Neutron-neutron quasifree scattering in neutron-deuteron breakup at 10 MeV
R. C. Malone, A. S. Crowell, L. C. Cumberbatch, B. A. Fallin, F. Q. L. Friesen, C. R. Howell, C. R. Malone, D. R. Ticehurst, W. Tornow, D. M. Markoff, B. J. Crowe, H. Witała
NNeutron-neutron quasifree scattering in neutron-deuteronbreakup at 10 MeV
R. C. Malone, ∗ A. S. Crowell, L. C. Cumberbatch, B. A. Fallin,F. Q. L. Friesen, C. R. Howell, C. R. Malone, D. R. Ticehurst, and W. Tornow
Department of Physics, Duke University and TriangleUniversities Nuclear Laboratory, Durham, NC 27708, USA
D. M. Markoff and B. J. Crowe
Department of Mathematics and Physics,North Carolina Central University Durham,North Carolina 27707 and Triangle Universities Nuclear Laboratory,Durham, North Carolina 27708, USA
H. Wita(cid:32)la
M. Smoluchowski Institute of Physics,Jagiellonian University, Krak`ow, Poland (Dated: April 1, 2020)
Abstract
New measurements of the neutron-neutron quasifree scattering cross section in neutron-deuteronbreakup at an incident neutron energy of 10.0 MeV are reported. The experiment setup was opti-mized to evaluate the technique for determining the integrated beam-target luminosity in neutron-neutron coincidence cross-section measurements in neutron-deuteron breakup. The measurementswere carried out with a systematic uncertainty of ± . . ± . ± . in comparison with the theory prediction of 20.1 mb/sr . These results validate our technique fordetermining the beam-target luminosity in neutron-deuteron breakup measurements. ∗ [email protected] a r X i v : . [ nu c l - e x ] M a r . INTRODUCTION The neutron-deuteron ( nd ) system is a robust platform for testing models of nucleoninteractions. Current calculations using ab-initio methods with state-of-the-art nucleon-nucleon ( NN ) potentials accurately predict most three-nucleon (3 N ) scattering observables[1]. However, some discrepancies between theory and data remain, such as for the neutron-neutron quasifree scattering ( nn QFS) cross section in nd breakup [2–5].Neutron-neutron QFS in nd breakup is the kinematic configuration in which the protonremains at rest in the laboratory frame during the scattering process. That is, the protonmay be considered as a spectator to the interaction between the two neutrons. Ab-initiocalculations illustrate that the nn QFS cross section is sensitive to the details of the nn interaction, even at low energies where the de Broglie wavelength of the incident neutronis comparable in size to the deuteron. This cross section depends on the nn effective rangeparameter ( r nn ) in the low-momentum expansion of the s -wave scattering amplitude [5].However, early measurements did not determine r nn with high enough precision to examinethe validity of charge symmetry in the NN interaction [6–10].The situation is significantly changed by recent cross-section measurements of nn QFSin nd breakup at incident neutron energies of 26 and 25 MeV. Rigorous nd breakup cal-culations underpredict these data by 18% and 16%, respectively [2, 3]. A third and earlierexperiment measured a similar discrepancy of 12% at 10.3 MeV [4]. A detailed analysis ofthe 26-MeV nn QFS data [2] using rigorous nd breakup calculations demonstrated that 3 N forces cannot account for the discrepancy between data and theory [5]. Also, the analy-sis showed that theory can be brought into agreement with data by scaling the magnitudeof the S nn interaction by a factor of 1.08. However, this remedy suggests substantialcharge symmetry breaking in the NN interaction manifested as either: changes to the S nn scattering length ( a nn ) to the extent of nearly creating a bound dineutron state, a sig-nificant deviation of r nn from the accepted value of the NN effective range parameter, or acombination of changes to the nominal values of a nn and r nn [5]. Possible explanations forthe nn QFS discrepancy include: (1) the NN system violates charge symmetry at a levellarger than generally accepted, (2) current 3 N force models do not properly account for all3 N force components that contribute to the reaction dynamics, and/or (3) the systematicuncertainties were underestimated in the reported measurements.A common feature of the comparisons of theory to data is that calculations describe theshape of the cross-section distribution along the kinematic locus well but fail to predict theabsolute magnitude of the data. This type of discrepancy is suggestive that the systematicuncertainty in the factors used to normalize the cross-section measurements might be un-derestimated. That is, an uncertainty of ±
