Reply to comment by Mayers et al. on "High energy neutron scattering from hydrogen using a direct geometry spectrometer"
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A p r Reply to comment by Mayers et al. on “High energy neutron scattering fromhydrogen using a direct geometry spectrometer”
C. Stock
1, 2 and R.A. Cowley
3, 4 NIST Center for Neutron Research, 100 Bureau Drive, Gaithersburg, Maryland 20899, USA Indiana University, 2401 Milo B. Sampson Lane, Bloomington, Indiana 47404, USA Oxford Physics, Clarendon Laboratory, Parks Road, Oxford, United Kingdom OX1 3PU, UK Diamond, Rutherford Appleton Laboratory, Chilton, Didcot, OX11 0QX, UK (Dated: December 6, 2018)
Our paper entitled “High-energy neutron scatteringfrom hydrogen using a direct geometry spectrometer”(Ref. 1) describes an investigation of the validity of con-ventional scattering theory on the cross section of hydro-gen using a direct geometry spectrometer. Contrary toprevious results using indirect geometry machines, whichobserve a 20-40% deficit in the cross section, we find thecross section is constant and therefore consider that theprevious results are an experimental artifact from theuse of indirect geometry spectrometers. The commentdescribed in Ref. 2 provides a detailed discussion regard-ing the resolution function in the case of direct and indi-rect geometry neutron scattering instruments at pulsedsources. Based on this analysis it is claimed that theconclusions made from data with a direct geometry spec-trometer (outlined in our publication) are invalid.In this reply, we point out several criticisms of theanalysis in Ref. 2 and show that the comment doesnot change the underlying conclusion presented by our-selves that there is no measurable deficit in the scatteringcross section of hydrogen. We therefore consider that ouroriginal conclusions are correct namely that the previousanomalies in the cross section are due to experimentaleffects related to the use of indirect geometry spectrom-eters.
1) Impulse Approximation
Ref. 2 calculates the energy widths by assuming thatthe Impulse approximation is valid. This assumption isalso made to obtain the energy profiles of a constant mo-mentum scan using an indirect geometry spectrometers.If as found in Ref. 2 the hydrogen cross section is notconstant with momentum transfer, then the impulse ap-proximation needs to be revaluated as done in severalof the theories and papers discussed in the comment.The result is that assuming the impulse approximationto analyse the data and then deriving a momentum de-pendent cross section is inconsistent and we believe thisanalysis needs to be reconsidered.In contrast with a direct geometry spectrometer, weare able to measure the widths directly from constantangle scans without using the impulse approximation be-cause the energy transfer of the scattering is centered at the free hydrogen energy.
2) Hydrogen cross section is constant as a functionof both momentum and energy transfer
Our published paper shows firstly that the cross sec-tion for all momentum transfers and energy resolutionsis constant. Secondly, the absolute value of the crosssection is that expected based on the Born and Impulseapproximations. We obtain the results independent ofthe incident neutron energy, independent of the energyresolution and independent of the scattering angle. Wenote that Ref. 2 agrees that the energy widths of the hy-drogen recoil line are comparable on both spectrometersat high scattering angles, yet we derive the same crosssection at these angles for a variety of incident neutronenergies and momentum transfers. We therefore do notagree with the comments in Ref. 2 as we have found thatthe hydrogen cross section is constant within the exper-imental error for all angles and momentum transfer andindependent of the incident neutron energy.
3) Jacobian
We do not agree with the semantics used by the au-thors when discussing resolution and believe it to be mis-leading. When discussing resolution applied to neutroninelastic scattering, there are two key points- first theraw width of the resolution ellipsoid in momentum andenergy, and second how the resolution ellipsoid cuts thedispersion surface of the excitation being measured. Thelater point is defined by the Jacobian discussed in Ref 1.We believe this second point is the origin of the broadpeaks observed in a constant angle scan obtained withan indirect spectrometer in Ref. 2.Ref. 2 present data based on simulated constant mo-mentum scans. We believe this analysis is invalid as theauthors have assumed the impulse approximation to ob-tain these plots. Also, the measurements deriving thechange in the cross section were obtained from constantangle scans and not by the scans presented.
4) Integrated intensity and sum rules
Ref. 2 refer in their comment to several theories stat-ing that the momentum cross section is tied to the energyresolution. Such a statement is consistent with sum ruleswhich state that the integral over energy must be a con-stant for different momentum transfers. The data setspresented are not consistent with this sum rule becausethe integral of the scattered intensity is not independentof the momentum transfer. The data taken on the directinstrument, MARI, is consistent with this basic notion ofneutron scattering.If the claim in Ref. 2 that the apparent inconsistencybetween the results of direct and indirect geometry ma-chines is due to the different energy resolutions of the ex-periments, then by integrating over all energies (or time)they should be able to obtain where the missing intensityhas reappeared. This analysis has never been performedto our knowledge.Theoretical work in Ref. 7 provided a possible expla-nation for some of the measurements described in Ref. 2.However, the suggestions in Ref. 7 are inconsistent withour measurements because we observe the same intensityfor different incident neutron energies and for a range ofscattering angles. If the theory in Ref. 7 was an ap-propriate description, the changes in the intensity wouldbe observed in our experiment and the missing intensitycould be derived from the data taken on Vesuvio.
5) Inconsistent Experimental results in the literature
A survey of the literature shows that the hydrogencross section has been measured using the instrumentVESUVIO at different times over the past 12 years andthe missing intensity has tended to become smaller. Weconsider that this suggests that the results are probablycaused by experimental issues, rather than physical ones such as the method of analyzing the data. This pointcan be explicitly seen in comparing the results of Refs. 3and 4 which report very different values and angular de-pendences of the hydrogen cross section in polyethylene.The results are also very different from the reports ofexperiments on various hydride materials. [5] This resultagain seems inconsistent with the Impulse approxima-tion where all the atoms should respond independently atshort times and therefore all hydrogen containing materi-als should give consistent results. We believe the authorsneed to give the experimental setup, particularly giventhe recent upgrade outlined in a recent publication. [6]Only when all of the data (widths and intensities) areprovided can we determine if there is a discrepancy be-tween experiments and theory in the current experimen-tal configuration.Based on these five points, we do not agree with thecomments described in Ref. 2 and consider that the con-clusion that the cross section of hydrogen varies withmomentum transfer to be an experimental artifact as-sociated with indirect geometry spectrometers. [1] C. Stock, R.A. Cowley, J.W. Taylor, and S.M. Benning-ton, Phys. Rev. B , 024303 (2010).[2] J. Mayers, N.I. Gidopoulos, M. Adams, G.F. Reiter, C.Andreani, R. Senesi, unpublished (arXiv:0909.2633).[3] M. Vos, C.A. Chatzidmimitriou-Dreismann, T. Abdul-Redah, and J. Mayers, Nucl. Instr. and Methods in Phys.Research B , 233 (2005).[4] R. A. Cowley and J. Mayers, J. Phys.: Condens Matter , 5291 (2006).[5] T. Abdul-Redah, M. Krzystyniak, J. Mayers, C.A.Chatzidimitriou-Dreismann, J. Alloys and Compounds, , 790 (2005).[6] J. Mayers, Measurement Science and Technology,22