Toward a new polyethylene scattering law determined using inelastic neutron scattering

Nuclear Instruments and Methods in Physics Research A 711 (2013) 166–179

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

Toward a new polyethylene scattering law determined using inelastic neutron scattering C.M. Lavelle a,n,1, C.-Y. Liu a, M.B. Stone b a b

Physics Department, Indiana University, Bloomington, IN 47408, USA Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 19 October 2012 Received in revised form 16 January 2013 Accepted 16 January 2013 Available online 4 February 2013

Monte Carlo neutron transport codes such as MCNP rely on accurate data for nuclear physics crosssections to produce accurate results. At low energy, this takes the form of scattering laws based on the dynamic structure factor, SðQ ,EÞ. High density polyethylene (HDPE) is frequently employed as a neutron moderator at both high and low temperatures, however the only cross-sections available are for ambient temperatures (  300 K), and the evaluation has not been updated in quite some time. In this paper we describe inelastic neutron scattering measurements on HDPE at 5 and 294 K which are used to improve the scattering law for HDPE. We review some of the past HDPE scattering laws, describe the experimental methods, and compare computations using these models to the measured SðQ ,EÞ. The total cross-section is compared to available data, and the treatment of the carbon secondary scatterer as a free gas is assessed. We also discuss the use of the measurement itself as a scattering law via the one phonon approximation. We show that a scattering law computed using a more detailed model for the Generalized Density of States (GDOS) compares more favorably to this experiment, suggesting that inelastic neutron scattering can play an important role in both the development and validation of new scattering laws for Monte Carlo work. & 2013 Elsevier B.V. All rights reserved.

Keywords: MCNP Neutron transport Inelastic neutron scattering Density of states NJOY Polyethylene

1. Introduction Moderated neutron spectra are often calculated using Monte Carlo techniques, such as those contained within the MCNP code [1]. The computation of leakage spectra and emission time distributions from High Density Polyethylene (HDPE) is of particular interest to a variety of disciplines, ranging from thermal and cold neutron moderators [2–5], neutronic assemblies [6], shielding, multiplying assemblies [7], and moderated detector design [8,9]. The accuracy of a Monte Carlo calculation depends in large part on the fidelity of the underlying nuclear physics data employed by the code as well as the accuracy of geometry. At low energy, this requires sampling the double differential crosssection many times over the history of the neutron’s random walk through the polyethylene (PE) layer. The double differential crosssection is [10,11] 2 d s s kf ¼ b SðQ ,EÞ dOdE 4p ki

n

Corresponding author. E-mail address: [email protected] (C.M. Lavelle). 1 Currently at: The Johns Hopkins University Applied Physics Laboratory.

0168-9002/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2013.01.048

ð1Þ

where sb is the scattering cross-section summed over all the nuclei and k represents the initial and final momentum of the neutron. For polyethylene, sb is dominated by the large incoherent scattering cross-section for hydrogen (20.48 b). The SðQ ,EÞ function describes the structure and collective dynamics of the medium. When written using dimensionless variables [12], SðQ ,EÞ is referred to as the ‘‘scattering law’’ for the material. The momentum transfer, Q, and energy transfer, E, between the initial and final state of the neutron are defined as Q ¼ 9ki kf 9 E ¼ Ei Ef

ð2Þ

where for the neutron mass m the free neutron dispersion is E¼

_2 Q 2 : 2m

ð3Þ

MCNP computations involving ambient temperature HDPE (  300 K) make use of a model for SðQ ,EÞ applied to neutron scattering below 4 eV. This scattering law has been ported forward [13,14] from the original, detailed in the 1969 report ENDF-269 [15], with evaluations at 296 and 350 K [1]. Accurate computations for different HDPE temperatures, such as are needed for cold moderators, would require a new scattering law to be produced. The usual procedure is to use the NJOY code [16], which computes the cross-section in the incoherent approximation.

C.M. Lavelle et al. / Nuclear Instruments and Methods in Physics Research A 711 (2013) 166–179

Incoherent inelastic neutron scattering defines the scattering in terms of a generalized density of states (GDOS), which represents the number of harmonic phonon excitations per unit energy. It is therefore essential to employ an accurate representation of the GDOS to calculate the scattering law. There are several different models for the HDPE scattering law presented in the literature [15,17,18]. These models are generally benchmarked against the rather limited set of measurements found in Ref. [15], which include the total cross-section measurement by Armstrong [19] and a suite of inelastic measurements of the double differential cross-section. The total cross-section measurement should probably be superseded by the more recent one by Granada et al. [20]. The inelastic measurements are on an absolute scale, but are of low energy resolution and cover a limited range of energy transfer about the  200 meV incident neutron energy. All comparisons have been made at ambient temperatures, which is not a sufficient test of a scattering law intended for use at low temperature, such as was done in Ref. [17]. The correction to the low energy portion of the scattering law suggested by Ref. [18] made comparisons to low temperature specific heat data as a test of the GDOS, a practice which also does not provide a sensitive check of the details of the model. Inelastic neutron scattering (INS), on the other hand, can provide an efficient means with which to examine the HDPE scattering law over a broad range of incident neutron energies and sample temperatures [21]. The main region of interest for the computation of thermal and cold neutron moderator spectra conventionally extends from 0.1 meV to 4 eV, which is well within the reach of a modern neutron spectrometer. INS measurements often lack an absolute scale, which should not be detrimental because the overall normalization of the calculated SðQ ,EÞ is provided by the bound atom cross-section and by the sum rules for SðQ ,EÞ [10]. This paper will review the results of an inelastic scattering experiment on HDPE performed using the wide angular range chopper spectrometer (ARCS) [22,23] at the Spallation Neutron Source (SNS) [24]. Our goal is to compose a HDPE GDOS which is consistent with the primary experimental data for 5 and 294 K. The analysis approach will be to compare several theoretical GDOS models to the experimental data in the presence of instrumental resolution. The models for the GDOS are chosen from a survey of the history of the current scattering laws and of the general literature. Throughout the remainder of the text, the models will be referred to using the acronyms identified by the bold letters

 ENDF: The current ENDF evaluation, with GDOS taken from





the details presented in Ref. [15]. The ENDF HDPE file comments [25] indicate it is derived from a treatment of an isolated molecular chain [26] and employs a Debye model below 20 meV [27]. KM: A composite model we have created using the GDOS presented by Lynch [28] for the high energy phonons, and the work of Kitagawa and Miazawa [29] at low energy. The KM work was generated in the same period as the ENDF law, but had moved beyond the isolated molecule treatment. Our model is derived by replacing the region below 71 meV in Ref. [28] by an equal area taken from the computation in Ref. [29] to describe the inter-chain modes. Data are extracted from digitized graphs. ST/LH: A composite model by Hill and Liu [17] using the GDOS correction suggested by Swaminathan and Tewari [18] for the low energy phonons, and the ENDF evaluation for the high energy phonons. ST/LH employs a different low energy density of states than is seen in KM. This is the most recent scattering





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law available in the literature. Data are extracted from digitized graphs. Barrera: A recent density functional theory (DFT) calculation for the inelastic scattering from HDPE [30]. This work represents an ab initio model considering the entire orthorhombic cell, without making a distinction between the intra- and inter- chain excitations, and without fitting parameters to measurements. The one phonon data presented at 20 K are extracted from the graph and inverted to obtain the density of states. Barrera does not show the relatively dispersionless C–H stretching modes, so we have modeled this feature as single Gaussian mode at 360 meV energy transfer with a FWHM of 10 meV and total area fraction of 0.31. This choice of area for the peak provides agreement with the measured value of the mean square displacement (u2 Þ. ARCS: The measured data is used in the one phonon approximation to the GDOS to examine whether the measurement itself can or should be used in lieu of a detailed calculation.