18% in the beam-target luminosity would bringmeasurements and theory into agreement within one standard deviation. In this paper, wereport new nn QFS cross-section measurements in nd breakup. Our experiment methoddiffers from previous measurements [2–4] in that the setup was optimized to evaluate thetechnique for measuring the absolute nn QFS cross section in nd breakup rather than forsensitivity studies of the strength of the nn interaction. Another important difference isthe method used to determine the integrated beam-target luminosity. In our experiment,the beam-target luminosity is determined from in-situ measurements of the yields for nd elastic scattering rather than from neutron-proton ( np ) scattering. This technique signifi-cantly reduces systematic error in the breakup cross section in comparison to previous nn QFS measurements. Our measurement was conducted at a neutron beam energy of 10.02eV, where theory predicts that the nn QFS cross section measured in the geometry ofour experiment has only modest sensitivity to the S nn interaction. That sensitivity isshown in Fig. 1 where the theoretical cross section averaged over the finite geometry ofour experiment is shown for calculations with and without scaling the S nn interactionby 1.08. The difference in the predicted cross section at the location of the QFS peak (S= 6 MeV) is only 1%. Additionally, a concurrent measurement of the integrated neutronbeam flux was made using np scattering to assess the systematic error in determining theluminosity via nd elastic scattering. S (MeV) M e V s r m b d S Ω d Ω d σ d No scaling 1.08 × S FIG. 1. Plot of the theoretical cross section for nn QFS in nd breakup as a function of arc lengthalong the kinematic locus ( S ) for an incident neutron energy of E n = 10 . θ = θ = 36 . ◦ and ∆ φ = 180 ◦ . Calculations were performed with unscaled interactionmatrix elements (solid black curve) and with the S nn matrix elements scaled by a factor of 1.08(dashed red curve). Both theory calculations have been averaged over the finite geometry of theexperiment using the Monte-Carlo simulation described in this paper. In this paper, we report the results of the 10-MeV measurement. We describe the setupof the experiment in Sec. II. In Sec. III we discuss the details of the data analysis. Our3esults are presented in Sec. IV and summarized in Sec. V.
II. DETAILS OF THE EXPERIMENT
The measurements were conducted at the tandem accelerator facility of the TriangleUniversities Nuclear Laboratory (TUNL) using standard neutron time-of-flight (TOF) tech-niques. The neutron beam was produced via the H( d, n ) He reaction with a pulsed deuteronbeam (period = 400 ns, FWHM = 2 ns) incident on a 3.16-cm-long gas cell filled with deu-terium to a pressure of 5 atm. The resulting neutron beam had a central energy of 10.0 MeVwith a spread of 330 keV (full width) due to energy loss by the deuterons in the deuteriumgas. The deuteron beam current on target was adjusted to optimize the ratio of the true nn coincidence rate to the accidental coincidence background rate.The experiment setup is shown in Fig. 2. A cylindrical scattering sample was mounted12.1 cm from the center of the gas cell with its axis vertical and centered in the beam atthe location of the pivot point about which the detectors rotate. Scattered neutrons weredetected by two heavily shielded liquid scintillators positioned on opposite sides of the beamaxis at equal angles of 36.7 ° . The left and right detectors are 5.08 cm long cylinders withdiameters of 12.7 cm and 8.89 cm filled with NE-213 and NE-218 liquid scintillator fluid,respectively. Each detector was housed inside a cylindrical shielding enclosure of lithium-doped paraffin with a double-truncated conical copper collimator [11]. Tungsten shadowbars were positioned to shield the detectors from directly viewing the neutron productioncell. The distance from the center of the sample to the center of each detector was 264.9cm for the left detector and 264.3 cm for the right detector. The neutron beam flux wasmonitored using two liquid scintillators not shown in Fig. 2. One monitor detector (5.08 cmdiameter × ° with respect tothe beam axis. The other detector (3.81 cm diameter × ° relative to the beam axis.Data were collected over the course of three runs for a total of 577 hours of beam on target.The integrated beam-target luminosity was determined from the nd elastic scattering yields,which were measured simultaneously with the data for the breakup reaction. The integratedincident beam flux was also measured using np scattering to check the systematic uncertaintyin our determination of the beam-target luminosity.All scattering samples used in these measurements were right cylinders. The mass anddimensions of each sample are given in Table I. The deuterium sample was composed of 98.4%isotopically enriched deuterated polyethylene, CD , where “D” denotes “ H” (CambridgeIsotope Laboratories, Inc., DLM-220-0). The np scattering measurements were performedusing the polyethylene (CH ) sample listed in Table I. The large and small graphite sampleswere used to measure the background from neutron scattering on carbon in the CD andCH samples, respectively. In addition to the samples listed in Table I, empty target holderswere used to measure backgrounds from air scattering.The energies of the detected neutrons