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Physica B 385–386 (2006) 1365–1370

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Precision neutron interferometric measurements of the n–p, n–d, andn–3He zero-energy coherent neutron scattering amplitudes

P.R. Huffmana,b,� , M. Arifc, T.C. Blackd, D.L. Jacobsonc, K. Schoene,W.M. Snowf, S.A. Wernere,c

aDepartment of Physics, North Carolina State University, Campus Box 8202, Raleigh, NC 27695, USAbOak Ridge National Laboratory, Oak Ridge, TN 37831, USA

cNational Institute of Standards and Technology, Gaithersburg, MD 20899, USAdUniversity of North Carolina at Wilmington, Wilmington, NC 28403, USA

eUniversity of Missouri-Columbia, Columbia, MO 65211, USAfIndiana University/IUCF, Bloomington, IN 47408, USA

Paper presented as part of the Festschrift honouring Samuel A. Werner

Abstract

We have performed high-precision measurements of the zero-energy neutron scattering amplitudes of gas phase molecular hydrogen,

deuterium, and 3He using neutron interferometry. We find bnp ¼ ð�3:7384� 0:0020Þ fm [K. Schoen, D.L. Jacobson, M. Arif, P.R.

Huffman, T.C. Black, W.M. Snow, S.K. Lamoreaux, H. Kaiser, S.A. Werner, Phys. Rev. C 67 (2003) 044005], bnd ¼ ð6:6649�0:0040Þ fm [T.C. Black, P.R. Huffman, D.L. Jacobson, W.M. Snow, K. Schoen, M. Arif, H. Kaiser, S.K. Lamoreaux, S.A. Werner, Phys.

Rev. Lett. 90 (2003) 192502, K. Schoen, D.L. Jacobson, M. Arif, P.R. Huffman, T.C. Black, W.M. Snow, S.K. Lamoreaux, H. Kaiser,

S.A. Werner, Phys. Rev. C 67 (2003) 044005], and bn3He ¼ ð5:8572� 0:0072Þ fm [P.R. Huffman, D.L. Jacobson, K. Schoen, M. Arif, T.C.

Black, W.M. Snow, S.A. Werner, Phys. Rev. C 70 (2004) 014004]. When combined with the previous world data, properly corrected for

small multiple scattering, radiative corrections, and local field effects from the theory of neutron optics and combined by the

prescriptions of the particle data group, the zero-energy scattering amplitudes are: bnp ¼ ð�3:7389� 0:0010Þ fm,

bnd ¼ ð6:6683� 0:0030Þ fm, and bn3He ¼ ð5:853� :007Þ fm. The precision of these measurements is now high enough to severely

constrain NN few-body models. The n–d and n–3He coherent neutron scattering amplitudes are both now in disagreement with the best

current theories. The new values can be used as input for precision calculations of few body processes. This precision data is sensitive to

small effects such as nuclear three-body forces, charge-symmetry breaking in the strong interaction, and residual electromagnetic effects

not yet fully included in current models.

r 2006 Elsevier B.V. All rights reserved.

PACS: 21.45.+v; 03.75.Dg; 61.12.�q

Keywords: Neutron interferometry; Scattering amplitude; Neutron optics; NN potentials; Three-nucleon force; Effective field theory

1. Introduction

The last decade has seen a revolution in the accuracywith which low-energy phenomena in nuclear few body

e front matter r 2006 Elsevier B.V. All rights reserved.

ysb.2006.05.185

ng author. Department of Physics, North Carolina State

pus Box 8202, Raleigh, NC 27695, USA.

3314; fax: +1 919 515 6238.

ss: paul_huffman@ncsu.edu (P.R. Huffman).

systems can be calculated. Insight into certain features offew-nucleon systems has come both from greatly improvedcalculations using potential models based on the measurednucleon–nucleon (NN) interaction [1] and also from thedevelopment of effective field theory (EFT) approachesbased on the chiral symmetry of QCD [2–4]. Such theorieshave been used to develop a physical understanding rootedultimately in QCD for the relative sizes of many quantitiesin nuclear physics, such as nuclear N-body forces [5] and in

ARTICLE IN PRESSP.R. Huffman et al. / Physica B 385–386 (2006) 1365–13701366

particular the nuclear three-body force (3N), which is nowinvestigated intensively. Although it is well understood that3N forces must exist with a weaker strength and shorterrange than the NN force, little else is known.