The paper is structured as follows. Section 2 reviews the basic theory behind the scattering law and gives an overview of the HDPE dynamics relevant to our measurements. Section 3 describes the samples and ARCS instrument, and provides a detailed discussion of the instrumental resolution. Section 4 describes the experimental results and analysis methods for elastic and inelastic scattering, and obtains the measured GDOS in the one-phonon approximation. Section 5 makes detailed comparisons between the SðQ ,EÞ calculated from each model, showing the KM and Barrera-based models provide the most accurate results. Results of the models are compared to the total cross-section at 294 K [31] before we conclude the analysis in Section 6.

2. Neutron scattering from HDPE 2.1. Incoherent neutron scattering Neutron scattering from HDPE is dominated by the large incoherent scattering cross-section for an ensemble of H nuclei [10]. We will describe our inelastic SðQ ,EÞ observations by considering the incoherent scattering only, a practice referred to as the incoherent approximation. We expect this description to be appropriate for HDPE, however in practice the same treatment is also used to develop cross-sections for strong coherent scatterers such as graphite [32]. The general reasoning to apply this treatment in all cases, besides its relative simplicity, is that even if the material was a strong coherent scatterer, the multi-phonon terms would spread out the scattering in (Q,E) space to such a strong degree that the coherence would not be recognizable [10]. Also, it is surmised that multiple scattering events within the layer would blur out any fine details resulting from well defined dispersion relations. Strongly coherent scatters now find wide use in moderated neutronic systems, such as cold moderators and ultra-cold neutron converters. Solid deuterium [73,33,74,34] and solid oxygen [35] are strong coherent scatterers used for the production of cold and ultra-cold neutrons (UCN,  100 neV). In these systems, the cryogenic solid can serve as both cold moderator and/or UCN converter. The arguments in favor of the incoherent approximation become strained at low temperature and low energy for a number of reasons. For example, the sampling of just one or two scattering events from a thin layer is a possibility. The consideration of coherent scattering in thin layers is also important for simulations which include the moderator vessel’s external structure. Recent

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concepts are also exploring the use of coherent scatterers within the moderator itself to increase cold neutron yield [36]. Finally, at low temperature and small energy transfer (E t 20 meV) the low order phonon scattering very clearly reflects the fine details of the GDOS. In a similar way, the fine details of the scattering law for a coherent scatterer would also be preserved. In the specific case of UCN production, dominated by one-phonon scattering, the incoherent approximation appears to fail by over predicting the intersection of the free neutron dispersion with SðQ ,EÞ [33]. A methodology based on the incoherent approximation would only account for phonon scattering alone. Other scattering mechanisms, such as small angle scattering [37], rotational excitations, and quasi-elastic scattering are neglected. These considerations are of growing interest for computing low energy neutron yields far from a cold neutron source for instrument design, or for investigating the potential of more exotic moderator materials which may exploit rotational excitations, confinement [37], amorphous dynamics, mixed moderator materials [38], or coherent scattering [36]. Future developments in scattering laws may wish to address these concerns. Nevertheless, in the case of HDPE, the incoherent approximation should be nearly exact. The incoherent SðQ ,EÞ, using the notation of Ref. [39], is written as the sum of an elastic and an inelastic term SðQ ,EÞ ¼ e2W dðEÞ þ e2W

1 X 1 2 2 n ðu Q Þ U n ðEÞ n! n¼1

ð4Þ

where the phonon excitation terms are Un(E) and the elastic scattering is represented by the delta function. The first phonon term is given by the GDOS, Z(E), as 8 > _2 ZðEÞ > > ðNðEÞ þ 1Þ for E Z 0 > < 2 mu2 E ð5Þ U 1 ðEÞ ¼ 2 > _ ZðEÞ > > NðEÞ for E o 0: > : 2 mu2 E The Debye–Waller factor, 2W ¼ u2 Q 2 , is temperature dependent and represents the effect of the atom’s mean square displacement, u2 . It is also calculated from the GDOS Z 1 _2 ZðEÞ ð2NðEÞ þ1Þ ð6Þ u2 ¼ dE E 2m 0 2

and m is the neutron mass (a useful value is _ =2m ¼ 2 2:07 meV A˚ Þ. The boson occupation number for b ¼ 1=kb T is NðEÞ ¼

1 ebE 1

and the GDOS is normalized as Z 1 ZðEÞ dE ¼ 1:

ð7Þ

ð8Þ

o

The multi-phonon overtones, n 4 1, are determined from successive convolutions of the one-phonon term Z 1 0 U n1 ðE0 ÞU 1 ðEE0 Þ dE : ð9Þ U n ðEÞ ¼ 1

At each stage, the Un terms are subject to the normalization condition Z 1 U n ðEÞ dE ¼ 1: ð10Þ 1

The scattering law must also obey the sum rules for the moments Z 1 SðQ ,EÞ dE ¼ 1 1

Z

1

ESðQ ,EÞ dE ¼ 1

_2 Q 2 2m

ð11Þ

and the detailed balance condition SðQ ,EÞ ¼ ebE SðQ ,EÞ:

ð12Þ

The measured dynamic structure factor, Smeas ðQ ,EÞ, is sometimes taken to represent an approximation to the one phonon term by inverting Eq. (4), averaging over Q, and assuming that U meas ðEÞ  U 1 ðEÞ Z Q þ dQ 1 Smeas ðQ ,EÞ U meas ðEÞ ¼ dQ : ð13Þ 2 dQ Q u2 Q 2 eu2 Q The cubic crystal approximation is usually invoked for scattering law calculations (i.e., a perfectly symmetric crystal). In this limit the mean square displacement is the same along all symmetry axes, and we have used u2 to represent the mean square displacement computed along any one axis. The mean square displacement averaged over all directions is then 3u2 . Our samples are not oriented, and we show good agreement using the cubic crystal approximation despite the highly anisotropic structure of HDPE. The HDPE measurements presented here will eventually be used to develop and validate improved data for thermal and cold moderator computations. However, it is the NJOY code, and the LEAPR module therein, which constructs the actual data file used within MCNP. Given the anticipated widespread usage of any new data concerning HDPE, it is advisable to move forward deliberately. We have therefore not employed the NJOY code as an intermediary to compute SðQ ,EÞ. Instead, we follow the equations above, using a fine linear grid in Q and E while simultaneously ensuring that the normalization conditions and sum rules are satisfied. Our future work will focus on obtaining an NJOY based calculation for SðQ ,EÞ, @2 s=@O@Ei , and stotal ðEi Þ which is consistent with the experimental data presented in this paper before introducing a new scattering law to the general community. 2.2. Structure and dynamics HDPE is composed of long chains of CH2 molecules arranged in an orthorhombic crystal. Regions of amorphous disorder are embedded within crystalline regions [40], such that samples are generally described in terms of their fractional crystallinity (70% crystalline in our case [41]). The dynamics can be roughly divided into modes corresponding to the interchain soft phonon modes o 71 meV, and to the intrachain excitations of an isolated chain 4 71 meV, although an additional intrachain mode has been identified near 24 meV [28,29]. The highly anisotropic shape of the molecule leads to distinct transverse and longitudinal components in both the inter- and intra-chain modes. The features of the different components have been confirmed by INS measurement on drawn samples (stretch oriented) [28,30,42]. The orthorhombic unit cell was determined from x-ray diffraction [43,44]. This result is shared by experiments involving electron diffraction on PE [45] and neutron diffraction experi˚ b ¼4.48 A, ˚ ments on deuterated PE [46]. The unit cell is a¼7.64 A, ˚ with the backbone of the molecule extending along and c ¼2.54 A, the c-axis. The first computations of the dynamics focused on an isolated ‘‘zig-zag’’ chain [26,47], a model which appears to provide an adequate description of the GDOS for the modes above  71 meV, but does not address the low energy phonons. One such computation [26] forms the basis of the ENDF model. A more detailed picture of the dynamics is found in the work of Shimanouchi [48], Tasumi and Krimm [49], and Kitagawa and Miazawa [29].

