# The First Two Decades of Neutron Scattering at the Chalk River Laboratories

## Abstract

**:**

**Q**technique, the fields of lattice dynamics and magnetism and their interpretation in terms of the long-range forces between atoms and exchange interactions between spins took a major step forward. Experiments were performed over a seven-year period on simple metals such as potassium, complex metals such as lead, transition metals, semiconductors, and alkali halides. These were analyzed in terms of the atomic forces and demonstrated the long-range nature of the forces. The first measurements of spin wave excitations, in magnetite and in the 3D metal alloy CoFe, also came in this period. The first numerical estimates of the superfluid fraction of liquid helium II came from extensive measurements of the phonon–roton and multiphonon parts of the inelastic scattering. After the first two decades, neutron experiments continued at Chalk River until the shut-down of the NRU reactor in 2018 and the disbanding of the neutron effort in 2019, seventy years after the first experiments.

## 1. Introduction

## 2. The Early Period 1949–1951

#### 2.1. Structure of Fluids and Solids

_{2}and CO

_{2}with 70 meV (1.06 Å) neutrons selected by a NaCl monochromator from the thermal spectrum at NRX. The experiment was designed to elucidate the atomic structure of the molecules, and the measurements were compared with calculations based on the scattering of the neutron by systems of two- and three-point molecules assuming no inelastic scattering. The calculation for O

_{2}matched the variation of the scattered intensity with angle reasonably well but lay above the experiment for CO

_{2}at low momentum transfers. It was surmised that since the CO

_{2}gas was quite close to the liquid phase, the vapor may have had characteristics of the liquid pattern, which would have removed the intensity at low momentum transfers. While care was taken to avoid scattering from the steel container, the weak scattering from the gases, including multiple scattering, was comparable with the measurements with no gas in the scattering chamber. The experiments were clearly right at the limit of what could be done at that time, and great care was taken to avoid systematic errors. A second paper [8] noted the surprising similarity of the quantum mechanical and the classical formulations of scattering by D

_{2}gas when the energy transfers are neglected, even though the mass of the scatterer is close to the mass of the neutron in that case.

^{2}to maximize the scattered intensity, and the distance from crystal to sample was about 72 in. to permit small take-off angles, 2θm from the monochromating crystal and hence access short wavelengths and high energies. With care, the upper and lower limits on incident neutron energy were from 50 eV to 1 meV, although the easily accessible range was 10 eV to 20 meV. The main bearing was from a Bofors anti-aircraft gun, which could support 500 lbs of counter and shielding at a distance of 72 in. The brass crystal table was precision made and could be read to 0.01° with a vernier. The counter arm, 2θm, and crystal table, θm, maintained a 2:1 angular relationship with spring tensioned steel belts and pulleys to better than 0.01° over a 90° rotation. The neutron energy (wavelength) was under motor control, via the take-off angle 2θm, with a minimum angular step of the arm of 1/16°. The beam was taken from just outside the heavy water core of the NRX reactor, and it provided a flux of all wavelengths of 4 × 10

^{7}neutrons/s at the monochromator with a horizontal divergence of 0.13° and a vertical divergence of 1.0°. The detector was a tubular BF3 proportional counter enriched to 96% with B10 with associated counter electronics all designed at Chalk River. The monochromator crystals, initially synthetically grown NaCl, LiF, and natural CaCO

_{3}, and eventually large crystals of aluminum and lead, were usually placed in transmission geometry so as to fill the whole beam with neutrons. It is clear that the competing demands of intensity and resolution were well understood as well as the effect of λ/2 and λ/3 contamination on any measurements. For example, the (222) reflection from LiF is nearly absent because of the opposite signs of the scattering lengths of Li and F. Careful studies were made of the resolution and bandwidth of the spectrometer. The spectrometer was in use from around 1949 up until the 1980s in various modifications, the latest of which was as the sample scattering axis on the C5 triple axis crystal spectrometer, which is a great credit to the original robust and accurate design.

#### 2.2. Nuclear Physics and the Consequences for Inelastic Scattering

_{3}counters arranged symmetrically around the neutron beam to give enhanced sensitivity. The scattering chamber was evacuated to avoid air scattering. The scattering from a Cd sheet thick enough to absorb nearly all the incident neutrons was compared with the scattering from a thin sheet of incoherently scattering V. In the case where the scattering cross-section is much smaller than the absorption, the ratio of the scattering to absorption cross-sections of Cd can be written

^{2}, and the constant K is related to the counter efficiency. The ratio, corrected for absorption in the V standard and for order contamination, was measured for neutron energies between 20 and 400 meV. This was compared with the Breit–Wigner formula for a single resonance at 176 meV and found to be in good agreement. Similar measurements were made by the same method on the ratio for Sm and Gd [14]. Sm was consistent with a single resonant level, but Gd could not be described well with a single resonance. Slight systematic departures from the Breit–Wigner theory suggested to Brockhouse and Hurst that inelastic scattering might be the cause via the change in wavelength upon scattering.