were determined from TOF measurements. Theincident neutron beam was pulsed at a repetition rate of 2.5 MHz, and the width of eachneutron bunch incident on the scatterer was about 2 ns FWHM. The arrival of the deuteronbeam pulse on the neutron production gas cell was sensed with a capacitive beam pickoffunit. A delayed signal derived from the beam pickoff unit was used as the time reference4 IG. 2. A diagram of the experiment setup (distances are to scale). The sample is 12.1 cm fromthe center of the neutron production cell, and the detectors are placed about 265 cm from thetarget at 36.7 ° on either side of the beam. More details are given in the text.TABLE I. Properties of the scattering samples used.Sample Mass (g) Diameter (mm) Height (mm)CD for measuring the TOF of each detected neutron. Pulse-shape discrimination techniqueswere used to reduce backgrounds from gamma rays. A detector pulse-height threshold of238.5 keVee ( × Cs Compton edge) was applied, where “keVee” denotes “keV electronequivalent”.For the neutron elastic scattering measurements, a TOF histogram was accumulated forthe neutrons that were independently detected in each of the two shielded detectors. Eventsfrom the nd breakup reaction were identified by the coincidence detection of neutrons inthe two shielded detectors. The nd breakup events were accumulated in a two-dimensionalhistogram of the TOF of the neutrons detected in the left detector (D ) versus the TOF5f those detected in the right detector (D ). The events corresponding to the nd breakupreaction lie along a contour defined by the reaction kinematics, i.e., the kinematic locus ofthe reaction or the S curve, as shown in Fig. 3. Arc length along the kinematic locus isdenoted by the variable S and measured in the counterclockwise direction starting at thepoint where the energy of the second neutron is a minimum [1]. (MeV) n1 E ( M e V ) n2 E S = 0increasing S
FIG. 3. Plot of the kinematic locus of allowed neutron energies in nd breakup for the centralgeometry of our experiment (see Fig. 2). The variable S measures the arc length along the locusin the counterclockwise direction starting from the point where E n is zero. A 100 ns wide time window was used to form the coincidences between the signals fromthe two neutron detectors. The accidental coincidence background was measured by formingcoincidences between detector signals caused by neutrons in two consecutive beam pulses.This was achieved by delaying the signal from one detector by 400 ns, which is one beam pulseperiod. With this technique, we detected and accumulated TOF spectra for two categoriesof events: (1) an admixture of nd -breakup events and accidental-coincidence events, and (2)purely accidental-coincidence events. 6 II. DATA ANALYSIS
The differential cross section along the S curve for the nd breakup reaction averaged overthe kinematic acceptance of our experiment setup was determined from the measured nn coincidence yields by Eq. 1: dσ ( θ , θ , ∆ φ ) d Ω d Ω dS = Y nn (cid:15) (cid:15) α α α N n ρ D d Ω d Ω dS . (1)The parameters in Eq. 1 are: the net number of detected nn coincidence events ( Y nn ); theefficiencies of the neutron detectors ( (cid:15) , (cid:15) ); the transmission of the incident neutrons tothe center of the CD sample ( α ); the transmission of the emitted neutrons through theCD sample and air to the face of each neutron detector ( α , α ); the number of neutronsincident on the CD sample ( N n ); the nuclear areal density of the deuterium sample ( ρ D innuclei/cm ); the solid angles of the neutron detectors ( d Ω , d Ω ); and the bin width alongthe S curve ( dS ). The scattering angles θ and θ are defined by the line that connects thecenter of the CD scatterer to the center of each neutron detector, D and D , respectively,shown in Fig. 2. The azimuthal opening angle ∆ φ is defined by the planes containingthe centers of D and D and the incident neutron beam axis. Detector solid angles werecalculated from the detector radii and distances from the sample to the detectors, assuminga point geometry. The Monte-Carlo simulation confirmed this assumption is accurate towithin 0.2%. A. Determination of Breakup Yields
A raw two-dimensional coincidence neutron TOF spectrum is shown in Fig. 4. Thekinematic locus is clearly visible, and the nn QFS region at the center of the locus (enclosedby the red dashed ellipse) is well separated from backgrounds. Accidental coincidences dueto elastic scattering from deuterium and carbon and inelastic scattering from the first excitedstate in carbon form bands parallel to the TOF axes; these are identified by the labels A, Band C in Fig. 4. The accidental coincidences above and to the right of the kinematic locusare due to coincidences between neutrons from nd breakup events in which only one neutronis detected and the elastic scattering of the continuum of neutrons produced via deuteronbreakup reactions in the neutron production cell.Events in a band around the