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Informed by INS experiments [28,42,50], a phonon treatment emerged [29] which predicts a complicated structure in the GDOS resulting from the significant dispersion in these low energy modes. A survey of the state of calculation methods at this time is found in Barnes and Franconi [51]. The INS experiments referenced in the Kitagawa work do not extend to energy transfers greater than 100 meV. A series of measurements by Parker [52,53] and Tomkinson and co-workers [54] also give insight into the low energy structure of the GDOS by investigating smaller chains of CH2 molecules, the n-alkanes. The dispersion curves derived from the measurements presented in Ref. [54] share many of the same details we observe in the measured GDOS of the full HDPE solid. The higher energy modes were measured with good resolution by Jobic [55] and by Parker [52,53]. Jobic notes that high Q excitations would be suppressed by the Debye–Waller factor, but were nevertheless observed [56]. Further study [57] of this effect lead them full circle to return to a variant of Wick’s short collision time approximation (SCT) [58]. A high phonon number summation of the incoherent scattering should also tend to the SCT [59]. We found no need to invoke special treatments or the SCT to describe SðQ ,EÞ when up to 100 phonons are included in the summation. 2.3. Crystallinity and low energy The incoherent approximation requires the GDOS to tend to zero as E2 (Debye-like crystal) to ensure convergence of the phonon sum and a finite Debye–Waller factor. On the other hand, the crystallinity of the sample itself can have a significant effect on the low energy dynamics [40,60]. Amorphous samples show an excess over Debye-like behavior [60,61], such that significant amorphous composition tends to enlarge the value of u2 [60], and can also contribute higher order powers of Q in the elastic component [40]. It has also been suggested that a fractional dimensionality Debye model could be used to represent the low energy portion of the GDOS [62]. The instrumental resolution of the ARCS spectrometer is about 1.6 meV FHWM for 50 meV incident neutron energy. Therefore these data do not permit qualitative statements to be made about the dynamics below a few meV. We enforce Debye-like E2 behavior for all models below 6 meV. Our samples are of a higher degree of crystallinity, so the Debye-like behavior should be sufficiently representative over the range of energy transfers accessible to these measurements. Ref. [60] demonstrates observational evidence for the type of GDOS modifications which could be made to accurately represent the low energy behavior of an amorphous HDPE, assuming the harmonic phonon approximation for SðQ ,EÞ remains viable. An increase in the low energy portion of the GDOS in amorphous HDPE is worth future consideration, since the increase might improve cold neutron yield from a low temperature moderator.

3. Measurements

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The HDPE samples were obtained from Goodfellow corporation in 10 mm thick sheet. The manufacturer reports the sample is produced from bulk by simultaneously inflating and drawing of the sample (blow film), which provides a crystallinity of 70–80% [41]. The primary measurements were conducted at 5 K to suppress multi-phonon processes, but we also collected data at other typical moderator operating temperatures (20, 77, and 294 K). This also permits us to examine whether a GDOS which is successful at low T will also give good results at high T. The high T GDOS can change if there is a phase change, or as is the case of HDPE, the influence of quasi-elastic modes and rotational excitations is enhanced at high T [61,63]. We acquired data on a range of samples, including 10021000 mm samples of Low Density PE and Ultra High Molecular Weight PE sheet, HDPE skived from the moderator material employed at the LENS facility [6], and the pelletized form of HDPE used at the UCNA ultra-cold neutron source [64]. These materials showed similar results, but we have not yet completed an assessment of the multiple scattering correction for these thicker samples. We therefore defer discussion of these measurements. Measurements were also performed on PMMA (Plexi-glass) and deuterated-PMMA, and we are exploring how best to make these data available to the wider community.

3.2. Instrument ARCS is a direct geometry time of flight (TOF) instrument [22,23]. Briefly, its operation is as follows. A pulse of fast neutrons undergoes thermalization in a decoupled, ambient temperature light water moderator. The neutrons emitted by this moderator are then highly collimated and mechanically chopped to increase resolution over the incident neutron energy. The combined effects of the moderator’s intrinsic emission time distribution and the chopper result in an asymmetric resolution function described [65]. The result of the resolution function is most clearly seen in the elastic scattering peak, especially for E o 0, shown for our measurement in Fig. 1 (see also [23] for additional examples). The thin film samples were mounted within a rectangular copper frame that was attached to the cold finger of a closed cycle 2

10

1

10

0

10

−1

10

3.1. Samples Hydrogen’s large incoherent cross-section requires the use of thin film samples to reduce the influence of multiple scattering. We also exercise caution in the analysis to reduce the degree of multiple scattering contamination by avoiding scattering angles through the thickness of the film. Because the mean free path decreases with decreasing neutron energy [20], we also decrease the multiple scattering by avoiding the use of data with small exit energies (i.e., E  Ei Þ.

−2

10

−10

0

10

20

30

40

50

Fig. 1. A background subtracted measurement acquired with Ei ¼ 50 meV neutrons for a fixed Q value at T¼ 5 K. The long tail of the elastic peak at E o 0 is due to instrumental resolution. The incoherent inelastic SðQ ,EÞ computed using the Barrera model and the background measurement are shown for comparison.

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Fig. 2. Energy transfer resolution. The full-width at half maximum (FWHM), sðEÞ, of the resolution for different incident neutron energies as a function of energy transfer.

Fig. 3. Resolution model compared to data for different incident energies. The widened resolution curve is the elastic peak at 50 meV broadened to the expected FWHM. The Gaussian width is determined from the expected FWHM.

refrigerator (CCR). The CCR operated between 5 K and 300 K. An aluminum thermal shield kept at approximately 50 K surrounded the sample and sample holder. The sample temperature was monitored at the corner of the frame furthest from the coldfinger. The incident beam side of the sample holder frame was shielded by boron carbide. The copper frame of the sample holder was outside of the incident beam dimensions and was oriented at an angle of 45 1 with respect to the incident beam. This effectively places the frame of the sample holder out of the path of the scattered neutron beam for the range of scattering angles sampled for the measurements presented. However, we later observed evidence that mounting the film sample at an angle to the beam introduced a Q dependence to the energy transfer resolution (see Section 3.3). The data is processed into NXSPE files at the instrument after a correction for detector efficiency by a vanadium measurement. This file is exported by the DAVE software [66] SðQ ,EÞ into a table for further post-analysis in MATLAB. An example measurement acquired with Ei ¼ 50 meV neutrons is shown in Fig. 1, where the long tail of the resolution function is evident in comparison to a best-fit gaussian at the elastic scatting position (E¼0). We therefore neglect negative energy transfers in most of this analysis. Fig. 2 shows the expected full width at half-maximum (FWHM) of the energy resolution function, sðEÞ. The resolution improves with increasing energy transfer. It is clear from these data that a measurement at a single large Ei is not sufficient to cover the entire dynamic range with good resolution. The models for the HDPE GDOS extend up to the C–H stretching modes near  360 meV [28], but the measured region needs to extend further if we are to include higher order multi-phonon processes in our analysis. We therefore will condense four measurements at Ei ¼50, 100, 250, and 675 meV to reconstruct SðQ ,EÞ across the entire dynamic range.