EFT has been used to solve the two and three nucleonproblems with short-range interactions [4,6]. For the two-body system, EFT is equivalent to effective range theory andreproduces its well-known results for NN forces [7–9]. Thechiral EFT expansion does not require the introduction of anoperator corresponding to a 3N force until next-to-next-toleading (NNLO) order in the expansion, and at this order itrequires only two low-energy constants [10,11], which aretaken to be the triton binding energy and the zero-energydoublet n–d scattering amplitude. There have also beensignificant advances in other approaches to the computationof the properties of few-body nuclei with modern potentialssuch as the AV18 potential [12,13], which includes electro-magnetic effects and terms to account for charge-indepen-dence breaking and charge symmetry breaking of the stronginteraction. These calculations accurately reproduce the well-measured energy levels of few-body bound states only withthe phonomological inclusion of a nuclear three-body force,so it is clear that more information on this force is needed forfurther progress.

Precision measurements of low-energy strong interactionproperties, such as the zero-energy scattering amplitudesand electromagnetic properties of small A nuclei, aretherefore becoming more important for low energy, stronginteraction physics both as precise data that can be used tofix parameters in the EFT expansion and also as newtargets for theoretical prediction. In this article, we brieflysummarize a series of precision measurements of the zero-energy coherent scattering amplitudes in the two, three,and four-body systems using neutron interferometrictechniques [14–16]. We summarize these zero-energycoherent scattering amplitude results and discuss possibi-lities for future measurements.

2. Neutron optics theory

The zero-energy coherent scattering amplitude is thelinear combination of scattering amplitudes that give riseto the optical potential of a neutron in a medium [17]. Thezero-energy bound scattering amplitude, b, is related to thefree scattering amplitude a by

b ¼mþM

Ma. (1)

Here, m is the mass of the neutron and M is the mass of theatom. For hydrogen, a is the linear combination of thesinglet and the triplet scattering amplitudes given by

anp ¼ ð1=4Þ1anp þ ð3=4Þ

3anp, (2)

for deuterium it is the linear combination of the doubletand quartet scattering amplitudes,

and ¼ ð1=3Þ2and þ ð2=3Þ

4and, (3)

and for 3He, it is the linear combination of the singlet andtriplet scattering amplitudes

an3He ¼ ð1=4Þ1an3He þ ð3=4Þ

3an3He. (4)

The bound zero-energy scattering amplitude is oneparticular linear combination of the triplet and singlet (ordoublet and quartet) scattering amplitudes. Knowledge ofsome other combination allows one to independentlyextract the individual bound (or free) scattering amplitudesfor each state.The phase shift measured in neutron interferometry is

proportional to the real part of the S-wave coherentscattering amplitude in the medium,

f ¼ kð1� nÞD ¼ �lNDb, (5)

where k is the incident wave vector, n is the index ofrefraction, N is the number density, D is the thickness ofthe sample, and l is the neutron wavelength. Thus toexperimentally measure b, the neutron optical phase shiftf, the atom density, the sample thickness, and the neutronwavelength must each be measured to high precision.

