e.g. cross-sections relevant for big-bangnucleosynthesis and stellar fusion [1,2]. Also thedetermination of fundamental properties of theneutron from experiments on few-nucleon systems

d.

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mandates a model-independent subtraction ofnucleon binding and meson exchange effects.Weinberg pointed out that chiral three-nucleon

(3N) forces appear naturally in wPT [3]. The mostrelevant processes are: a two-pion exchange, a 2Ncontact interaction with pion exchange, and a 3Ncontact interaction [4]. The contact interactionsparameterise the short-range physics. As in Fer-mi’s theory of weak interaction, they are char-acterised by effective couplings, called low-energyconstants (LECs). They have to be fixed bymeasured data of two independent low-energy3N observables. Very recently, a first completeanalysis of nd scattering at next-to-next-to leadingorder has been performed with impressive results[5]. All observables are expanded in powers ofmomenta and the pion mass over the wPT-break-down scale of about 800 MeV:An even simpler approach to the nuclear few-

body problem is an effective field theory withoutpionic degrees of freedom [6,7]. This theory startsout from point-like interactions between nucleons,which only have to respect the symmetries ofQCD. Like wPT; it describes phenomena in asystematic way, but is applicable only at energieswell below a breakdown scale set by the pion mass.Again, only two LECs characterising 3N forces arerequired to predict observables with an accuracyof less than 1% in processes involving threenucleons (Fig. 1).However, this accuracy can only be achieved if

the experimental inputs are known correspond-

Fig. 1. The strengths H0 of the point-like three-body forces of

the effective field theory in which pions are integrated out (top)

must be determined from experimental three-body data. At

momenta above mpc; they are partially resolved as pion-

exchange (bottom), but the core-strengths eHH0 still need the

input from three-nucleon observables. Solid (dashed) lines

denote nucleons (pions).

ingly well. The binding energy of the triton and thedoublet nd scattering length a2 are particularlywell suited to determine the LECs for wPT and forthe pion-free theory. First, there are no Coulombeffects to be considered. Second, a2 is verysensitive to 3N forces: in the quartet channel, theincident neutron has its spins parallel to the onebound in the deuteron, so that the Pauli principleprohibits 3N forces to play any sizeable role at lowmomentum transfer. In contradiction, the s-wavein the doublet channel allows for a momentum-independent 3N interaction. This turns out evennecessary to achieve results which are not sensitiveto physics at high energy scales, or respectively, atshort distances beyond the range of applicabilityof the theory. While the triton binding energy isknown with an accuracy of 5� 10�7; the experi-mental knowledge of the nd doublet scatteringlength is only 6%.

2. Present situation and accuracy goal

The scattering length of a neutron with spin s

and a nucleus with spin I is given by

a ¼I þ 1

2I þ 1aþ þ

I

2I þ 1a� þ

2ðaþ � a�Þ2I þ 1

s � I

¼ ac þ2aiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

IðI þ 1Þp s � I ð1Þ

where aþ and a� denote the scattering lengths inthe state with total spin I þ 1

2; respectively, I � 1

2:

Since one cannot prepare the latter state, it is notpossible to measure a� directly. Experimentallyaccessible are the spin-independent, coherentscattering length ac; and the factor ai whichparameterises the spin-dependence (sometimescalled ‘‘incoherent’’ scattering length). For theneutron–deuteron system, the doublet and quartetscattering lengths, a� ¼ a2; respectively aþ ¼ a4;are given as the linear combinations

a2 ¼ ac;d �ffiffiffi2

pai;d; a4 ¼ ac;d þ

1ffiffiffi2

p ai;d: ð2Þ

The best experimental value of a2 was obtained 30years ago [8], using a combination of a measure-ment of the scattering cross-section of the free