3 meV, and the resulting resolution peak is limited to 10 to 3 meV, then normalized to unity. The measured FWHM of this curve is so ¼ 1:6 meV, consistent with the expected value. A scaling law translates the model’s elastic peak resolution to the resolution of a different energy transfer

3.3. Resolution model To facilitate comparisons of calculated SðQ ,E) to the data, a model for the non-Gaussian experimental resolution is needed. We obtain this function from the elastic peak shown in Fig. 1 after subtraction of the inelastic component. For the Ei ¼ 50 meV data, the inelastic phonon background is normalized to the data above

RðE, sðEÞÞ ¼ aRðaE, so Þ:

ð14Þ

where a ¼ so =sðEÞ. The resolution convolution is applied to the data at fixed Q Smeas ðQ ,EÞ ¼ RðE, sÞ  SðQ ,EÞ Z 1 0 dE RðEE0 , sÞSðQ ,E0 Þ: ¼

ð15Þ

1

As shown in Fig. 3, the modeled resolution curve satisfactorily describes the elastic peak of the 50 and 100 meV incident energy data. For the 250 and 675 meV incident energies, the tail at E o 0 is better fit to a Gaussian at the expected FWHM, but the positive energy transfer is extended a bit. The extension is likely due to the inclusion of the E 40 dynamics within the resolution of the elastic peak. The resolution function used at 250 and 675 meV is therefore a Gaussian of FWHM set by the data presented in Fig. 2. We found the computed FWHM shown in Fig. 2 for the instrumental resolution accurately describes the elastic peak width for Q =k0 t0:5, but the peak width increased with increasing Q, as shown in Fig. 4. This effect, with its initial rise, fall, and rise again at larger scattering angles has recently been shown [67] to result from the use of thin film samples oriented at an angle to the incident beam. 3.4. Analysis strategy The one-phonon approximation provides an efficient means to compare our measurements to the different GDOS models in present use. However, our analysis shows that the GDOS cannot be determined precisely by the measurement due to the combined effects of the multi-phonon component and instrumental resolution. The deconvolution algorithm suggested by Ref. [39] was attempted, but the result deformed the density of states to such a great extent that caution was warranted. Additionally, because multiple incident energies were used to accurately measure a large range of (Q,E) space, it is not clear how

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Fig. 4. Elastic peak width as a function of Q as determined from the different incident energy measurements.

the variable instrumental resolution would affect the total area under different features of the GDOS following deconvolution over such a broad energy range. Our general approach will therefore be more conservative. We compare resolution convoluted computations of the incoherent SðQ ,EÞ directly with the primary scattering data for each of the different GDOS models. Several considerations factor into the points in (Q,E) space chosen for detailed analysis. For example, the effect of multiple scattering within the sample is most likely to occur when the exit energy is small, but the energy resolution improves with increasing E. It is also advisable to prepare SðQ ,EÞ by averaging over low Q values to limit multi-phonon contamination for evaluation of the one-phonon dynamics, but the information at large Q is a useful test of the multi-phonon portion of the scattering law. We therefore present results multiple ways. An initial comparison is made to the GDOS models using the one-phonon approximation. We then evaluate the models at fixed Q for each scan. Finally, we select a GDOS model based on the Barrera calculation [30] to make a final overlay of the data and calculated SðQ ,EÞ as a contour plot in (Q,E) space. No multiple scattering corrections have been performed to the data presented here.

4. Results 4.1. Elastic scattering results The mean square displacement u2 is needed to successfully average over a range of Q values when applying Eq. (13). Good knowledge of u2 can also provide a valuable check of Z(E) via Eq. (6). We obtain u2 from the Ei ¼ 50 meV data by subtracting the inelastic component at each Q and integrating the remainder between 10 r Er 3 meV to derive S(Q). We then compare that 2 2 result to eu Q . The S(Q) result and fit is shown in Fig. 5, for 2 u2 ¼ 0:020 70:001 A˚ . This value is commensurate with other values found in the literature [28,60,61]. Multi-phonon terms will therefore become an increasingly dominant part of SðQ ,EÞ as

Fig. 5. S(Q), the elastic scattering structure factor, showing the described fit 2 yielding u2 ¼ 0:020 A˚ . Data are integrated as described in the text for the Ei ¼ 50 meV measurement. The residual shows the HDPE lattice [110] peak at 1 Q ¼ 1:5 A˚ , which is clearly distinct from the sample frame background. Other expected peak locations are shown as the red vertical lines, with height proportional to intensity relative to the [110] peak [44,46] (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.) .

1

u2 Q 2 \1, or Q 47 A˚ . The low temperature value is closer to the crystalline value reported by Ref. [60]. Repeating the procedure at 2 300 K gives u2 ¼ 0:0557 0:002 A˚ , also commensurate with crystalline PE. As discussed earlier, non-Q2 behavior has been observed in the exponent of S(Q) for HDPE [40,61], which is attributed to amorphous dynamics leading to anisotropies in the assumed Gaussian distribution of displacements about the mean value [40]. Above about 150 K, additional mean square displacement due to rotational excitations have been observed [61,63]. Nevertheless, the fitted value for u2 is found with a narrow confidence range and we do not observe significant departure from the Gaussian behavior in S(Q). This is likely the result of working at low temperature with a highly crystalline form of HDPE. Subtracting off the incoherent S(Q) from the data in Fig. 5 reveals the underlying coherent scattering. Fig. 5 appears to show the most intense [110] diffraction peak from the HDPE lattice, 1 which is expected near Q ¼ 1:5 A˚ . The expected location and relative intensity of additional diffraction peaks are indicated as the thin red lines on the figure with height in proportion to the [110] peak intensity. The intensities measured by Ref. [46] for deuterated-PE, but using the unit cell dimensions for hydrogenated-PE [44], were employed to calculate the Q values for the diffraction peaks. Several possible HDPE diffraction peaks are seen despite the strong incoherent scattering. Most of these peaks are not clearly set apart from the background of the sample holder. Additionally, the empty sample frame background was acquired only at 294 K. The diffraction peaks from the sample holder would be sharpened and enhanced by the Debye–Waller factor at low temperature as compared to those measured at 294 K. Thus, we observe a small incommensurate background subtraction in the elastic scan. The [110] peak, however, sits clearly apart from the background peaks and was observed in all our other measurements on LDPE, UHMWPE, and the HDPE from other sources. As shown in Fig. 5, the signal to background ratio is high enough that the

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measurement of background at a higher temperature should not affect the inelastic portion of the measurement.