3. Experimental procedure

Scattering amplitude measurements were performed atthe National Institute of Standards and Technology(NIST) Center for Neutron Research (NCNR) Interfe-rometer and Optics Facility [18]. A cold monochromaticneutron beam (E ¼ 11:1meV, Dl=l�1%) is collimatedusing a 2mm slit and enters the perfect silicon crystalneutron interferometer and is coherently divided via Braggdiffraction into two beams that travel along paths I and II

as shown schematically in Fig. 1. These beams are againdiffracted and then coherently recombined to form theinterference pattern. A detailed description of the facility,experimental arrangement, and procedures for the deter-mination of zero-energy neutron scattering amplitudes canbe found in Ref. [14].A secondary sampling method is used to measure the

phase shift f due to the gas sample. This is accomplishedby positioning a rotatable quartz phase shifter across thetwo beams as shown in Fig. 1. The intensities of the beamsthat arrive at the two 3He detectors are a function of thephase shifter angle d and are given by

IOðdÞ ¼ AO þ B cosðCf ðdÞ þ fgas þ fcellÞ,

IHðdÞ ¼ AH þ B cosðCf ðdÞ þ fgas þ fcell þ pÞ. ð6Þ

The values of AO, AH, B, and C are extracted from fits tothe data. The function f ðdÞ depends on the Bragg angle yBand is a measure of the neutron optical path lengthdifference between the beams induced by the phase shifterand is given by

f ðdÞ ¼sinðyBÞ sinðd� d0Þ

cos2ðyBÞ � sin2ðd� d0Þ. (7)

The hydrogen, deuterium, or 3He gases are housed in acell specifically designed to minimize the phase shift fcell

ARTICLE IN PRESS

φgas + φcell

0

100

200

300

400

500

600

–1–2 0–3

I O (

coun

ts/m

in)

δ (deg)

cell in +3Hegascell out

1 2 3

Fig. 2. A typical pair of interferograms with 3He present in the cell. The

oscillations arise from the change in path lengths created as the phase

shifter is rotated (see Fig. 1). Data are shown for both the cell in and out of

the interferometer.

path I

path

II

neutron beam

3He detectors

cell in

H b

eam O

beam

vacuum

n, d, or 3He gas

Si (111)phase shifter

quartz plate

cell out

1 cm

A

BC

D

δ

Fig. 1. A schematic view of the experimental setup as the neutron beam

passes through the perfect crystal silicon interferometer. Parameters

associated with the neutron optics are discussed in the text.

P.R. Huffman et al. / Physica B 385–386 (2006) 1365–1370 1367

due to the aluminum walls of the cell (see Fig. 1). The phaseshifts arising from the presence of the gas and cell (fgas andfcell) were determined by collecting � 103 interferogrampairs with the cell positioned both within the interferometerand removed from the beam paths. Each of thesemeasurements took approximately 42min, with typicalcount rates averaging around 300–400 counts per minute.The phase difference between cell-in/cell-out sets ofinterferograms is extracted for each pair, with a typicalset shown in Fig. 2. The phase shift from the cell, fcell, wasdetermined using an evacuated cell.

The atom density was determined using the measuredpurity of the gas and the ideal gas law with virial coefficientcorrections up to the third pressure coefficient. Theabsolute temperature was continuously monitored usingtwo calibrated 100O platinum thermometers that have anabsolute accuracy of 0.023% at 300K. The pressure wascontinuously monitored using a calibrated silicon pressuretransducer capable of measuring the absolute pressure tobetter than 0.01%. The wavelength of neutrons traversingthe interferometer was measured using a pyrolytic graphite(PG 002) crystal. This analyzer crystal, calibrated sepa-rately against a Si crystal with a precisely known latticeconstant, was placed in the H-beam of the interferometerand rotated so that both the symmetric and anti-symmetricBragg reflections were determined. In a separate test, thestability of the wavelength over the measurement time wasshown to be 0:001%. More details on the measure-ment techniques and systematic uncertainties involvedin the determination of the neutron wavelength, atomdensity, temperature, and cell thickness can be found inRefs. [14,16].

The value of the bound zero-energy scattering amplitudewas calculated for each data set on a run-by-run basis.These values were combined using a weighted average toobtain b for each gas species. Our reported results are bnp ¼

ð�3:7384� 0:0020Þ fm [14], bnd ¼ ð6:6649� 0:0040Þ fm[14,15], and bn3He ¼ ð5:8572� 0:0072Þ fm [16].

