Triple-axis Neutron Spectrometer Research Articles

Inelastic neutron scattering from solidAr36

The phonon dispersion relations of $^{36}\mathrm{Ar}$ have been measured in the [100], [110], and [111] symmetry directions at 10 K using a triple-axis neutron spectrometer. A comparison of the measured dispersion curves and the best available theoretical calculations based on the Bobetic-Barker (BB) potential shows small systematic discrepancies for the [100]L, [110]${\mathrm{T}}_{2}$, [110]L, and [111]L branches. Otherwise the agreement between experiment and theory is remarkably good. The data are well described by several harmonic Born-von K\'arm\'an models the most general of which (a three-neighbor general-force model with nine parameters) has been used to deduce values of zero-sound elastic constants. At 10 K (lattice constant $a=5.313\ifmmode\pm\else\textpm\fi{}0.002$ \AA{}), these are ${c}_{11}=(424\ifmmode\pm\else\textpm\fi{}5)\ifmmode\times\else\texttimes\fi{}{10}^{8}$, ${c}_{12}=(239\ifmmode\pm\else\textpm\fi{}5)\ifmmode\times\else\texttimes\fi{}{10}^{8}$, and ${c}_{44}=(225\ifmmode\pm\else\textpm\fi{}1)\ifmmode\times\else\texttimes\fi{}{10}^{8}$ dyn ${\mathrm{cm}}^{\ensuremath{-}2}$ which imply a deviation from the Cauchy relation with $\ensuremath{\delta}=\ensuremath{-}0.06\ifmmode\pm\else\textpm\fi{}0.02$. From the energies of phonons of small wave vector measured just below the melting point at 82 K ($a=5.463\ifmmode\pm\else\textpm\fi{}0.002$ \AA{}), the zero-sound elastic constants were determined to be ${c}_{11}=(248\ifmmode\pm\else\textpm\fi{}6)\ifmmode\times\else\texttimes\fi{}{10}^{8}$, ${c}_{12}=(153\ifmmode\pm\else\textpm\fi{}5)\ifmmode\times\else\texttimes\fi{}{10}^{8}$, and ${c}_{44}=(124\ifmmode\pm\else\textpm\fi{}4)\ifmmode\times\else\texttimes\fi{}{10}^{8}$ dyn ${\mathrm{cm}}^{\ensuremath{-}2}$ ($\ensuremath{\delta}=\ensuremath{-}0.19\ifmmode\pm\else\textpm\fi{}0.04$). In both the 10- and 82-K measurements, particular care has been taken to remove the systematic errors introduced by finite instrumental resolution. The measured elastic constants have been compared with several calculations and with results obtained from ultrasonic and Brillouin scattering experiments. Calculations of elastic constants based on the BB potential with the Axilrod-Teller three-body interactions are in good agreement with our data at both temperatures. The energies of some typical phonons have been measured at temperatures of 10, 35, 55, and 75 K. In the only case in which these measurements can be compared with an existing calculation (the [100]T mode at $\ensuremath{\zeta}=0.4$, our results show a slightly larger shift of the phonon energy than that computed by Klein et al. from the BB potential.

A computer controlled triple axis neutron spectrometer (TAS) with CAMAC instrumentation and a high level computer language control program

Hard- and software of a computer controlled triple axis spectrometer are described. The hardware system consists largely of CAMAC units, an internationally standardized modular system 1,14). For the software, a first approach to use a high level language has been made by attaching an overlay to FOCAL, which enables FOCAL to execute CAMAC commands. It is pointed out that for flexible process control a compilation to generate a fast code would be desirable. A lexical scanner phase for FOCAL, to get an immediately executable string of codes and pointers, is briefly discussed.

Gaussian approximation to the transmission function of a triple-axis neutron spectrometer

A previously derived system of equations for a point-by-point calculation of the triple-axis neutron spectrometer transmission function is approximated by another system of equations which is more economic in computer time and convenient for some special applications. This system is equivalent to Cooper-Nathan's equations with the additional advantage of more symmetry and transparency. The accuracy of the approximations underlying this system is estimated in a particular case and weighted against uncertainties arising from other sources.