4.2. S(Q,E) from multiple measurements The measured value of SðQ ,EÞ at a fixed Q across the entire dynamic range of HDPE is obtained by combining the measurements at different incident energies. A balance between the desire for low Q but sufficient coverage of the dynamics leads us to 1 average from 5.75 to 6:25 A˚ for incident neutron energies of 50 1 and 100 meV, and from 5.75 to 7:25 A˚ for the 250 and 675 meV incident energy measurements. The results of different energy scans are shown in Fig. 6, normalized to the same integral area between the regions indicated by the black vertical lines. The normalization of the 50 meV measurement is carried forward in each successively higher incident energy used. For example, the measurement at 100 meV is scaled to agree with the measurement taken at 50 meV. The measurement taken at 250 meV is normalized to the newly scaled measurement taken at 100 meV, and so forth. The application of the resolution convolution should allow the transformation of a lower energy measurement to the broader resolution of a higher energy measurement. In this case, we assume that the result of successive convolutions add in quadrature, such that to broaden a measurement of width sA to a width sB , the FHWM of the resolution function should be

sðEÞ2 ¼ sB ðEÞ2 sA ðEÞ2 :

ð16Þ

Agreement for the overlapping ranges of energy transfer is satisfactory (Fig. 6), indicating that the resolution functions derived in Section 3.3 are appropriate. The curves between 0 and 500 meV with different Ei are connected piecewise at 40, 80, and 220 meV as shown in Fig. 6(d). The full SðQ ,EÞ resulting from this combination of measurements is displayed in Fig. 7, also showing the regions of accessible phase space at each incident energy. 3 2 1 0 5

0 5

0 5

0

0

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40

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100

0

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100

150

200

250

0

100

0

50

200

100

150

300

200

400

250

300

350

400

450

Fig. 6. Normalization procedure between different scans of SðQ ,EÞ. The Ei ¼ 50 and 1 100 meV scan are averaged between Q¼ 5.75 and 6:25 A˚ , and the Ei ¼250 and 1 675 meV scans are averaged between Q¼ 5.75 and 7:25 A˚ . The integral area of a higher incident energy is normalized to that of the lower one within energy regions indicated by the black vertical lines. Subplots (a), (b), and (c) show that the resolution convolution of the lower energy scan faithfully reproduces the higher energy scan. This gives confidence that the resolution convolution applied to calculated values of SðQ ,EÞ can be compared to the experimental data. The crosscalibrated combined measurements are shown in subplot (d).

Fig. 7. The SðQ ,EÞ for HDPE measured at different incident energies, normalized together as described in the text and Fig. 6. The phase space accessible to incident energies of 50, 100, 250, and 675 meV is indicated by the black lines. Data are only 1 shown for Q o 17 A˚ .

4.3. Comparison to GDOS models Fig. 8 shows the various models compared to the one-phonon approximation for the GDOS obtained from the combined measurements. The data and models are shown for energy transfers between 0 and 80 meV (Fig. 8(a)) and 0 and 400 meV (Fig. 8(b)).The multiphonon terms contribute to the normalization, and artificially suppress the measured GDOS in comparison to the other models. Because the integral normalization of each Un(E) term is to unity, we can estimate the additional area under the normalization due to multi-phonon terms Z 1 Z X 1 2 2 n1 ðu Q Þ U n ðEÞ dE U Meas ðEÞ dE ¼ n! n¼1 ¼ 1þ

1 X 1 2 2 n1 ðu Q Þ : n! n¼2

ð17Þ

Therefore, the GDOS obtained directly from the measurement and shown in Fig. 8 is normalized to a larger value than unity for comparison to the other models. However, the GDOS remains normalized to unity when computing SðQ ,EÞ so that the sum rules are satisfied. The areas under each basic region of the GDOS and resulting u2 for each model are given in Table 1. Several features are shared between the models and experimental data. The lack of dispersion in the high energy modes leads to peaks in the GDOS referred to as ‘‘Van Hove Singularities’’ [21]. The measured peaks, which are near 80, 140, 160, 170, and 360 meV, are broader than implied by instrumental resolution because observation of these modes is made in the presence of multi-phonon convolution, or ‘‘phonon-wings’’ [21]. The high energy peaks in the ENDF model therefore appear overly broad after accounting for these effects. The ‘‘double hump’’ structure at low energy results from the significant dispersion of the low energy modes [53,54]. This feature is captured by the KM, ST/LH, and Barrera models. The details of these dispersions are well illustrated by Tomkinson et al. [54]. The ST/LH model’s cutoff frequencies are found to be too low, and the overall shape of the resultant GDOS is not representative of the ARCS measurement. Fig. 8 also reveals the magnitude of the deviation between the ENDF model’s Debye approximation and the measurement (i.e., for energy transfers below 20 meV). The models based on the more detailed treatment by KM or Barrera are more successful at low energy. The flat region in the

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0.02 0.015 0.01 0.005 0

0

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0

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350

400

0.06 0.05 0.04 0.03 0.02 0.01 0

Fig. 8. Comparison of suggested models for Z(E), including the current ENDF model, details in text. The experimental data presented in the one phonon approximation clearly excludes both the ENDF and the ST/LH models. The double hump structure near 20 meV results from the dispersion of the translation excitations, and the constant value between 30 and 70 meV results from the one-dimensional nature of the longitudinal mode. At  350 meV, we see the Gaussian approximation in the ENDF model is qualitatively correct, but smooths the away fine details found in more recent calculations.

Table 1 Characteristic areas under each part of the GDOS for each scattering law model, and u2 at 5 K. The value of u2 from the 5 K S(Q) elastic window scan is

5. Calculation and experiment 5.1. Low temperature, 5 K

2

0:020 7 0:001 A˚ . The remainder corresponds to the remaining spectral weight outside of the three regions listed in the table. Model

u2 ðA˚ Þ at 5 K

Area: o 71 meV

Area: 71– 200 meV

Area: 340– 380 meV

Remainder

ENDF KM ST/LH Barrera ARCS, one phonon

0.024 0.022 0.027 0.021 0.017

0.112 0.105 0.121 0.114 0.085

0.577 0.602 0.571 0.576 0.397

0.306 0.293 0.306 0.310 0.229

0.005 – 0.002 – 0.289

2

observed data extending from  25271 meV is attributed to be the one-dimensional Debye GDOS of the longitudinal mode. The measured value exceeds the model values because the multiphonon overtones are superimposed over this region, such as were computed in Ref. [50]. The region from 220 to 360 meV shows no excitations in any model, but a broad area is seen in the measured GDOS resulting from multi-phonon terms. The C–H stretch modes near 360 meV in the measurement show a width commensurate with instrumental resolution and phonon-wing broadening, but are shifted to a slightly different energies in the ENDF and ST/LH treatments. The KM model predicts two sharp peaks close together in this region, resulting from the symmetric and anti-symmetric stretch of the C–H bond. The two peaks are within instrumental resolution, such that we observe only a single broad peak. The effects of the area and cutoff assignments will be quantitatively assessed in Section 5.

5.1.1. Mean square displacement, u2 Results for u2 obtained from the various models (Eq. (6)) and measurements at 5 K are shown in Table 1. These data provide another point of comparison between GDOS models and the measurement. At 300 K, the measured value would also contain a component of the mean square displacement which is due to rotational motion at temperatures above  200 K [63], making the high temperature value a potentially less stringent test of the GDOS since other effects may be present. The Barrera and KM based GDOS models predict u2 results consistent with the low temperature measured value. The ST/LH model’s excess area at low E leads to an increased value of u2 relative to the measured value, with similarly inflated results for the ENDF model. The ARCS GDOS, based on the one-phonon approximation to the measurement, displays excess area at large E, leading to a deflated value for the mean square displacement.

5.1.2. Inelastic dynamics The various GDOS models are compared to the 5 K data in Fig. 9. The data are averaged over a range of Q to improve statistics and normalized to the same integral area as the data between the ranges indicated by the dashed vertical lines. Similar to the u2 result, agreement is most satisfactory when either the Barrera or KM based GDOS are used. The 50 meV scan shows low energy structure and the one-dimensional Debye-like behavior from 25 to 70 meV for the longitudinal inter-chain modes is also in good agreement when the multi-phonon terms are included. As expected from the examination of the one-phonon

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Fig. 9. Comparison of various scattering law models to data at 5 K. The measured data and calculations for each incident energy are averaged over the Q range indicated above each panel. The calculations are convolved with instrumental resolution, then normalized to the same integral area as the measured data in the range between the dashed vertical lines.

approximation, the ST/LH and ENDF models show regions of disagreement which are most pronounced in the o 20 meV region and in the region from 100 to 150 meV, where the intensities of the Van Hove singularities are under-assigned. However, despite the disagreement, the ENDF based models have the appearance of performing well when compared to the low resolution 675 meV incident energy scan. The resolution of this measurement is similar to the benchmarks presented to validate the original ENDF scattering law [15]. The Barrera model successfully reproduces most of the observed features of SðQ ,EÞ. To explore the dynamics more closely, we examine the results for SðQ ,EÞ predicted by the Barrera model apart from the other models in the next section.