4. Results and discussion

Since these measurements of the n–p, n–d, and n–3Hezero-energy scattering amplitudes were performed in anidentical manner using the same apparatus (cell, neutronwavelength analyzer, and pressure and temperature moni-tors), one can take the ratio of the measurement values toobtain results which are even less sensitive to any potentialremaining systematic errors. Using bnp as the reference, weobtain the ratios

bn3He=bnp ¼ ð�1:5668� 0:0021Þ,

bnd=bnp ¼ ð�1:7828� 0:0014Þ.

Although these ratios possess slightly larger statisticaluncertainties, it will be independent of any unknownsystematic uncertainty to first order and can provide amore robust target for comparison to theories whichattempt to calculate all of the scattering lengths within acommon theoretical framework.Recently, theoretical predictions of the n–d zero-energy

scattering amplitude were performed using the high-precision NN forces CD Bonn 2000, AV18, Nijm I, IIand 93 in combination with 3N force models [19]. Theresults of these calculations are summarized in Fig. 3alongside the experimental data. For NN forces alone withand without electromagnetic interactions, they recoveredthe approximate correlation between the triton binding

ARTICLE IN PRESS

-8.6

-8.4

-8.2

-8.0

-7.8

-7.6

E 3 H

(M

eV)

7.06.96.86.76.6

bnd (fm)

-8.60

-8.55

-8.50

-8.45

-8.40

6.666.646.62

ExperimentalValue

Fig. 3. Theoretically determined values for the n–d scattering amplitude

bnd and the triton binding energy using different NN and 3N forces [19].

The experimental value consists of our measurement of bnd and the very

precisely known value of the triton binding energy, ð8:481855�0:000013ÞMeV [20].

-2.50

-2.48

-2.46

-2.44

-2.42

-2.40

-2.38

-2.36

-2.34

b n3H

e in

cohe

rent

(fm

)

6.106.056.005.955.905.855.805.75

bn3He coherent (fm)

ExperimentalValue

AV18AV18 + UIX + V*

3

AV18 + UIX

R-Matrix

Fig. 4. Comparisons of the n–3He coherent and incoherent scattering

amplitudes as determined from our measurement and the measurement of

Zimmer et al. [22] to theoretical calculations of these same parameters

using either an R-matrix formalism or the AV18 NN potential with two

types of 3N forces [21].

P.R. Huffman et al. / Physica B 385–386 (2006) 1365–13701368

energy and 2and known as the Phillips line. The approx-imate correlation between these observables is now under-stood to be a generic feature for systems such as thedeuteron and other low A nuclei whose size is significantlylarger than the 1 fm scale of the NN interaction range.However, it is understood that this correlation is only anapproximation and should not be obeyed exactly: forexample 3N forces will introduce corrections. Although theaddition of 3N forces of a wide variety of types does shiftthe values closer to the observed 2and, none of thesecalculations agrees with the new high-precision measure-ments. The authors note that the 2and zero-energyscattering amplitude must be considered as an independentlow-energy observable for future calculations in few bodysystems.

A second group has recently published new calculationsof the spin-dependent n–3He scattering amplitudes usingthe resonating group method and a variety of modern NNand 3N potentials [21]. Comparisons of the experimentalfree nuclear singlet and triplet scattering amplitudes asdetermined from our measurement and the n–3He inco-herent scattering amplitude from Zimmer et al. [22] withtheoretical calculations of these parameters were per-formed using a R-matrix formalism and the AV18 NNpotential with two types of 3N forces and are shown inFig. 4. It is clear that the theoretical predictions from eventhese impressive, state-of-the-art calculations using reso-nating group techniques still lie outside the range ofexperimental uncertainties for the 3He binding energy andthe singlet and triplet n–3He scattering amplitudes.