ideal point-geometry kinematic locus ( S curve) definedby the central scattering angles of the experiment θ , θ , and ∆ φ were projected into binsalong the locus. The width of the band was determined by the energy spread and angularacceptance of the experiment. Events were projected using the method of Finckh et al. [12].The S curve was discretized in steps of 50 keV and each event was projected to the closestpoint on the locus. Every event can be represented by a point ( k expn , k expn ) in the k n − k n momentum plane, where k n is the momentum of a neutron in the laboratory frame. Also,any point along the S curve can be represented in momentum space as ( k idealn , k idealn ). Foreach event, the squared distance in momentum space between the event and every point onthe S curve was calculated: d = (cid:0) k idealn − k expn (cid:1) + (cid:0) k idealn − k expn (cid:1) . (2)For each event, the bin on the S curve corresponding to the minimum value of d wasincremented by one count. After projecting onto the S curve the yields were rebinned in7 Neutron 1 TOF (ns) N eu t r on T O F ( n s ) nn QFS C C(n,n) A: H H(n,n) B: C C(n,n’) C: A AB BC C
FIG. 4. Raw two-dimensional neutron TOF coincidence spectrum accumulated with the setupshown in Fig. 2 and described in Sec. II. The vertical scale (i.e., the z-axis) is from a minimumof 1 count to a maximum of 50 counts per bin. The kinematic locus is clearly visible with the nn QFS region circled by the red dashed curve. The main backgrounds from accidental coincidencesare labeled by the blue arrows. This histogram was accumulated in 178 hours of data collection. nd breakup. The net nd breakupyields were computed bin-by-bin along the S curve by subtracting the accidental coincidencecounts from the raw spectrum in each bin, see Fig. 5. The cross section was determinedfrom the net coincidence yields in each bin along the S curve. B. Detector Efficiency Measurements
Detector efficiencies were determined in a separate experiment by measuring the neutronyield from the H( d, n ) He reaction at zero degrees [13]. Measurements were taken for8
S (MeV) C oun t s RawSpectrumAccidentalSpectrum
FIG. 5. Raw and accidental neutron coincidence counts projected onto the S curve. This histogramwas accumulated in 577 hours of data collection. neutron energies between 4 and 10 MeV in 1 MeV steps. The detector efficiency curves weresimulated using the code neff7 [14] between neutron energies of 0 and 20 MeV in 50 keVsteps. The results of the neff7 simulation were scaled to fit the measured efficiencies, asshown in Fig. 6. The simulated efficiencies agreed well with the data; the efficiency curvesfor D and D were scaled up by 0.9% and 0.5%, respectively, to fit the measured efficiencies.The scaled efficiency curves were used in the Monte-Carlo simulation (see Sec. III C).At each end of the S curve in the nn QFS configuration, one of the breakup neutrons hasa very low energy. The simulated energy of each neutron as a function of S is plotted in Fig.7 for our experiment setup. The bands represent the energy spread of neutrons projectedinto each bin along the S curve (one standard deviation). As shown in Fig. 6, the efficiencycurves of the neutron detectors rise sharply from the threshold energy of about 1 MeV upto about 2.3 MeV where the slope of the efficiency curve starts to flatten as a function ofneutron energy. Because the uncertainty in the detector efficiency is greater than ±
50% nearthe threshold energy, events that have a neutron with an energy of less than 2.45 MeV were9 .050.10.150.20.250.30.35 E ff i c i en cy D Neutron Energy (MeV) E ff i c i en cy D FIG. 6.
Top : Plot of efficiency for D . Bottom : Plot of efficiency for D . All efficiencies shownare for a pulse-height threshold of 238.5 keVee ( × Cs Compton edge). Measured efficiencies areindicated by the points. The vertical error bars include statistical and systematic uncertainties.The horizontal error bars show the calculated full energy spread due to deuteron energy loss inthe gas cell used to produce the neutrons. Simulated detector efficiencies are shown by the curves.The simulation results for D and D have been scaled by 1.009 and 1.005 to fit the data. rejected. The energy threshold cut is indicated by the horizontal line in Fig. 7. This cutselects the S -curve region from 4.4 to 7.9 MeV for reporting cross-section data, as indicatedby the vertical lines in Fig. 7. 10 S (MeV) ( M e V ) n E M e V s r m b d S Ω d Ω d σ d FIG. 7. Plot of the simulated neutron energies as a function of S for nn QFS in nd breakupat 10.0 MeV. The detector setup is shown in Fig. 2. The energy of the neutrons is given on theleft vertical axis. The bands show the energy spread (one standard deviation) about the averageneutron energy in each bin. The solid blue band is the energy of the first neutron and the hashedred band is the energy of the second neutron. Also, the finite-geometry averaged cross section as afunction of S is shown by the solid black curve. The cross-section values are given on the verticalscale on the right side of the plot. The horizontal line shows the energy threshold of 2.45 MeV andthe vertical lines show the region of the data passing this threshold cut. C. Monte-Carlo Simulation