_{3}counters. The incident neutron energy was 350 meV. The intensity transmitted by the Cd was measured for each of the ten thicknesses. The Cd transmission for the Al sample is shown in Figure 3. The solid line in the body of the diagram corresponds to zero energy change on scattering, and the open circles indicate the actual transmission and correspond to a net increase in neutron energy upon scattering. The deviation from the solid line increases in the order Pb, Al, C, corresponding to larger energy transfers as the mass of the scattering nucleus decreases.

^{113}with spin 1 and for Sm with a single resonance in Sm

^{149}with spin 7/2. Gd had contributions from two resonant isotopes. For Dy, the ratio $\frac{{\sigma}_{s}}{{\sigma}_{a}}$ was 0.2, which is too large for the approximations made in the analysis to be accurate. The resonances in Rh and In were at energies of 1260 and 1450 meV, which were well outside the accessible neutron energies available for the experiment.

## 3. The Period Prior to the Invention of Constant-Q (1951–1957)

#### 3.1. Lattice Vibrations in Aluminum

**k**and

**k′**are the initial and final neutron wavevectors,

**Q**=

**k**−

**k′**, is the wavevector transfer or scattering vector,

**q**is the phonon wavevector, and

**τ**is a reciprocal lattice vector, then wavevector conservation between the neutron and the phonon may be written,

**k**−

**k′**− 2π

**τ**+

**q**= 0.

^{2}k

^{2}/2m) − (ħ

^{2}k′

^{2}/2m) = hν

_{j}

_{j}is the frequency of the phonon in the jth normal mode. When these two conditions are met, a peak may be seen in the scattered intensity plotted as a function of the final neutron energy, E′, or the energy transfer to the crystal, E–E′. For a monotonic lattice, the frequencies are expected to separate into three branches, corresponding in the low-frequency limit, to the familiar longitudinal and two transverse sound waves.

_{B}, for scattering from the (333) reflection when the [111] direction is along the scattering vector,

**Q**, is 95.1°. Measurements were made with the crystal angle rotated by increments, δ, from the angle, Ψ

_{B}, the angle between the [002] axis of the crystal, and the incident wavevector,

**k**

_{B}, for Bragg scattering. The energy distribution was measured with a second crystal spectrometer, thus constituting a rudimentary triple-axis crystal spectrometer. The set-up with fixed E and variable E′ is equivalent to a chopper spectrometer with fixed incident energy and variable scattered energy determined by the time of flight of the neutrons from the sample to the counter. Then, it is difficult to be sure which phonons will satisfy the energy and momentum conditions or whether they will be in high symmetry directions. Sometimes, the branch of phonons, transverse acoustic (TA) or longitudinal (LA), could be assigned, but often, they are not. Since these were the first recorded phonons, the scattered neutron intensities, referred to as neutron groups, as well as the first experimental ν(

**q**) dispersion relation are shown in Figure 4 and Figure 5. Four points on the TA branch were established and one was established on the LA branch, but the remainder were of unknown polarization. The position of the TA phonons matched a sinusoidal curve with an initial linear slope corresponding to the transverse velocity of sound in aluminum. The authors noted that similar inelastic neutron scattering experiments were being carried out by Jacrot in Paris [21] and by Carter et al. at Brookhaven [22].

_{κ}is the mass of the atom of type κ in the l th unit cell, u

_{x}is the displacement of the atom from equilibrium in the x direction, and Φ

_{xy}(κκ′:ll′) is the force in the direction x on the κth atom in the lth unit cell when the κ′th atom in the l′th cell is moved one unit distance in the y direction. The Φ

_{x}

_{,y}are the interatomic forces of interest. The solutions to the equation of motion are plane waves of frequency, υ, wavevector,

**q**, and polarization vector,

**U**(

**q**), where all wavevectors

**q**within the Brillouin zone of the reciprocal lattice are permitted.

**U**(

**q**), in three selected orthogonal directions,

**R**

_{l}is the vector distance of the atom in cell l from the given atom, and Φ

_{αβ}(

**R**

_{l}) is the force on any given atom in the α-direction when the neighbor at

**R**

_{l}moves a unit distance in the β-direction. The generalization to the cases where there are more than one atom per unit cell is straightforward and has been given in detail in [24]. The equation only has a solution for $U(q)$ when the frequency satisfies the determinantal equation

**q**, there is a polarization vector, an eigenvector, whose components satisfy Equation (7) above.

**q**lies in a mirror plane, for example the $(1\overline{1}0)$ plane of a cubic crystal, one may take one component of the polarization vector, say U

_{3}, in the z-direction along the plane normal, and then the other two components, U

_{1}and U

_{2}, must lie in the plane but are otherwise not fixed. Then, the determinant, Equation (7), factors into two equations, namely for j = 3

**q**lies in a second mirror plane, U

_{1}and U

_{2}are also fixed by symmetry, ${D}_{xy}^{2}$ = 0, and Equation (9) also factorizes. Then, the squares of υ

_{1}and υ

_{2}are also linear in the force constants, which may then be fitted by linear least squares methods to the squares of the frequencies. For these cases, the equations may be written in terms of interplanar force constants, which are equivalent to the forces along the length of a simple linear chain, as was pointed out by Foreman and Lomer [25]. The equation for one direction may be written

_{coh}and m are the coherent cross-section and mass of the scattering nucleus, and e

^{−2W}is the Debye–Waller factor. The temperature factor N

_{λ}is given by

_{λ}= [e

^{hν/kT}− 1]

^{−1}.