5.2. Barrera based model, 5 K Fig. 10 shows only the Barrera based model from Fig. 9 for a more clear comparison. The one phonon term is also shown, but without convolution by the instrumental resolution, and the elastic peak is not included in the calculations. The inclusion of the phonons within the broad instrumental resolution is seen in the 250 and 675 meV incident energy scans, justifying our selection of a Gaussian resolution function based only its agreement in the E o0 region. Overall, the description is quite good, however, the singularities near 150 and 200 meV appear to be shifted to slightly higher energies compared to the measurement. This discrepancy is shared with the other measurements and Raman spectra and was noted in the original paper [30]. The multi-phonon wings appear to be of the correct width, and the wings near 500 meV also are present in the model calculation, appearing just before the data drops off at the edge of the accessible measurement range for this Q value.

We overlay a calculation without resolution convolution with the data to assess the computation’s multi-phonon component in Fig. 11. Again, agreement is satisfactory, indicating good description of the scattering in both Q and E. 5.3. High temperature, 294 K Results at 294 K are shown in Fig. 12. We find that the multiphonon terms smooth away virtually all of the fine details of the GDOS, and general agreement for all models is good. The ENDF based high energy phonons are found to be slightly underassessed in the 100–150 meV region, and over-populated in the sub 20 meV region. Given the wide use of the ambient temperature ENDF scattering law, there is a desire to assess the sensitivity of the cross-section and the effect these differences would have on neutron transport computations. Such an analysis exceeds the scope of this paper, however, recent work by Muhrer [68] facilitates a brief discussion. This study looked at the liquid water scattering law, where some of the bonds are modeled as Einstein oscillators in the NJOY code. The complete removal of a singularity-type excitation significantly reduced the total cross-section in the region of the excitation. The energy spectrum of neutron leakage flux computed for a scattering law without the Einstein oscillator showed a depletion of the flux in the vicinity of the missing excitation, and a  5% increase in the low energy neutron spectrum. In another component of the study, it was shown an increase in low energy excitations leads to an increase in the low energy leakage flux from the neutron moderator. Because the GDOS integral area is unity by definition, an oscillator’s missing area is redistributed across the entire GDOS spectrum. This redistribution increases the area under the lower energy excitations, which increases the cross-section for neutron down-scattering. We therefore surmise that the ENDF scattering

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will over-predict the low energy neutron yield at 300 K, and show a corresponding under-prediction of leakage flux near 100– 150 meV. However, in the particular case of the ENDF law at high

175

temperature, the singularities are not missing entirely, they are only underassessed. Therefore, the overall magnitude of these corrections is likely to be small.

Fig. 10. The Barrera model compared to the experimental data at 50, 100, 250 and 675 meV incident neutron energy. The model curve shows the calculation using 100 phonons in the sum, and convoluted with experimental resolution. The calculations are normalized the data between the vertical dashed lines. The black line shows the one phonon term without resolution convolution.

50

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Fig. 11. A contour plot of the measured data at each incident energy at 5 K, on a logarithmic intensity scale. The Barrera calculation is shown without resolution convolution and normalized to the data as the overlaying lines. The contour lines for the calculation are drawn at the same levels as the measurement. There is a reasonable agreement in both Q and E, with the exception of the slight misplacement of the Van Hove singularities near 175 meV seen in the Ei ¼ 250 meV figure.

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Fig. 12. Comparison of scattering law models to data at 294 K, normalized to the same area between the vertical dashed line. Note the underassessment of the crosssection in the  100 meV region by the ENDF model. The combined smoothing effect of multi-phonon terms and instrumental resolution lead to a qualitative agreement of all models when resolution is broad (as in the Ei ¼ 675 meV scan).

5.4. Using the one-phonon approximation Finally, we assess the consequences of applying the onephonon approximation (Eq. (13)) to produce the GDOS for the scattering law directly from measurement. At low temperature, the one-phonon approximation actually appears to work reasonably well. The low energy structure is captured by this model, and both the singularities and the smooth regions from 100 to 250 meV are accurately described. However, wherever multiphonon contamination is present in the measurement, the resulting SðQ ,EÞ tends to be too large. This results from what is essentially the double inclusion of the multi-phonon contamination of the GDOS. Multi-phonon terms are present in the Zmeas(E) in Eq. (13), and then included again when the multi-phonon expansion is calculated from Zmeas(E). The inclusion of the multi-phonon component in the Zmeas(E) also leads to an incorrectly small value for u2 by shifting some of the area under Z(E) to higher energies. It may be possible to correct for some of these discrepancies by assessing the multiphonon contamination by deconvolution [39], and by using the measured value for u2 to help determine whether the deconvolution has been successful. One might also consider using the measured u2 in place of the one calculated from Zmeas(E), but if the measured value is significantly different from the calculated one, the sum rules on SðQ ,EÞ would be violated.

can be solved by a change of variable to Q and E. The inelastic portion of the total cross-section is Z @2 s 2p dm dEf sInel ðEi Þ ¼ ð18Þ @O@E

sInel ðEi Þ ¼

sb

Z

2 2ko

1

Q dQ 0

Z

Emax

SðQ ,EÞ dE

where the kinematic range is   _2 ko Q Q 1 Emax ¼ m 2ko Emin ¼ 

_2 ko Q m

ð19Þ

Emin

  Q 1þ 2ko

ð20Þ

ð21Þ

The inelastic cross-section is computed numerically. The elastic scattering can be solved analytically

sel ðEi Þ ¼ sb

1e4z 4z

ð22Þ

5.5. Total cross-section

where z ¼ ð2m=_2 Þu2 Ei . The elastic and inelastic cross-sections computed from each model GDOS are shown in Fig. 13. At low temperature and low energy, the scattering cross-section is dominated by the elastic component. At high temperature, the upscattering cross-section begins to dominate as Eo kb T. As E b kb T, the cross-section tends toward the limiting value of the free atom cross-section.

5.5.1. Mathematical treatment There are no available data for the cross-section of HDPE at 5 K, or at low temperature for that matter, so we must compare calculations of the total cross-section to ambient temperature measurements. The total cross-section is calculated by integrating Eq. (4) over all final energies and all angles. Following [69], this

5.5.2. Comparison to measurement Fig. 14 shows the total cross-section computed using the GDOS models and compared to the 300 K cross-section measurement of Granada and one by Herdade, both available on the EXFOR database [31]. The material in the total cross-section measurement is of 53% crystallinity and somewhat thicker at 0.9 to

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177

commonly suggested practice. We will discuss the effect of the free gas treatment in the next section. The agreement with data for the calculated total cross-section at 294 K is improved using the Barrera or KM models compared the other models, which are typically too large at low energy. It is interesting to note such good agreement at low energy even at high T, which is presumably due to the improved description of the low energy portion of the GDOS. It has been suggested that this region should have a rotational component to the SðQ ,EÞ, which is absent at low temperature [60,61,63]. We find improved agreement despite neglecting any amorphous or rotational aspects of the SðQ ,EÞ. The GDOS derived from the one phonon approximation is the exception, performing poorly at all energies by over-estimating the cross-section in the higher energy phonon region and correspondingly underestimating it in the low energy phonon region. This observation is consistent with the redistribution of GDOS area away from low energy via multi-phonon contamination at higher energies.