The precision of these measurements is now high enoughto severely constrain these few-body models. Both the n–dand n–3He coherent neutron scattering amplitudes are in

disagreement with the best current theories. Only modelswhich correctly take into account nuclear three-bodyforces, charge-symmetry breaking, and residual electro-magnetic effects have the possibility to successfullyconfront the data.We can combine these measurements with the previous

world’s data to obtain new values for the coherentscattering amplitudes. We note that some of the pastmeasurements of high precision using a gravity reflect-ometer performed in Garching a few decades ago [23] mustbe corrected for local field effects [24], which are significantfor the coherent neutron scattering amplitude in hydrogenbnp as measured in the gravity refractometer but arenegligible in neutron interferometry. This correction shiftsthe value from bnp ¼ ð�3:7409� 0:0011Þ fm [23] to bnp ¼

ð�3:7390� 0:0011Þ fm [24], an approximately two sigmaeffect. In addition, there is another two sigma correction tothe neutron interferometry measurement in hydrogen gasdue to multiple scattering and correlation corrections tothe neutron index of refraction which are negligible for thegravity refractometer measurements. These correctionshave already been applied to obtain the value of bnp ¼

ð�3:7384� 0:0020Þ fm [14] quoted above. After thesecorrections, both independent measurements are in ex-cellent agreement. We can therefore present an improvedvalue of the coherent neutron scattering length of hydrogenof bnp ¼ ð�3:7389� 0:0010Þ fm. These corrections turn outto be negligible for deuterium.

5. Future measurements

As mentioned above, the measured value of bnp

was corrected for multiple scattering and correlation

ARTICLE IN PRESSP.R. Huffman et al. / Physica B 385–386 (2006) 1365–1370 1369

corrections to the neutron index of refraction. A moreprecise interferometric measurement in comparison withthe gravity refractometer value would be able to isolatethis term experimentally. Such a measurement wouldconstitute the first experimental observation of correctionsto the neutron index of refraction due to virtual excitationsand multiple internal scattering within a molecule. Sucheffects were predicted by Nowak 20 years ago [25]. Thecalculation of Nowak, which was done in the longwavelength limit kR0 ! 0 (R0 ¼ bond length of H2 ¼

0:74611 nm, for D2 R0 ¼ 0:74164 nm [26]), must beextended to the conditions of the experiment (whichcorrespond to kR0 ¼ 1:73) to make a precise prediction.It would be possible to measure bnp using H2 gas at least afactor of two more precisely, which may be accurateenough to isolate this neutron optics effect experimentallyfor the first time.

Our measurement of bnd will hopefully soon becomplemented by a measurement in progress at PSI ofthe incoherent n–d scattering amplitude using pseudomag-netic precession in a polarized deuterium target [27], whichis also an interferometric technique that operates inneutron spin space as opposed to real space. Thismeasurement will allow one to separate the two spinchannels and determine the interesting doublet amplitude,2and, with high precision. We have argued that the quartetamplitude should be independent of 3N forces and thetheoretical value should be reasonably robust [28] and werethus able to extract a value of 2and ¼ ð0:645�0:003ðexptÞ � 0:007ðtheoryÞÞ fm [16], however an experi-mental determination is badly needed. We expect that thesenew measurements will improve 2and by an order ofmagnitude to � 10�3. The doublet amplitude is veryimportant from a theoretical point of view: in the EFTapproach it fixes one of the two low-energy constants in theexpansion to NNLO and its determination will allow moreprecise predictions to be made for other few-body systems.

It is also interesting to consider how the experimentaldeterminations of the n–3He scattering amplitudes can befurther improved. Our measurement of the zero-energyscattering amplitude is dominated by statistical uncertain-ties in the measurement of the phase shifts and conse-quently there is still room for improvement. In the case ofthe incoherent scattering amplitude (bi) determinationfrom pseudomagnetic precession performed by Zimmer etal. [22], the accuracy is unfortunately limited by the poorexperimental knowledge of the relative contributions ofsinglet and triplet channels to the n–3He absorption cross-section. A better measurement of this ratio, currentlyknown to � 1% [29,30], could be immediately combinedwith the Zimmer et al. measurement to improve theaccuracy of bi by as much as a factor of three. In addition,a new measurement currently being planned at NIST todirectly measure the spin-dependent n–3He scatteringamplitudes is independent of this ratio [31]. Dramaticallyreduced uncertainties for 1a

n3Heand 3a

n3Hein n–3He are

therefore possible.