Derivation and experimental verification of the normalized resolution function for inelastic neutron scattering

Based upon the work of Cooper & Nathans, an expression for the normalized resolution function of a triple-axis neutron spectrometer is derived and tested experimentally. The formalism is extended to show the explicit dependence of the integrated intensity of a sharp excitation spectrum on all of the relevant instrumental parameters. Extensive measurements of the integrated intensities of phonons in copper have been carried out for a wide range of all adjustable parameters. The experimentally determined intensities are found to be in good agreement with the calculated values.

The resolution function of a triple-axis neutron spectrometer

The amplitude and the shape of the resolution function of a triple-axis neutron spectrometer were calculated in the Gaussian approximation. Formulae containing the explicit analytical dependence of the resolution function on the incoming and scattered neutron wave vectors as well as on instrumental parameters in forms suitable for computation were obtained. The case of the double-crystal monochromator was also treated.

Lattice Dynamics of Cesium Chloride

The phonon-dispersion curves have been measured for CsCl at 78 and 298 \ifmmode^\circ\else\textdegree\fi{}K on a triple-axis neutron spectrometer and analyzed in terms of a rigid-ion model and various shell models. A simple shell model with eleven parameters is sufficient to reproduce the measured dispersion curves. Frequency distributions and ${\ensuremath{\Theta}}_{D}$-vs-$T$ curves were calculated based on the parameters determined from the two sets of data.

Resolution Effects in the Measurement of Phonons in Sodium Metal

In the course of making a detailed study of the phonons in Na metal using a triple-axis neutron spectrometer we have observed anomalous, asymmetric, and, in some cases, extraordinarily broad or split resonances. In this paper we account for these effects quantitatively by performing a complete resolution analysis. This analysis includes fully the combined effects of vertical resolution, a highly anisotropic dispersion surface, the variation of the polarization vectors and the temperature factor inside the resolution volume, and the sample mosaic spread. The application of this calculation to the measurement of phonons in other materials is straightforward.

The (Q,ω) transmission function of a triple-axis neutron spectrometer

Formulae are given for the counting rate and the transmission function of a triple-axis neutron spectrometer for three-dimensional and two-dimensional momentum transfers starting from the known transmission functions of the monochromating and the analysing part respectively. Some problems of relating these formulae to actual spectrometers and of obtaining the necessary functions and parameters are discussed. The general formulae are then specialized to a model spectrometer with Gaussian collimators and crystals. This results in a system of non-linear equations for a point-by-point calculation of the transmission function. Special care has been taken to keep the formulae absolute, i.e. to avoid unknown proportionality factors. This feature allows a detailed treatment especially of the counting rate in relation to resolution and the characteristic features of the sample cross section (liquid, phonon and Bragg-peak type). Two kinds of intensity factors in the counting rate are distinguished: the `trivial' factors are directly related to the magnitude of the illumination spot in (Q, ω) space and thus to resolution. The other factors are called `non-trivial'. These two kinds of intensity factors are identified in the formulae. Some considerations are given on the influence of experimental conditions and parameters on each of them.

The lattice dynamics of calcium oxide

Dispersion relations of phonons propagating along three symmetry directions in calcium oxide have been measured by the triple axis neutron spectrometer method. The experimental results have been interpreted in terms of a shell model showing that the electronic polarizability of both ions must be taken into consideration. The parameters of the most satisfactory model obtained by least squares fitting to neutron, elastic and dielectric data have been used to calculate the frequency distribution at room temperature. The results of the analysis are compared with those of similar analyses of the related crystal, magnesium oxide.