A Monte-Carlo (MC) simulation of the experiment was developed for four purposes:(1) to allow for direct comparisons between the experiment and theory by averaging thetheoretical point-geometry cross sections over the finite geometry and energy resolution ofthe experiment; (2) to determine the average neutron transmission factors and detectorefficiencies used to convert the measured coincidence yields into a cross section (see Eq. 1);(3) to determine quantitatively the effects of multiple scattering of neutrons in the target;11nd (4) to quantify sources of background relevant to extracting the nd elastic yields.The MC simulation was used to average the breakup cross section over the finite geometryof the experiment and to determine the average detector efficiencies and transmission factorsin Eq. 1. Scattering events were simulated by tracing individual neutrons from their originin the gas cell to the detection of one or two neutrons in the liquid scintillators. A forcedscattering routine was used for computational efficiency. Details of the MC simulation aredescribed in the Appendix. Theoretical point-geometry nd breakup cross sections used inthe simulation were calculated by solving the three-body Faddeev equations [15] with theCD-Bonn NN potential [16] using the technique described by Gl¨ockle et al. [1]. Neutrondetector efficiencies were determined using the efficiency curves calculated with the code neff7 as discussed in Sec. III B. Finite-geometry averaged values for the product of detectorefficiencies (cid:15) (cid:15) as a function of S are shown in Fig. 8. Neutron transmission factors werecalculated using total neutron scattering cross sections from the ENDF/B-VII.1 database[17]. Finite-geometry averaged values for the product of neutron transmission factors α α as a function of S are shown in Fig. 9.Elastic scattering processes were also simulated for all four scattering samples (see TableI). The elastic scattering simulation used the same input data as the nd breakup simulationfor detector efficiencies and neutron transmission calculations. Cross sections for nd elasticscattering were calculated using the CD-Bonn NN potential. Cross sections for np scatteringwere obtained from the program said using the Bonn potential [18]. Cross sections for elasticand inelastic neutron scattering from carbon were taken from Refs. [17, 19].The simulation was also used to study the effect of multiple scattering of neutrons inthe target on the extraction of nd breakup and elastic scattering yields from the measuredneutron TOF spectra. It was found that multiple scattering accounts for about 9.9% of thebreakup yields near the QFS peak (see Fig. 10) and only 5.0% of the total yields in the nd elastic scattering peak (see Fig. 11). In both cases, the measured yields were corrected toaccount for multiple scattering.Significant background was due to reactions induced with neutrons produced by the H( d, n ) He reaction on deuterons implanted in the tantalum beam stop at the end of theneutron production gas cell. Simulations revealed neutrons produced in the beam stop makeup less than 0.1% of the nn coincidence yields and about 2.7% of the counts in the nd elasticscattering peak, as shown in Fig. 11.As shown in Fig. 11, there is a small background in the region of the nd elastic TOFpeak due to neutron scattering from protons in the approximately 1.6% CH contaminant inthe CD sample. Because of the mass difference in hydrogen and deuterium, less than halfof these events fall within the window of the nd elastic TOF peak. Overall, the simulationsindicate that the np scattering events contribue 0.8% of the total yields in the nd elasticscattering window.Another background quantified by the MC simulation was nd breakup events for whichonly one neutron was detected. As shown in Fig. 11, the energy reach of neutrons from thenon-coincident breakup events is insufficient to contribute to the yields in the window for theelastic TOF peak. These events do contribute significantly to the background at long timesin the TOF spectra measured with the CD sample. However, no such events are presentin TOF spectra measured with the graphite sample. This must be carefully understood toensure proper normalization of TOF spectra for the graphite sample (see Sec. III D).12 S (MeV) ∈ ∈ FIG. 8. Plot of the product of detector efficiencies (cid:15) (cid:15) as a function of S averaged over theexperiment geometry and energy spread using the MC simulation described in the text. D. Luminosity Determination