Measurements in other light nuclei are also quitefeasible. Two targets that we are presently exploring are4He and tritium. From a theoretical point of view atpresent, a 4He measurement is not that interesting becausethe framework for solving five-body problems is not yet inplace. The tritium measurement on the other hand is quiteinteresting theoretically because the small inelastic effectsmake it much easier to calculate than n–3He, but it isexperimentally difficult because of the radioactive nature ofthe target. We are investigating the possibility of perform-ing a measurement of bnt using the same techniques used inthe current measurements. We expect that the limitingfactor in these measurements will be how well we candetermine the isotopic composition of the radioactivetritium gas.Perhaps the single most interesting scattering length to

measure is the neutron–neutron scattering length bnn; nodirect measurements presently exist. An experiment that iscurrently being designed seeks to determine bnn by viewing ahigh-density neutron gas near the core of a pulsed reactorthat produces an extremely high instantaneous neutrondensity and measuring the quadratic dependence of theneutron fluence on source power [32]. A second experimentthat relies on neutrons from an extracted beam to scatterfrom each other has also been considered [33]. A precisionmeasurement of bnn—even at the few percent level—wouldbe very valuable, since it could provide new information onthe size of charge symmetry breaking effects of the stronginteraction. Charge symmetry is the discrete subset ofisospin symmetry corresponding to a 180� rotation inisospin space. A calculation of bnn using low-energy effectivefield theories of QCD would seem to be out of reach atpresent since there are so few constraints on the nature ofcharge symmetry breaking in the strong interaction fromother systems. An EFT analysis to extract bnn fromneutron–neutron final state effects in few-body reactionsare possible, however, and an EFT analysis of the p�d !

nng reaction to extract bnn has recently appeared [34].One additional measurement using neutron interferome-

try is presently underway at NIST, a determination of theneutron’s mean square charge radius by measuring theneutron–electron scattering amplitude, bne [35]. This experi-ment is expected to lead to a five-fold improvement in theprecision of bne. This precision may be sensitive to radiativecorrections calculable in a recent EFT approach [36].Theoretical advances over the last decade have made

few-nucleon systems into a quantitative testing ground forlow-energy QCD. Low-energy neutrons can be used toperform high-precision measurements of scattering ampli-tudes. We look forward to new high-precision measure-ments in this field in the next few years.

Acknowledgments

We acknowledge the support of the National Institute ofStandards and Technology, U.S. Department of Commerce,in providing the neutron research facilities used in this work.

ARTICLE IN PRESSP.R. Huffman et al. / Physica B 385–386 (2006) 1365–13701370

This work was supported in part by the U.S. Department ofEnergy under Grant No. DE-FG02-97ER41042 and theNational Science Foundation under Grants No. PHY-9603559 at the University of Missouri, No. PHY-9602872at Indiana University, and No. PHY-0245679 at theUniversity of North Carolina at Wilmington.

References

[1] J. Carlson, R. Schiavilla, Rev. Mod. Phys. 70 (1998) 743.

[2] P.F. Bedaque, U. van Kolck, Phys. Lett. B 428 (1998) 221.

[3] H.W. Hammer, arXiv:nucl-th/9905036, 1999.

[4] P.F. Bedaque, U. van Kolck, Ann. Rev. Nucl. Part. Sci. 52 (2002) 339.

[5] J. Friar, in: R. Seki, U. van Kolck, M.J. Savage (Eds.), Nuclear

Physics with EFT, World Scientific, Singapore, 2001.

[6] S.R. Beane, M.J. Savage, Nucl. Phys. B 636 (2002) 291.

[7] U. van Kolck, Nucl. Phys. A 645 (1999) 273.

[8] D.B. Kaplan, M.J. Savage, M.B. Wise, Phys. Lett. B 424 (1998) 390.

[9] J. Gegelia, arXiv:nucl-th/9802038, 1998.

