Neutron Doppler broadening studies of tantalum and tungsten metal

doi:10.1016/S0168-583X(02)00480-9

Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms

Volume 192, Issue 3, May 2002, Pages 318-330

https://doi.org/10.1016/S0168-583X(02)00480-9 Get rights and content

Abstract

With the aim of providing experimental verification of the validity of the neutron resonance spectroscopy method for measuring the moments of the phonon frequency spectra, we have determined some of the phonon spectral parameters for W and Ta in the bcc phase of these metals by measuring the Doppler broadening of neutron resonances as a function of temperature. We find the mean phonon energy (first moment of the phonon frequency spectrum) of W to be 〈hν〉=20.2±0.4 meV and the second moment to be 〈(hν)²〉=(4.9±0.2)×10⁻⁴ eV². The Boltzmann-weighted moments from all the measurable resonances of tungsten are quite consistent with a Debye model with a Debye temperature of 315±4 K. For Ta, 〈hν〉=14.3±0.1 meV, 〈(hν)²〉=(2.2±0.2)×10⁻⁴ eV². The fitted Debye model temperature for the tantalum data is 217±2 K. In both cases the first and second moments agree well with those determined from neutron coherent inelastic scattering. Results with reasonable accuracy were extracted from resonances up to a few tens of eV. These results give high confidence that neutron resonance Doppler broadening spectroscopy is a viable and useful method for determining the characteristics of phonon frequency spectra.

Introduction

Neutron resonance Doppler broadening spectroscopy is in principle a valuable tool for making direct measurements of phonon spectrum parameters. It can be particularly useful when suitable single crystal samples are not available for making neutron coherent inelastic scattering measurements, from which detailed phonon spectra can be deduced, or when the thermal neutron absorption cross-section is too high to observe inelastic scattering. It also has the important advantage that it is element (or, more precisely, nuclide) specific. Most reports that have exploited the technique have followed the original method [1], in which the data have been fitted to the parameters of a specific phonon model (for example, an Einstein or generalized Nernst–Lindemann model). Some years ago, we published a method [2] for obtaining the principal moments of the phonon spectrum from resonance data and tested this theoretical approach with numerical experiments. These indicated that the first moment could be extracted with an accuracy better than 1%, the second moment to about 2% and the third moment to about 10%. These accuracies were not dependent on having a good prior knowledge of the nuclear resonance parameters.

The numerical trials of [2] did not include the experimental complications of background and resolution that are to be found in real data. These are particularly acute, even for low energy resonances, in transmission measurements that are done with high intensity spallation neutron sources requiring current-mode detector systems [3], [4] to cope with the high data collection rates. Although the moment method has been used successfully to obtain information on the plutonium phonon spectrum of a δ-phase Pu–Ga alloy [5], this was a special case in that the 1.04 eV resonance of the $^{240}$ Pu cross-section was selected for analysis; the resonance parameters were already very precisely known [6], thus allowing the determination of the spectrum moments with good confidence. It remains the case that the moment method requires experimental testing and demonstration for general use. This is carried through in the present paper.

For this purpose we have selected two materials for which the phonon frequency spectra are already well-known from atomic force models deduced from phonon dispersion curves measured by coherent neutron inelastic scattering, and have suitable narrow, low energy resonances in their neutron cross-sections. These are tungsten and tantalum metals. These materials have body centered cubic lattices at normal pressure and temperatures ranging from zero to well above room temperature. Both have had their phonon dispersion curves along principal crystallographic axes measured by coherent inelastic scattering [7], [8], [9]. The dispersion curves in turn have been fitted to Born–Von Karman force models (up to 7th neighbor for tantalum [7], third neighbor for tungsten [9]), from which the phonon frequency spectrum has been calculated. The measurements were made at room temperature. A secondary aim is to establish some of the ranges of physical and experimental conditions (neutron resonance energies and widths, neutron resolution parameters etc.) within which reliable results can be expected.