The product of N n and ρ D in Eq. 1 was determined from the yields for nd elasticscattering, which were measured concurrently with the nd breakup nn coincidence yields.The integrated beam-target luminosity is given by: N n ρ D = Y nd (cid:15) nd α α nd dσd Ω d Ω . (3)The parameters in Eq. 3 are: the net yields for nd elastic scattering ( Y nd ); the efficiencyof the neutron detector at the energy of neutrons from nd elastic scattering ( (cid:15) nd ); thetransmission of the incident neutrons to the center of the sample ( α ); the transmission ofthe scattered neutrons through the sample and air to the face of the neutron detector ( α nd );the differential scattering cross section for nd elastic scattering ( dσd Ω ); and the solid angle ofthe neutron detector ( d Ω). 13
S (MeV) α α FIG. 9. Plot of the product of neutron transmission factors α α as a function of S averaged overthe experiment geometry and energy spread using the MC simulation described in the text. An accurate extraction of the nd elastic scattering yields requires a detailed understandingof the backgrounds in the region of the nd elastic scattering peak in the neutron TOFspectrum as shown in Fig. 11. Two major sources of background were neutrons scatteringfrom air and neutrons scattering elastically from carbon. Scattering from air was measuredusing an empty target holder and the background due to carbon was measured using agraphite sample. The TOF spectra measured with the various samples were normalized toeach other using the integrated beam current, the data acquisition system live time, andthe gas pressure in the neutron production cell. The empty sample TOF spectrum wassubtracted from the spectra measured with the CD and carbon samples.The yields in the inelastic carbon scattering peak were used to finely adjust the normal-ization factor of the spectrum obtained with the graphite sample to the spectrum measuredwith the CD sample. The backgrounds due to neutron multiple scattering in the CD sample, non-coincident nd breakup events, neutrons scattering from hydrogen in the CD sample, and neutrons produced via the H( d, n ) He reaction on deuterons implanted in the14
S (MeV) E v en t s / N M u l t i p l e S c a tt e r N FIG. 10. Plot of the fraction of simulated breakup events in which a neutron scattered twice asa function of S . beam stop were calculated using the MC simulation and subtracted from the measuredspectra.We measured an integrated beam-target luminosity of [4 . ± . ± . × cm − in the left detector and [4 . ± . ± . × cm − in the rightdetector. The average of these values was used in Eq. 1 to calculate the breakup crosssection: (cid:104) N n ρ D (cid:105) = (cid:112) ( N n ρ D ) ( N n ρ D ) . (4)A geometric mean was chosen to better cancel systematic uncertainties in the final result.The value of (cid:104) N n ρ D (cid:105) used to calculate the breakup cross section was [4 . ± . ± . × cm − .Sources of systematic uncertainty in the luminosity determination are listed in Table II.Uncertainties in the yields for nd scattering are mainly due to background subtraction er-rors. Uncertainty in the absolute detector efficiencies is due primarily to the uncertaintiesin the number of deuterium nuclei in the gas cell and the background subtraction in the15 Neutron TOF (ns) × C oun t s sample CDGraphite sampleEmpty target holderSimulated multiplescatteringSimulated beam stopneutronsSimulated non coincident breakupSimulated hydrogenscattering
FIG. 11. Plots of measured TOF spectra for scattering of 10.0 MeV neutrons from the CD sample, graphite sample, and empty target holder at θ = 36 . ◦ . From left to right, the peaksin the spectrum are from elastic scattering on carbon, deuterium, and inelastic scattering fromcarbon. The plots include an overlay of the sum of simulated TOF spectra for multiple scatteringof neutrons in the target, scattering of neutrons produced in the beam stop, neutrons scatteringfrom hydrogen in the sample, and neutrons from non-coincident nd breakup events. efficiency measurements, as well as the uncertainties in the evaluated H( d, n ) He reactioncross sections used to calculate the efficiencies [13, 20]. The uncertainty in the relative de-tector efficiency is based on the variance between the simulated detector efficiency curvesand measured efficiencies (see Fig. 6). A significant contribution to the uncertainty in thedetector efficiency is due to drifts in the detector threshold (or gain) over time. Uncertaintiesin neutron transmission are due to uncertainties in the total cross section data [17]. The un-certainty in the cross section for nd scattering comes from the differences in the values givenby different NN potentials [21]. The uncertainty in the solid angle is mainly due to mea-surement errors in the distances from the sample to the detectors. The uncertainties for theneutron transmission factors and the uncertainties for the absolute detector efficiencies are16orrelated. They must be summed before adding in quadrature with the other uncorrelateduncertainties. This is accounted for in Table II. TABLE II. Sources of systematic uncertainty in the measurement