Vertically bent pyrolytic graphite crystals applied to triple-axis neutron spectrometry

This paper demonstrates the great value of verticallt bent pyrolytic graphic crystals used as monochromator or analyzer in triple-axis neutron spectrometers. A simple model is presented for estimating the flux gain to be achieved by bending the monochromator or analyzer, and the effects this bending has on instrumental resolution are also considered. Experiments with three monochromators bent to 50, 35, and 21 inch radii over a wide range of neutron energy (5 to 40 meV) gave a maximum flux gain factor of 3.5 as predicted by the model, with no measurable change in resolution. In addition, if vertical resolution is not crucial, a further intensity gain can be achieved by using a bent analyzer.

Lattice Dynamics of Yttrium at 295 K

Phonon-dispersion curves along the [$10\overline{1}0$], [$11\overline{2}0$], and [0001] symmetry directions of the hcp metal yttrium have been measured by inelastic neutron scattering using the triple-axis neutron spectrometer at the Ames Laboratory Research Reactor. The dispersion curves are not strikingly different from those of other hcp metals with a similar $\frac{c}{a}$ ratio. A careful search has been made for Kohn-type anomalies, and evidence for two such anomalies has been obtained. A modified axially symmetric force-constant model has been used to fit the phonon frequencies, and it has been found that although forces acting normally to the basal plane are not large beyond first neighbors, forces acting parallel to the basal plane are long-ranged, and interactions up to at least sixth neighbors have to be taken into account. The fitting has been done in a linear manner using Fourier analyses of the dispersion curves and other linear constraints such as the elastic constants. A frequency-distribution function has been calculated and used to calculate the specific heat and anisotropic Debye-Waller factor. Good agreement is obtained on comparison with the experimental data.

Measurements of the frequencies of the normal modes of zinc

A triple-axis neutron spectrometer has been used to measure the frequencies of the normal modes in zinc with wave vectors along the direction [011, macron0] and polarization vectors perpendicular to the basal plane. Previous measurements of this polarization branch were in disagreement. The present measurements confirm and extend those of Borgonovi et al.

Critical magnetic fluctuations in MnF2

The critical spin fluctuations in MnF2 have been studied near the critical temperature TN using a triple axis neutron spectrometer with high resolution. Due to the anisotropy in MnF2, the longitudinal and the transverse component of the spin fluctuations can be separated experimentally. The longitudinal component of the static susceptibility diverges at TN, while the time fluctuations undergo the expected thermodynamic slowing down. The spectral width above TN varies in proportion to the temperature power law {(T-TN)/TN}0·62±0·02. The temperature variation of the static susceptibility and the inverse correlation range of the longitudinal fluctuations are in agreement with the predictions of the Heisenberg model. The divergence of the transverse component is suppressed by the anisotropy, and this component is therefore finite at TN. The variation above TN was interpreted in terms of the transverse transition temperature Tperpendicular, introduced by Moriya. In relation to this temperature, the static properties behave in an almost classical manner, that is in agreement with the molecular field theory. The transverse spectral width above TN varies as {(T-Tperpendicular)/Tperpendicular}0·53±0·05. Below TN the transverse fluctuations consist of two spin waves. The spin-wave renormalization with temperature was measured in the long wavelength limit, and was found to vary in proportion to the staggered magnetization. The dynamical scaling law prediction for antiferromagnets of a q3/2 dependence of the spectral width at TN was verified for the transverse component.

On-line computer control of a triple-axis neutron spectrometer

A description is given of an instrumentation for control of a triple-axis neutron spectrometer used in inelastic scattering experiments with single crystals. The instrumentation includes an on-line PDP-8/S computer. Details are given of the principle of operation with the computer.

Phonon dispersion curves for zinc

Using a triple-axis neutron spectrometer all branches of the phonon dispersion curves along the three major axes of zinc have been measured. The results for the Δ and Σ axes are in fair agreement with those of previous workers. Results for the T, Tprime axes have not previously been reported. All results have been fitted by a model with seventeen independent interatomic force constants. Though the fit to the measured phonons is adequate, the agreement with elastic constants is quite unsatisfactory. The more basic pseudopotential models cannot fit the data.