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deuteron

ss;d ¼ 4pðjac;dj2 þ jai;dj2Þ ð3Þ

with a gravity refractometric measurement of thebound coherent nd scattering length bc;d (throughthe relation of bound and free scattering lengths b;respectively, a by

a ¼M

M þ mb ð4Þ

with corresponding indices. M is the mass of thenucleus and m the neutron mass). The experi-mental values were ss;d ¼ 3:39070:012 barn andbc;d ¼ 6:67270:007 fm; leading to

a2 ¼ 0:6570:04 fm: ð5Þ

Recently, a new measurement of bc;d wasperformed at the NIST interferometer at Wa-shington, DC [9]. Including the result, bc;d ¼6:664970:0040 fm; the present world average is

bc;d ¼ 6:668370:0030 fm: ð6Þ

However, this improvement does not significantlyreduce the experimental uncertainty of a2; sincethis is dominated by the unsufficient knowledge ofai;d: On the other hand, the authors of Ref. [9]argue that, because a4 should have a very smalldependence on 3N forces, ai;d may be derived fromthe experimental value stated in Eq. (6) and atheoretical value of a4; using Eqs. (2) and (4). Thatway, they obtain a semi-experimental value a2 ¼0:64570:00370:007 fm:The goal of the present experiment is a direct

measurement of ai;d; which does not rely on anynuclear few-body theoretical input. As a minimumaim we hope to achieve an accuracy of 10�3: UsingEq. (2) together with Eq. (6), this shall provide anew value of a2 with an uncertainty of 0:004 fm:

3. Method

The spin-dependent scattering length induces aspin-dependence of the neutron refractive index.As a result, bi can be determined directly witha polarised neutron beam passing through apolarised target, via detection of pseudo-magnetic neutron precession around the axis ofnuclear polarisation [10,11]. The pseudomagnetic

precession angle is given by

j� ¼ 2ldX

k

ffiffiffiffiffiffiffiffiffiffiffiffiffiIk

Ik þ 1

rPkNkbi;k ð7Þ

where Nk is the number density, Ik the nuclear spinand Pk the nuclear polarisation, with the sumindex k extending over the nuclear species withspin. l is the de-Broglie wavelength of theneutrons, and d is the thickness of the sample.The angle j� can be measured with an accuracy ofat least one degree, using the method described inRefs. [12–14] based on Ramsey’s well-knownresonance technique with two separated oscillatoryfields. The target is situated in the homogeneousmagnetic field between the two high-frequency p=2coils.The uncertainty of the deuteron polarisation Pd

would impose a severe limitation in accuracy if thedeuterons were the only nuclei with spin in thetarget. This difficulty can be considerably relaxedin a measurement of bi;d relative to bi;p of theproton, which is known with the high accuracy of2:4� 10�4: Using a single target which containsboth deuterons and protons at hydrogen sites,absolute polarisation measurements can beavoided [15]. Many materials are suitable to applythe method of dynamic nuclear polarisation(DNP), by which both isotopes can be polarisedsimultaneously under still rather convenientconditions [16].The method combines several measurements

described in the following. First, the sample ispolarised via DNP. After freezing the nuclearpolarisation one determines the pseudomagneticprecession angle. According to Eq. (7) and includ-ing an additional, instrumental phase j0; it is givenby

f1 ¼ j�d þ j�p þ j0 ð8Þ

with

j�d ¼ffiffiffi2

pldPdNdbi;d

j�p ¼2ffiffiffi3

p ldPpNpbi;p: ð9Þ

Further, using hf-saturation, one can selectivelydepolarise either the protons or the deuterons,without significantly affecting the polarisation of

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the other spin species. The subsequent crossrelaxation between the two spin systems can beheld sufficiently slow by keeping the temperaturesufficiently low. Saturating, e.g. first the protons,one can then measure

f2 ¼ j�d þ j0: ð10Þ

After a subsequent depolarisation of the deuteronsone measures

f3 ¼ j0: ð11Þ

Combining the measured values and using Eq. (9),we obtain

bi;d ¼

ffiffiffi2

3

rf2 � f3f1 � f2

PpNp

PdNdbi;p: ð12Þ

The method is completed with measurements ofdeuteron and proton NMR signals Ik; taken asintegral of the corresponding RF-absorption linebefore and after each measurement of a pseudo-magnetic precession angle:

Ik ¼ CkPkNk: ð13Þ

Apart from natural constants, Ck is given as theproduct of g2k=Ik (with gk denoting the g-factor)[17] and a factor accounting for the sensitivity ofthe resonance circuit. The latter would be difficultto determine absolutely. However, since only theratio of PkNk for the two spin species occurs inEq. (12), replacing this by the ratio of NMRsignals via Eq. (13), the common instrumentalfactor cancels out. This requires using the same,linear resonance circuit at the same frequency inthe measurements of Id and Ip; which thereforehave to be performed at different main magneticfields to account for the different gyromagneticratios.Combining the various measurements into

ratios, we thus finally obtain

bi;d ¼1ffiffiffi6

p g2dg2p

f2 � f3f1 � f2

Ip

Idbi;p: ð14Þ

It is the gist of this method that we do not supposeany exact knowledge of neither Nd; Np; Pd; Pp; d

or l; nor of any absolute calibration factors.