of the beam-target luminosity N n ρ D . See text for details.Source Magnitude (%)Yields in nd elastic peak 2.3Absolute detector efficiency 3.9Relative detector efficiency 1.1Detector gain drift 0.5Neutron transmission 1.1Cross section for nd elastic scattering 1.5Solid angle 0.4Total 5.1 As a benchmark on our method for determining the beam-target luminosity, the nd elasticscattering cross section was determined relative to the np scattering cross section at 32 ° inthe lab. This angle was chosen to kinematically separate neutrons scattering on hydrogenfrom neutrons scattering on carbon. The np scattering yields were extracted from TOFspectra in the same way as the nd elastic scattering yields. The np scattering cross sectionsused in this work were obtained from Ref. [18].We measured an nd elastic scattering cross-section value of 213 . ± . ± . . ± . ± . NN potential. TABLE III. Sources of systematic uncertainty in the measurement of the nd elastic scatteringcross section.Source Magnitude (%)Yields in nd elastic peak 3.2Yields in np peak 1.4Finite geometry correction 2.2Relative detector efficiency 2.1Detector gain drift 1.4Number of deuterium nuclei 1.0Number of hydrogen nuclei 0.4Neutron transmission 0.8Cross section for np scattering 0.4Live time correction 0.6Total 5.1 The sources of systematic errors in our measurement of the nd elastic scattering cross sec-tion are listed in Table III. Uncertainties in the yields for np scattering are due to background17ubtraction errors. There is substantial uncertainty in the finite-geometry correction usedto account for the difference in the average flux seen by the CD and CH samples becauseof their sizes relative to their distance from the neutron production gas cell. Drifts in thedetector bias over time change the detector efficiencies, resulting in a significant uncertainty.The uncertainty in the number of nuclei in the CD sample is due to the unknown chemicalpurity of the sample, which is listed as > . ± . sample is mainly due to the uncertainty in the measured massof that sample. The uncertainty in the cross section for np scattering is the difference in thevalues given by different NN potentials and partial-wave analyses [18]. The data acquisitionsystem live time was measured in two ways: (1) the number of event triggers passing thedata acquisition system (DAQ) veto were compared to the total number of event triggers,and (2) the number of pulses from a 60 Hz clock passing the DAQ veto were compared tothe total number of clock pulses. The DAQ veto was the logical or of the analog-to-digitalconverter busy signal, time-to-digital converter busy signal, and DAQ computer readoutsignals. The live time measured by the triggers was used to compute the nd elastic crosssection, and the associated uncertainty is the difference between the live times determinedby the two methods. All errors are uncorrelated and added in quadrature. IV. RESULTS
Our cross-section data for nn QFS in nd breakup at 10.0 MeV are plotted as a functionof S in Fig. 12. The curves are predictions of rigorous 3 N calculations based on the CD-Bonn potential that have been averaged over the finite geometry and energy resolution ofthe experiment using the MC simulation. The error bars on the data points represent onlystatistical uncertainties; there is also a systematic uncertainty of ± nd elastic scattering and nn QFS breakup for small drifts in the gain. All other sources of uncertainty are the same asthose discussed in Sec. III D for the determination of the integrated beam-target luminosity.Combining Eqs. 4, 3, and 1 leads to a reduction of several uncertainties. Specifically, theneutron transmission factors α for the incident neutron cancel, resulting in elimination ofthat uncertainty. Uncertainties in the neutron transmission factors and the absolute detectorefficiencies for each detector are correlated and must be added linearly with opposite signsfor factors in the numerator and denominator. The square root in Eq. 4 results in a factor of ≈ √ multiplying uncertainties associated with the luminosity. Summing all uncertainties inquadrature with the appropriate factors (accounted for in Table IV) gives a total systematicuncertainty of ± . χ per datum = 0.97).Integrating the cross section from S = 4.4 MeV to 7.9 MeV gives an integrated measuredcross section of 20 . ± . ± . , which is consistent with the simulatedvalue of 20.1 mb/sr . Scaling the nn S interaction by a factor of × .
08 slightly increasesthe chi-squared value of the comparison between data and theory ( χ per datum = 0.98);however, this change is not significant. 18 S (MeV) M e V s r m b d S Ω d Ω d σ d DataMonte Carlo 1.08 × S FIG. 12. A plot of the measured nn QFS cross section (circles) and the result of the MCsimulation (solid curve) for nd breakup at 10.0 MeV. The experiment setup is shown in Fig. 2.The red dashed curve is the result of the MC simulation performed with the S nn interactionmatrix elements scaled by × .