Section snippets

General theory

The detailed nature of the Doppler broadening of neutron resonances depends on the environment in which the nucleus is embedded, i.e. the solid, liquid or gaseous state and its temperature and pressure. Information on the nuclear environment can therefore be extracted by careful measurement of the form of the neutron cross-section in the resonance region.

The basic papers are those by Bethe [10], [11], on Doppler broadening in a gaseous medium, and Lamb [12] on the basic formalism for crystals.

Experimental procedures

Neutron energies are measured by the time-of-flight method on a 58 m flight path (flight path 5) from the MLNSC target of the 800 MeV proton linear accelerator at the Los Alamos Neutron Science Center (LANSCE). Full details of the main characteristics of our time-of-flight transmission spectrometer can be found in [5]. The main difference in the system used in the work described here lies in the detector. The active material of the detector (which was designed and loaned by Professor H.

The resolution function and background

Our analysis of the data consists essentially of least-squares fitting to Eq. (14) of the neutron rate measured by the detector, the variables in the fitting being some or all of the nuclear resonance parameters and the more important phonon moments. The characteristics of the resolution function Z(E,E^′) are very important in this fitting process, and must be considered carefully. Basic parameters governing the resolution function are the proton pulse-width, the timing channel width, the angle

Discussion

For both tantalum and tungsten the phonon moments measured by neutron resonance Doppler spectroscopy are within a few percent of the corresponding quantities deduced from Born–Von Karman atomic force models fitted to phonon dispersion data. Our measured mean phonon spectrum energy for Tantalum (14.3±0.1 meV at 17 K) is slightly lower than that calculated (14.6 meV at 296 K) from the spectrum of [3], while for tungsten the measured value (20.2±0.2 meV at 15 K) is slightly higher than the

Conclusions

We have measured some important phonon spectrum moments for the bcc metals tantalum and tungsten by means of neutron resonance Doppler spectroscopy. We initially established resonance parameters from the data taken at 300 and 310 K for several of the important resonances up to 85 eV in Ta and 48 eV in W. These are in reasonable agreement with the parameters previously reported in the literature (see Ref. [17]), but our values are generally considerably more precise. Our primary measured

Acknowledgements

We thank Bard Bennett, Gene Farnum and Steve Sterbenz for helpful discussion and their continued support and interest in this work. The experiments were carried out at the Manuel Lujan Jr. Neutron Scattering Center at Los Alamos (MLNSC), which is funded jointly by Defense Programs and Energy Research (Office of Basic Energy Sciences) of the US Department of Energy (DOE). This work was done at Los Alamos under the auspices of the DOE (under contract W-7405-ENG-36) and funded through its Science

References (22)

J.E. Lynn et al.
Nucl. Instr. and Meth. B
(1996)
J.D. Bowman et al.
Nucl. Instr. and Meth. A
(1990)
Y.-F. Yen
Nucl. Instr. and Meth. A
(2000)
S.H. Chen et al.
Solid State Comm.
(1964)
H.J. Groenewold et al.
Physica
(1947)
C. Coceva et al.
Nucl. Inst. and Meth.
(1983)
H.E. Jackson et al.
Phys. Rev.
(1962)
J.E. Lynn et al.
Phys. Rev. B
(1998)
R.R. Spencer et al.
A.D.P. Woods
Phys. Rev.
(1964)

A. Larose et al.

Can. J. Phys.

(1976)

Cited by (21)