4. Some practical comments

The choice of the sample is governed by severalfactors. First, we consider the isotopic composi-tion. Best sensitivity is attained for j�dEj�p : DNPkeeps the spin temperatures of protons anddeuterons equal [18]. Using the Brillouin functionsin the high-temperature limit,

Pd

PpE4gd3gp

E0:2: ð15Þ

From Eq. (9),

Nd

NpE5

Pp

Pd

j�dj�p

: ð16Þ

Thus, j�dEj�p for Nd=NpE4%: On the otherhand, to keep the systematic uncertainties inducedby non-linearities of the NMR resonance circuitsmall, requires IdEIp; which happens to be truefor about the same ratio. Considering a typicalmaterial with number density 6:7� 1022 cm�3 ofhydrogen sites and the measurements being doneat l ¼ 0:4 nm; the corresponding pseudomagneticprecession angles are

j�d ¼ 90 rad Pd j�p ¼ 440 rad Pp: ð17Þ

Since the Ramsey technique is sensitive to preces-sion angles of at least 1; a nuclear polarisation ofa few percent is sufficient already for a target only3 mm long.Relaxation times are known to be strongly

temperature dependent. Using a magnetic field of2:5 T and a cryostat providing TE100 mK; thenuclear spin systems are frozen. Typical spin–lattice relaxation times of 500 h and cross relaxa-tion time of 100 h; obtained for ðCH2OHÞ2 dopedwith Cr(V) paramagnetic centres (radicals), werepublished in Ref. [18]. These are sufficiently long,since, for the given neutron intensity, a single runof the experiment will take much less than 1 h:Operating with an amount of radicals below theoptimum value to reach maximum polarisation(which is not required here), spin relaxation maybe suppressed even further.Apart from stability requirements, keeping low

the amount of radicals is also of interest tosuppress systematic errors which may be due

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to their local magnetic fields. Nuclear spins closeto the paramagnetic electrons contribute topseudomagnetic precession but usually stay un-detected by NMR, since their resonance frequen-cies are strongly shifted with respect to the NMRline of the nuclei in the bulk. Two measures maybe taken. First, the radical concentration shouldbe kept as low as possible. Second, deuteratedradicals should be used, in order to keep the fewprotons of the sample away from the radicals. Alsofor practical reasons, a good candidate of a targetmaterial is a plastic deuterated to the requiredabundance and doped with deuterated nitroxylradicals [19]. Note in addition that, using a novelstroboscopic technique of simultaneous small-angle neutron scattering and NMR measurements,detailed information can be obtained about thepolarisation state of protons close to radicalsdissolved in a deuterated matrix [20]. That way onecan determine a correction, if necessary. Note also,that the paramagnetic electrons are always po-larised very close to 100%, and their fields cause asmall magnetic neutron precession, which isincluded in the instrumental phase j0:The NMR detection scheme is a crucial part of

the setup and deserves special consideration. Ananalysis of several possibilities is described in aseparate publication [21]. Apart from its specialrole in the measurement, NMR can also be used tomonitor the polarisation state of the spin systemsat any stage of the experiment. Many systematicvariations of parameters can be performed tocheck the independence of the measurement withrespect to the conditions which have dropped outin the derivation of Eq. (14). Different sampleswith different chemical composition, varied degreeof deuteration and polarisation will be used. Asystematic variation of target and beam para-meters should be able to demonstrate the relia-bility of the measurement (this strategy was alsoadopted in a recent determination of the spin-dependent n3He scattering length [22]).

Acknowledgements

This work has been funded by BMBF (contractnumber 06MT 197).

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