08. Error bars represent statistical uncertainties only; there is alsoa systematic uncertainty of ± . V. CONCLUSIONS
We have measured the nn QFS cross section in nd breakup at an incident neutron beamenergy of 10.0 MeV using standard neutron TOF techniques. Our setup used a pulsed beamwith an open neutron source and heavily shielded neutron detectors. This was the firstmeasurement of this quantity using this detector and source arrangement. The theoreticalprediction agrees well with the data within the uncertainty of the experiment. The goodagreement between our data and the 3 N calculations indicates that the technique of using nd elastic scattering to determine the beam-target luminosity works well for this type ofmeasurement. With this method we were able to determine the beam-target luminosity toan accuracy of ± . ABLE IV. Sources of systematic uncertainty in the measurement of the nn QFS cross section.See text for details.Source Magnitude (%)Coincidence yields 1.0Absolute detector efficiency 3.9Relative detector efficiency 2.4Detector gain drift 1.1Neutron transmission 0.8Yields in nd elastic peak 2.3Cross section for nd elastic scattering 1.5Solid angle 0.4Total 5.6 discrepancies between previously reported data [2–4] and theory for the cross section for nn QFS in nd breakup. The nn QFS dilemma remains unresolved, suggesting the possibilityof significant charge-symmetry breaking in the NN system. New measurements of nn QFSin nd breakup should be performed at higher energies using a collimated neutron beam formaximum sensitivity to the S nn interaction and using nd elastic scattering to determinethe beam-target luminosity, a technique validated in this work. The measurements shouldbe carried out with a systematic uncertainty less than ± ACKNOWLEDGMENTS
The authors thank the TUNL technical staff for their contributions. The authors ap-preciate the use of the supercomputer cluster of the JSC in J¨ulich, Germany, where partof the numerical calculations were performed. This work is supported in part by the U.S.Department of Energy under grant Nos. DE-FG02-97ER41033 and DE-SC0005367 and bythe Polish National Science Center under grant No. DEC-2016/22/M/ST2/00173.
Appendix
Here we discuss the details of the nd breakup MC simulation and note that the elasticscattering simulation follows a similar procedure. The steps for simulating a single nd breakup history are outlined below.1. A point was randomly selected in each: the neutron production cell, the scatteringsample, and both detectors. These points fix the scattering angles θ , θ , and ∆ φ forthe event. The incident neutron energy E was calculated from the incident deuteronenergy and the kinematics of the H( d, n ) He reaction. The deuteron energy in thegas cell was approximated as a linear function of distance along the axis of the gas celldue to the energy loss in the gas by the deuterons.2. An intensity factor for the incident neutrons, I ( E ), was calculated from the H( d, n ) Hereaction cross section [13] and neutron transmission from the production point in the20as cell to the breakup point in the CD sample.3. The breakup cross section was determined in steps of 50 keV along the S curve us-ing a multiparameter interpolation over a library of theoretical point-geometry crosssections. The incident neutron energy E , the scattering angles θ , θ , ∆ φ , and theposition along the S curve were used as interpolation parameters.4. The product of detector efficiencies (cid:15) (cid:15) and neutron transmission factors α α werecalculated for points at 50 keV intervals along the S curve of the simulated event.5. A weight factor w ( S ) used to calculate the average breakup cross section was tabulated.The weight factor for each breakup event is: w i ( S ) = I ( E ) (cid:15) (cid:15) α α . (A.1)6. The TOF of each neutron was computed at each point along the S curve. The simu-lated TOF t sim was used with the center-to-center distance from the sample to detector d (cid:48) to calculate the energy E (cid:48) of each breakup neutron in the same way as the experi-ment: E (cid:48) = 12 m n (cid:18) d (cid:48) t sim (cid:19) . (A.2)7. The energies E (cid:48) , E (cid:48) of the two breakup neutrons were used to project each simulatedevent onto the point-geometry kinematic locus in the same way as the experimentaldata (see Eq. 2). For each simulated event, the weight factor, the values of thebreakup cross section, the detector efficiencies, and the neutron transmission factorswere stored in bins along the point-geometry S curve.After simulating a sufficient number of histories ( ∼ ), the finite-geometry averagedvalues of the breakup cross section, detector efficiencies, and neutron transmission factorsas a function of S -curve length were calculated using the weight factors from step 5 above.The formula for calculating the average breakup cross section is given by: (cid:28) d σ ( S ) d Ω d Ω dS (cid:29) MC = (cid:80) i w i ( S ) σ i ( S ) (cid:80) i w i ( S ) , (A.3)where σ i ( S ) is the breakup cross section, w i ( S ) are given by Eq. A.1, and the index i runsover events. The average product of detector efficiencies and neutron transmission factorsare calculated similarly.Theoretical point-geometry cross sections in the interpolation library were calculated withthe CD-Bonn NN potential [16]. The library was a five-dimensional array indexed by theincoming neutron energy E , the scattering angles θ , θ , and ∆ φ , and the position along the S curve. The range of the library indices spanned all possible scattering configurations forthe geometry of our experiment and the step size in each dimension was chosen to minimizethe variance between points on the grid while keeping the library to a reasonable size ( ∼ points).Some modifications to the simulation procedure outlined above are necessary to simulatevarious backgrounds. To simulate multiple scattering, a second point was randomly chosenwithin the scattering volume. Processes with more than two neutron scattering sites were21ot considered. The simulation process was otherwise the same as described above. Inthe case of elastic scattering, all permutations of scattering from two nuclei in the samplewere simulated. 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