On the Methodology to Calculate the Covariance of Estimated Resonance Parameters
2015, Nuclear Data Sheets
Principles to determine resonance parameters and their covariance from experimental data are discussed. Different methods to propagate the covariance of experimental parameters are compared. A full Bayesian statistical analysis reveals that the level to which the initial uncertainty of the experimental parameters propagates, strongly depends on the experimental conditions. For high precision data the initial uncertainties of experimental parameters, like a normalization factor, has almost no impact on the covariance of the parameters in case of thick sample measurements and conventional uncertainty propagation or full Bayesian analysis. The covariances derived from a full Bayesian analysis and least-squares fit are derived under the condition that the model describing the experimental observables is perfect. When the quality of the model can not be verified a more conservative method based on a renormalization of the covariance matrix is recommended to propagate fully the uncertainty of experimental systematic effects. Finally, neutron resonance transmission analysis is proposed as an accurate method to validate evaluated data libraries in the resolved resonance region.
Neutron resonance transmission spectroscopy with high spatial and energy resolution at the J-PARC pulsed neutron source
2014, Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, Detectors and Associated Equipment
Citation Excerpt :
In this paper we limit the theoretical estimation of the sample transmission to a simple convolution with a neutron double pulse of a given width (changing with the energy of neutron) and compare the measured spectra with the experimental data for samples of various materials placed directly on the detector input face (~15 mm from the active area). A more sophisticated fitting of the transmission spectra [19,20,38,39] can be implemented to increase the accuracy of the reconstructed elemental thicknesses within the sample. Calibration of transmission spectra with samples of known thickness can also be used for data interpretation.
The sharp variation of neutron attenuation at certain energies specific to particular nuclides (the lower range being from ~1 eV up to ~1 keV), can be exploited for the remote mapping of element and/or isotope distributions, as well as temperature probing, within relatively thick samples. Intense pulsed neutron beam-lines at spallation sources combined with a high spatial, high-timing resolution neutron counting detector, provide a unique opportunity to measure neutron transmission spectra through the time-of-flight technique. We present the results of experiments where spatially resolved neutron resonances were measured, at energies up to 50 keV. These experiments were performed with the intense flux low background NOBORU neutron beamline at the J-PARC neutron source and the high timing resolution (~20 ns at epithermal neutron energies) and spatial resolution (~55 µm) neutron counting detector using microchannel plates coupled to a Timepix electronic readout. Simultaneous element-specific imaging was carried out for several materials, at a spatial resolution of ~150 µm. The high timing resolution of our detector combined with the low background beamline, also enabled characterization of the neutron pulse itself – specifically its pulse width, which varies with neutron energy. The results of our measurements are in good agreement with the predicted results for the double pulse structure of the J-PARC facility, which provides two 100 ns-wide proton pulses separated by 600 ns, broadened by the neutron energy moderation process. Thermal neutron radiography can be conducted simultaneously with resonance transmission spectroscopy, and can reveal the internal structure of the samples. The transmission spectra measured in our experiments demonstrate the feasibility of mapping elemental distributions using this non-destructive technique, for those elements (and in certain cases, specific isotopes), which have resonance energies below a few keV, and with lower resolution for elements with relatively high resonance energies in the 1–30 keV range.
Determination of Resonance Parameters and their Covariances from Neutron Induced Reaction Cross Section Data
2012, Nuclear Data Sheets
Cross section data in the resolved and unresolved resonance region are represented by nuclear reaction formalisms using parameters which are determined by fitting them to experimental data. Therefore, the quality of evaluated cross sections in the resonance region strongly depends on the experimental data used in the adjustment process and an assessment of the experimental covariance data is of primary importance in determining the accuracy of evaluated cross section data. In this contribution, uncertainty components of experimental observables resulting from total and reaction cross section experiments are quantified by identifying the metrological parameters involved in the measurement, data reduction and analysis process. In addition, different methods that can be applied to propagate the covariance of the experimental observables (i.e. transmission and reaction yields) to the covariance of the resonance parameters are discussed and compared. The methods being discussed are: conventional uncertainty propagation, Monte Carlo sampling and marginalization. It is demonstrated that the final covariance matrix of the resonance parameters not only strongly depends on the type of experimental observables used in the adjustment process, the experimental conditions and the characteristics of the resonance structure, but also on the method that is used to propagate the covariances. Finally, a special data reduction concept and format is presented, which offers the possibility to store the full covariance information of experimental data in the EXFOR library and provides the information required to perform a full covariance evaluation.