[10] E. Epelbaum, H. Kamada, A. Nogga, H. Wita"a, W. Glockle, Ulf-G.

Meissner, Phys. Rev. Lett. 86 (2001) 4787.

[11] E. Epelbaum, A. Nogga, W. Glockle, H. Kamada, U.G. Meissner, H.

Wita"a, Phys. Rev. C 66 (2002) 064001.

[12] R.B. Wiringa, V.G.J. Stoks, R. Schiavilla, Phys. Rev. C 51 (1995) 38.

[13] S.C. Pieper, R.B. Wiringa, Ann. Rev. Nucl. Part. Sci. 51 (2001) 53.

[14] K. Schoen, D.L. Jacobson, M. Arif, P.R. Huffman, T.C. Black,

W.M. Snow, S.K. Lamoreaux, H. Kaiser, S.A. Werner, Phys. Rev. C

67 (2003) 044005.

[15] T.C. Black, P.R. Huffman, D.L. Jacobson, W.M. Snow, K. Schoen,

M. Arif, H. Kaiser, S.K. Lamoreaux, S.A. Werner, Phys. Rev. Lett.

90 (2003) 192502.

[16] P.R. Huffman, D.L. Jacobson, K. Schoen, M. Arif, T.C. Black,

W.M. Snow, S.A. Werner, Phys. Rev. C 70 (2004) 014004.

[17] H. Rauch, S. Werner, Neutron Interferometry: Lessons, in: Experi-

mental Quantum Mechanics, Oxford University Press, Oxford, 2000.

[18] M. Arif, D.E. Brown, G.L. Greene, R. Clothier, K. Littrell, Proc.

SPIE 2264 (1994) 21.

[19] H. Wita"a, A. Nogga, H. Kamada, W. Glockle, J. Golak, R.

Skibinski, arXiv:nucl-th/0305028, 2003.

[20] A.H. Wapstra, G. Audi, Nucl. Phys. A 432 (1983) 1.

[21] H.M. Hofmann, G.M. Hale, Phys. Rev. C 68 (2003) 021002(R).

[22] O. Zimmer, G. Ehlers, B. Farago, H. Humblot, W. Ketter, R.

Scherm, EPJdirect A 1 (2002) 1.

[23] L. Koester, H. Rauch, E. Seymann, Atom. Data Nucl. Data Tables

49 (1991) 65.

[24] V.F. Sears, Z. Phys. A 321 (1985) 443.

[25] E. Nowak, Z. Phys. B 49 (1982) 1.

[26] in:R.C. Weast (Ed.), CRC Handbook of Chemistry and Physics, 68th

ed., Cleveland, Ohio, CRC Press, Boca Raton, FL, 1987.

[27] B. van den Brandt, H. Glattli, H. GreiXhammer, P. Hautle, J.

Kohlbrecher, J.A. Konter, O. Zimmer, Nucl. Instr. and Meth. A 526

(2004) 91.

[28] J.L. Friar, D. Huber, H. Witala, G.L. Payne, Acta. Phys. Pol. B 31

(2000) 749.

[29] L. Passell, R.I. Schermer, Phys. Rev. 150 (1966) 146.

[30] S.B. Borzakov, H. Malecki, L.B. Pikel’ner, M. Stempinski, E.I.

Sharapov, Sov. J. Nucl. Phys. 35 (1982) 307.

[31] F.E. Wietfeldt, private communication.

[32] W.I. Furman, et al., J. Phys. G Nucl. Part. Phys. 28 (2002) 2627.

[33] Yu.N. Pokotolovskii, Phys. Atom. Nucl. 56 (1993) 524.

[34] A. Gardestig, D.R. Phillips, arXiv:nucl-th/0501049, 2005.

[35] F.E. Wietfeldt, M. Huber, T.C. Black, H. Kaiser, M. Arif, D.L.

Jacobson, S.A. Werner, These proceedings.

[36] A. Pineda, arXiv:hep-ph/0412142, 2004.