Fig. 13. Comparison of the elastic and inelastic total cross-section for H in PE for the different GDOS models at 300 K (solid lines), and at lower temperatures for the Barrera-based model (dashed lines). The inelastic (Eq. (19)) and elastic (Eq. (22)) components of the cross-section are indicated by the arrows. At low temperature and low energy, the elastic cross-section dominates. Data are normalized to per H atom.

5.6. Scattering from carbon The scattering law discussed thus far pertains only to the scattering from hydrogen. The conventional in MCNP is to treat the secondary scatterers using the free gas scattering law. Since we have confidence in the scattering law and total cross-section developed from the Barrera GDOS model, we have the opportunity to test the efficacy of this practice against the total crosssection measurement. The free gas scattering law is essentially a gaussian in energy transfer, centered about the recoil momentum. The width of the gaussian is temperature dependent 1 2 2 2 SðQ ,EÞ ¼ pffiffiffiffiffiffi eðbEð2sr ÞÞ =2sr 2psr

ð24Þ

where the recoil width about the energy transfer is s2r ¼ _2 Q 2 =mAb. The free gas total cross-section is [11]   pffiffiffiffiffiffiffiffiffiffi s 2 pffiffiffiffiffiffiffiffiffiffi AbEi AbEi e sFreeGas ðEi Þ ¼ free ð2AbEi þ1Þerf ð AbEi Þ þ ð25Þ 2AbEi p

Fig. 14. The total cross-section per CH2 molecule in HDPE at 300 K, computed using the various models and compared to the Granada and Herdade measurements taken from the EXFOR database [31]. The KM and Barrera based models show better agreement at low E, presumably due to their better description of the low energy phonon GDOS. The inset shows the hump in the 100–300 meV region in closer detail, which is not precisely reproduced by any model (see also Fig. 15). The outlier at 17.6 meV in the Herdade data is present in the EXFOR data.

1.8 mm compared to our 70% crystalline, 10 mm thick samples, and normalized to per CH2 molecule. This normalization is accomplished in Ref. [20] data by adding the free atom cross-section of carbon, scarbon ¼ 4:7 b.

stotal ðEi Þ ¼ 2  ðsInel ðEi Þ þ sEl Þ þ scarbon

ð23Þ

We have also included scattering from carbon, however, we have used the energy dependent free gas cross-section to represent the carbon atom. This is the default treatment in MCNP, and a

where A is the mass of the scatterer in units of neutron mass, and sfree is the free atom cross-section. Following the subtraction of the hydrogen scattering foreground, the residual difference between the total cross-section measurement by Granada et al. and the calculated cross-section is shown in Fig. 15. The remainder is not completely consistent with a free gas treatment, giving some indication that the dynamical scattering from carbon and/or the neglected coherent scattering is playing a role at the  3 barn level. The free gas total cross-section is within the uncertainty of the cross-section measurement, however, the presence of additional dynamics in the 10–100 meV region appears to be indicated by the structure of the residual. However, it appears that below 10 meV (i.e., 5kT=25) the up-scattering portion of free-gas crosssection of carbon is unrealistically large. As a fraction of the total cross-section, however, this effect is a small contribution.

6. Conclusion We have employed INS data to guide our selection of a new HDPE GDOS, from which a new scattering law can be developed. Going forward, the INS measurement can also serve as an effective and detailed benchmark for a new scattering law. However, in the course of this work, we have subsequently found other good experimental studies which would be useful for an

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would likely translate into vast improvements to the scattering laws. Greater focus on the microscopic dynamics will lead to a more complete description of the scattering law, the effect of which on future neutronic designs is difficult to anticipate, but would likely prove beneficial.

Acknowledgment

Fig. 15. The residual of the measured cross-section of Granada et al. and calculations shown in Fig. 14. If the incoherent approximation for the H atom is accurate, the residual should be due to scattering from the carbon atom in the HDPE. Both the ENDF and Barrera-based models are shown for comparison. The uncertainty bounds are shown about the free atom carbon cross-section (4.8 b). The residual total cross-section is consistent with either the free gas treatment of the carbon atom or the free atom cross-section. However, between 10 and 100 meV there appears to be evidence for additional dynamics at the two-error bar level.

improvement of the GDOS [52,55], some of which have been available for some time [28,29]. The performance of new measurements has proven to possess distinct advantages. Foremost, the measurements taken at different incident neutron energies where conducted on the same instrument, the same exact sample, and under identical sample environment conditions. The spectrometer resolution and the experimental conditions are both known and controlled, and a detailed comparison of the calculation to primary experimental data can be performed. In doing so, we found that the recent model of Barrera et al. [30], and a composite model based on the work of Kitagawa et al. [28,29], reproduced the INS data more accurately than the other models. This kind of direct comparison is simply not possible when constructing a scattering law based on a survey of the literature. It should also be noted that a successful experimental data driven methodology has also been developed by Granada [69,70]. This method uses total cross-section measurements to infer the GDOS as well as a ‘‘synthetic scattering kernel’’ which reproduces the basic neutron transport parameters in the absence of detailed knowledge of the microscopic dynamics. On the other hand, many of the most commonly encountered materials and temperatures requiring scattering laws for cold moderator calculations are of a finite number (i.e., the ortho- and para- forms of deuterium, methane, water, etc.). The measurements described herein were completed in about one day. Given the central role played by legacy scattering laws in the design of expensive neutronic systems, the devotion of some INS beam time toward the improvement in scattering laws may be both fruitful and cost effective. The latest computational methods and data on materials of interest are not being translated into widely available scattering laws. This, despite calls by the IAEA [71] and the cold moderator community [72] citing a need for improved data. We have shown good success in reproducing the HDPE scattering law by applying the GDOS models of KM or Barrera, so it seems that even minimal coordination between materials scientists and neutronics experts

We gratefully acknowledge assistance from Yiming Qiu for providing a version of the DAVE software which enabled scripting to analyze the large number NXSPE files produced in the course of this work. We thank Jose Ignacio Marquez Damian of the Neutron Physics Department at Centro Atomico Bariloche for alerting us to the work of Barrera, and the IAEA discussions of this subject. We also thank David Baxter for providing the LENS samples and Albert Young for providing samples from the UCNA experiment. The Research at Oak Ridge National Laboratory’s Spallation Neutron Source was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U. S. Department of Energy.

Appendix A. Supplementary data Supplementary data associated with this article can be found in the online version at http://dx.doi.org.10.1016/j.nima.2013.01. 048.