Resolution of a triple axis spectrometer

A new method for obtaining the resolution function for a triple-axis neutron spectrometer is described, involving a combination of direct measurement and analytical calculation. All factors which contribute to the finite resolution of the instrument may be taken into account, and Gaussian or experimentally determined probability distributions may be used. The application to the study of the dispersion relation for excitations in a crystal is outlined.

Coherent Inelastic Neutron Scattering in NH4Cl

Some normal modes of vibration have been measured in undeuterated ammonium chloride by coherent inelastic neutron scattering utilizing a triple-axis neutron spectrometer. The behavior of the low-energy modes near the $\ensuremath{\lambda}$-type order-disorder phase transition at 242\ifmmode^\circ\else\textdegree\fi{}K is correlated with the anomalous behavior of the elastic constants in this region. A splitting of the torsional mode has also been observed above the phase transition.

Polarization Analysis of Thermal-Neutron Scattering

A triple-axis neutron spectrometer with polarization-sensitive crystals on both the first and third axes is described. The calculation of polarized-neutron scattering cross sections is presented in a form particularly suited to apply to this instrument. Experimental results on nuclear incoherent scattering, paramagnetic scattering, Bragg scattering, and spin-wave scattering are presented to illustrate the possible applications of neutron-polarization analysis.

An inexpensive triple-axis neutron spectrometer

A simple design is suggested for a triple-axis neutron spectrometer. Although the instrument does not have the flexibility of sophisticated machines, it is capable of the measurements required by most phonon problems. As it has only one working configuration it should be very simple to use. The basic design problem of these instruments is the control of a heavily shielded counter at the extremity of three variable arms. In the suggested instrument the counter is free only to rotate about the vertical. Support and control of the heavy shielding is, therefore, much simplified. The geometry is altered by varying the lengths of two of the arms, which is possible to high accuracy.

Lattice Dynamics of Lithium Hydride

Crystalline lithium hydride, with four electrons per primitive unit cell, has the simplest electronic configuration of all ionic crystals with the sodium-chloride-type structure, and thus represents an excellent test case for existing theories of lattice dynamics of these substances. Lattice vibrations in this system have been studied by measuring the phonon dispersion curves along the three high-symmetry directions in a single crystal of $^{7}\mathrm{LiD}$, using the techniques of coherent inelastic scattering of thermal neutrons. The isotopes $^{7}\mathrm{Li}$ and D were chosen over normal Li and H because of their more favorable neutron cross sections. All measurements were taken at room temperature, using a triple-axis neutron spectrometer operating in the constant-Q mode. Uncertainties in the measured frequencies are estimated to be no greater than 3-5%. The transverse zone center frequency is in excellent agreement with infrared absorption data. The experimental dispersion curves have been least-squares-fitted to several versions of the rigid-ion and shell models of lattice dynamics. A seven-parameter shell model fits the data to within the experimental accuracy. In this model, the ionic charge was variable and only the hydride ions were polarizable. Short-range forces were assumed between nearest neighbors (${\mathrm{Li}}^{+}$ and ${\mathrm{H}}^{\ensuremath{-}}$) and between negative second neighbors (${\mathrm{H}}^{\ensuremath{-}}$ and ${\mathrm{H}}^{\ensuremath{-}}$). The ionic charge obtained from the fit indicates that the bonding in $^{7}\mathrm{LiD}$ is about 88% ionic, in good agreement with estimates obtained from electronegativity considerations. The second-neighbor bond-stretching force constant was found to be about 7% of that obtained for the nearest neighbors. The model parameters obtained for $^{7}\mathrm{LiD}$ were used to calculate dispersion curves for $^{7}\mathrm{LiH}$, and frequency distribution functions for both substances. The frequency distribution for $^{7}\mathrm{LiH}$ has a gap between the acoustic and optic modes, whereas that for $^{7}\mathrm{LiD}$ does not.