Experimental evidence sustaining a compact-extended model for doppler broadening of neutron absorption resonances
2011, Annals of Nuclear Energy
Citation Excerpt :
We find this model being used in different sort of applications such as Neutron Resonance Absorption Spectrometry (Kiyanagi et al., 2005), Remote Sensed Thermometry (Mayers et al., 1989; Stone et al., 2005) and above all in Reactor Physics (Dresner, 1960) (where it is used without the effective temperature refinement but as a free gas model). More refined models based upon the previous cited work of Nelkin and Parks are also found (Lynn et al., 2002). There are indeed examples (particularly for thin resonances in tightly bound systems like uranium oxides) for which departures from the ETGM are observed (Borgonovi et al., 1970).
Recently we presented a new model to handle solid state effects on Doppler broadening of neutron absorption resonances (DBNAR) in a simplified manner. Our model extends the commonly used effective temperature gas model (ETGM) to regions of lower energies and temperatures where the latter produces non-negligible errors; at the same time, our model reduces to the ETGM in energy and temperature regions where this model has doubtlessly proved its validity. All this is achieved with a minimum amount of information needed to describe the solid system, which is thus approximately represented by a Debye spectrum, compatible with a great degree of accuracy in the calculation. In this work a robust validation of our model is achieved when it is contrasted against experiment.
Indicator to estimate temperature sensitivity of resonance in temperature measurement by neutron resonance spectroscopy
2011, Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms
Citation Excerpt :
The phenomenon of Doppler broadening of neutron resonance plays an important role in pursuing sample properties. On the one hand, neutron resonance method has been used to determine the resonance parameters simultaneously with phonon moments of the sample [1–5]. On the other hand, this method has successful applications in experiments aiming to measure temperature.
Through series of simplification and simulation, we find that it is the ratio of z (=Δ/Γ) and σ₀ (cross-section at the exact resonance energy in the absence of temperature broadening) that contains the most information of temperature sensitivity for a resonance. We consequently define a factor h called the effective fitness indicator to represent the lower limit of temperature error for each resonance. h bears succinct forms and is tested against numerical simulations as well as gathered experimental data. Our further analysis and simulation justify the use of h at high temperatures (above several hundreds degrees centigrade). When the transmission of a resonance is free from suffering a flat-bottom (without ntσ₀ ≫ 1), h can be used to estimate the temperature sensitivity of individual resonance with an analytic formula constructed from fitting, telling a relation between the temperature error and h. Moreover, spread of emission time caused by the moderator, phosphorescent decay of the scintillator, and background fraction are all included in numerical simulations to reveal their influences.
Synthetic model for Doppler broadening of neutron absorption resonances in molecular fluids
2010, Annals of Nuclear Energy
A general and systematic approach expressed in modern language, accounting for molecular motion effects on Doppler Broadening of Neutron Absorption Resonances (DBNAR) is given the form of a new model. It relies on well validated hypothesis: The separability of atomic from nuclear degrees of freedom, the use of the Van Hove scattering formalism and the fact that a conceptually identical approach produced experimentally proved predictions when applied to DBNAR in solid systems. We treat the molecular internal degrees of freedom approximately as harmonic oscillators. As a second contribution of this work, a synthetic model is presented in order to make the more complete model mentioned above suitable for neutron calculation codes. This second synthetic model reduces to the exact expressions of the complete model in the low and high neutron energy regimes and provides a plausible transition in between. Numerical results are presented for a general hypothetical case to show its strengths and limitations. Also, both models are applied to a real case of the ²³⁸U 6.674 eV resonant effective broadened absorption cross-section in UF6 (uranium hexafluoride). A direct experimental validation of our models is still necessary for which a special high resolution neutron transmission experiment ought to be devised at low temperatures and pressures on a gaseous system. It is showed how the synthetic model can be used to make thermometric predictions in an improved fashion in comparison to the effective temperature gas model at low temperatures.

View all citing articles on Scopus

View full text

Neutron Doppler broadening studies of tantalum and tungsten metal

Abstract

Introduction

Section snippets

General theory

Experimental procedures

The resolution function and background

Discussion

Conclusions

Acknowledgements

Nucl. Instr. and Meth. B

Nucl. Instr. and Meth. A

Nucl. Instr. and Meth. A

Solid State Comm.

Physica

Nucl. Inst. and Meth.

Phys. Rev.

Phys. Rev. B

Phys. Rev.

Can. J. Phys.