References [1] MCNPX 2.5.0, MCNPX Users Manual, Technical Report LA-CP-05-0369, Los Alamos National Laboratory, 2005. [2] K.F. Graham, J.M. Carpenter, Nuclear Science and Engineering 49 (1972) 418. [3] D.F.R. Mildner, et al., Nuclear Instruments and Methods 152 (1978) 437. [4] S. Ikeda, J.M. Carpenter, Nuclear Instruments and Methods in Physics Research Section A 239 (1985) 536. [5] Y. Kiyanagi, Journal of Nuclear Science and Technology 22 (1985) 934. [6] C.M. Lavelle, et al., Nuclear Instruments and Methods in Physics Research Section A 587 (2008) 324. [7] S.C. Frankle, A Suite of Criticality Benchmarks for Validating Nuclear Data, Tech. Report LA-13594, Los Alamos National Laboratory, 1999. [8] M.S. Gordon, et al., IEEE Transactions on Nuclear Science NS-51 (2004) 3427. [9] K. Amgarou, V. Lacoste, Journal of Instrumentation 5 (2010) P09002. [10] G.L. Squires, Introduction to the Theory of Thermal Neutron Scattering, Cambridge University Press, Cambridge, New York, 1978. [11] M.M.R. Williams, The Slowing Down and Thermalization of Neutrons, NorthHolland Publishing Company, 1966. [12] R.E. MacFarlane, Neutron Slowing Down and Thermalization Handbook of Nuclear Engineering, Springer, USA, 2010, pp. 189–277. [13] M.B. Chadwick, et al., Nuclear Data Sheets 107 (2006) 2931. [14] LANL Group XCP-5, Nuclear Data Team. Private Communication. [15] J.U. Koppel, D.H. Houston, Reference Manual for Thermal Neutron Scattering Data, Technical Report ENDF-269, 1969. [16] R.E. MacFarlane, A.C. Kahler, Nuclear Data Sheets 111 (2010) 2739. [17] R.E. Hill, C.-Y. Liu, Nuclear Instruments and Methods in Physics Research Section A 538 (2005) 686. [18] K. Swaminathan, S.P. Tewari, Nuclear Science and Engineering 91 (1985) 95. [19] S.B. Armstrong, Nuclear Science and Engineering 23 (1965) 192. [20] J.R. Granada, et al., Nuclear Instruments Methods in Physics Research Section A 261 (1987) 573. [21] P.C. Mitchell, et al., Vibrational Spectroscopy with Neutrons, Series on Neutron Techniques and Applications, World Scientific Publishing Company Pvt. Ltd., 2005. [22] D.L. Abernathy, Notiziario Neutroni E Luce di Sincrotrone 13 (2008) 4. [23] D.L. Abernathy, et al., Review of Scientific Instruments 83 (2012) 015114-11. [24] T.E. Mason, et al., Physica B 385–386 (2006) 955. [25] ENDF/B-VII Thermal Neutron Scattering Library (CH2) /http://www.nndc. bnl.govS. [26] T.P. Lin, J.L. Koenig, Journal of Molecular Spectroscopy, 9. [27] D. Sprevak, J.U. Koppel, Nukleonik 12 (1969) 87. [28] J.E. Lynch, et al., Journal of Chemical Physics 48 (1968) 912. [29] T. Kitagawa, T. Miyazawa, Neutron Scattering and Normal Vibrations of Polymers, Advances in Polymer Science, vol. 9, Springer, Berlin/Heidelberg, 1972. [30] G.D. Barrera, et al., Macromolecules 39 (2006) 2683.

C.M. Lavelle et al. / Nuclear Instruments and Methods in Physics Research A 711 (2013) 166–179

[31] Data taken from the EXFOR database, retrieved from the IAEA Nuclear Data Services website. Granda data: EXFOR file 30748, dated May 13, 1987; Herdade data: EXFOR file 30229, dated 1970. [32] Al-Qasir, I.I., Thermal Neutron Scattering in Graphite, Ph.D. Thesis, North Carolina State University, 2008. URL /http://www.lib.ncsu.edu/resolver/ 1840.16/3386S. [33] C.-Y. Liu, et al., arXiv:1005.1016. [34] A. Frei, et al., Europhysics Letters 92 (2010) 62001. [35] C.-Y. Liu, C.M. Lavelle, C.M. Brown, arXiv:1103.5091. [36] D.V. Baxter, et al., Physics Procedia 26 (2012) 153. [37] G. Muhrer, et al., Nuclear Instruments and Methods in Physics Research Section A 681 (2012) 91. [38] Y. Ogawa, Y. Kiyanagi, N. Watanabe, Journal of Nuclear Science and Technology 36 (1999) 105. [39] J. Dawidowski, et al., Nuclear Instruments and Methods in Physics Research Section B 195 (2002) 389. [40] T. Kanaya, et al., Physical Review B 61 (2000) R6451. [41] Goodfellow Corp., Private Communication. [42] W. Myers, G.C. Summerfield, J.S. King, Journal of Chemical Physics 44 (1966) 184. [43] C.W. Bunn, Transactions of the Faraday Society 35 (1939) 482. [44] M. Laridjani, P. Leboucher, PLoS ONE 4 (2009) e6228. [45] D.L. Dorset, B. Moss, Polymer 24 (1983) 291. [46] Y. Takahashi, T. Kumano, Journal of Polymer Science Polymer Physics 42 (2004) 3836. [47] B. Wunderlich, Journal of Chemical Physics 37 (1962) 1207. [48] T. Shimanouchi, Pure and Applied Chemistry 7 (1963) 131. [49] M. Tasumi, S. Krimm, Journal of Chemical Physics 46 (1967) 755. [50] T. Kitagawa, Polymer Letters 6 (1968) 83. [51] J. Barnes, B. Fanconi, Journal of Physical and Chemical Reference Data 7 (1978) 1309. [52] S.F. Parker, Journal of the Chemical Society Faraday Transactions 92 (1996) 1941. [53] S.F. Parker, Macromol. Symp., 119: 227–234.

179

[54] J. Tomkinson, et al., Physical Chemistry Chemical Physics 4 (2002) 716. [55] H. Jobic, Journal of Chemical Physics 76 (1982) 2693. [56] H. Jobic, R.E. Ghosh, A. Renouprez, Journal of Chemical Physics 75 (1981) 4025. [57] A. Griffin, H. Jobic, Journal of Chemical Physics 75 (1981) 5940. [58] G.C. Wick, Physical Review 94 (1954) 1228. [59] NJOY 99.0 Code System for Producing Code System for Producing, Pointwise and Multigroup Neutron and Photon Cross Sections from ENDF/B Data, Technical Report PSR-480, Los Alamos National Laboratory, 2000. [60] T. Kanaya, et al., Chemical Physics Letters 150 (1988) 334. [61] B. Holzer, et al., Colloid and Polymer Science 272 (1994) 1396. [62] J.W. Halley, et al., Physical Review B 32 (1985) 2132. [63] G. Hohlweg, et al., Macromolecules 25 (1992) 6248. [64] R.E. Hill, et al., Nuclear Instruments and Methods in Physics Research Section A 440 (2000) 674. [65] C.K. Loong, S. Ikeda, J.M. Carpenter, Nuclear Instruments and Methods in Physics Research Section A 260 (1987) 381. [66] R.T. Azuah, et al., Journal of Research of the National Institute of Standards and Technology 114 (2009) 341. [67] R. Zorn, Nuclear Instruments and Methods in Physics Research Section A 674 (2012) 85. [68] G. Muhrer, Nuclear Instruments and Methods in Physics Research Section A 664 (2012) 38. [69] J. Dawidowski, J.R. Santisteban, J.R. Granada, Physica B 271 (1999) 212. [70] J.R. Granada, Nuclear Instruments and Methods in Physics Research Section B 266 (2008) 164. [71] A. Plompin, Summary Report of the Technical Meeting on Long-term Needs for Nuclear Data Development, Technical Report INDC(NDS)-0601, IAEA INDC International Nuclear Data Committee, 2011. [72] J.M. Carpenter, E.B. Iverson (Eds.), Proceedings of the International Workshop on Cold Moderators for Pulsed Neutron Sources, OECD. [73] J.W. Schmidt, M. Nielsen, W.B. Daniels, Physical Review B 30 (1984) 6308. [74] J.R. Granada, Europhysics Letters 86 (2009) 66007.