_{λ}+ 1, and for phonon annihilation, the intensity is proportional to N

_{λ}. The Jacobian factor is

^{12}neutrons per cm

^{−2}s

^{−1}) at Chalk River on two crystals of Al oriented with [$1\overline{1}0$] or [100] vertical axes with either a crystal spectrometer equipped with Al monochromators defining the incident and scattered beams or with a filter-chopper. With the crystal spectrometer set-up, the background from cosmic rays or fast neutrons coming down the incident beam or through the shielding was measured by turning the analyzer crystal off its Bragg position. The resolution in

**k**and

**k′**was estimated from the collimator geometry and the mosaic spreads of the monochromating crystals. The incoherent scattering from vanadium at the elastic position gave the energy resolution.

**k**, is short. The difference count also serves to subtract the fast neutron background. The velocities of neutrons scattered at Φ = 90° were measured with a simple Fermi chopper which produces 240 pulses per second, with width in time of 140 μs. The time to traverse the 4.45 m flight path to the counter permits the scattered energy E′ to be measured. The scattering process is neutron energy gain by phonon annihilation. The energy resolution of the set-up was of order of the width of the observed neutron groups.

**q**were specified. Then, this process was iterated to obtain values of

**q**close to symmetry directions.

**q**) dispersion relation was plotted for the [ζ00], [ζζ0], and [ζζζ] directions. The data showed that the phonon intensities were largest near the reciprocal lattice points varying such as kT/ν

^{2}at relatively high temperatures, as given by Equations (11) and (12).

**Q**.

**U**

_{j}]

^{2}in Equation (11) is a selection rule for the intensities of the phonons. In the ($1\overline{1}0$) and (001) mirror planes, one of the polarization vectors has to lie perpendicular to the mirror plane, i.e., along the [$1\overline{1}0$] and [001] directions, respectively, while the other two polarization vectors have to lie in the respective planes. Only the transverse mode in the plane is allowed because of the selection rule. However, the TA phonon with [$1\overline{1}0$] polarization can be seen in the (001) plane. For [100] phonons, the transverse displacements in the [010] and [001] direction are equal, so the TA modes are degenerate.

#### 3.2. Lattice Dynamics of Germanium, Silicon, and Diamond

**q**goes to zero, and 3n − 3 are optical branches, longitudinal LO or transverse TO, where ν is finite at

**q**= 0. In special directions, [111] and [001], in a mirror plane, the normal modes are strictly transverse or longitudinal. For

**q**in a mirror plane such as ($1\overline{1}0$), there are two branches with polarization vectors normal to the plane and four branches with eigenvectors in the plane. The scattered neutrons occur in groups satisfying Equations (2) and (3). In the Born–von Kármán theory, the frequencies ν

^{2}are eigenvalues of a 3n × 3n determinant, which in symmetry directions factorizes into n × n and 2n × 2n determinants, making the calculations simpler. The cross-section for the creation or annihilation of one phonon integrated over energy is similar to Equation (11)

_{j}(

**q**,

**τ**), which is a generalization of (

**Q.U**

_{j})

^{2}in Equation (11), may be written for the four branches visible in the $(1\overline{1}0)$ plane as

**k**, was fixed, and the scattering angle and crystal angle were varied to locate the phonon frequencies near the symmetry directions around various reciprocal lattice points,

**τ**. Once an estimate of the phonon frequency was found, it was straightforward to alter the conditions slightly to come closer to the symmetry direction.

#### 3.3. Incoherent Scattering and the Phonon Density of States, g(ν)

_{coh}= 0.018, σ

_{inc}= 5.10, σ

_{abs}= 5.08 bn). By alloying Mn and Co together for a concentration of 0.42, the average coherent cross-section can also be made small, namely (σ

_{coh}= 0.007, σ

_{inc}= 4.08, σ

_{abs}= 26.7 bn). The theory of incoherent inelastic scattering was given by Cassels [34]. A slow neutron may be scattered inelastically when a neutron excites or de-excites one phonon or more than one phonon (multiphonon) modes. For an incoherent scatterer, the momentum change of the neutron does not impose any conditions on which phonon modes are excited. Then, the probability that a neutron changes its frequency by energy hυ is proportional to the number of vibrational modes between $\nu $ and $\nu $ + d $\nu $.

_{3}counters. The Be-Pb difference pattern as a function of average scattered wavelength is shown in Figure 9. The region marked “elastic scattering” is a frame overlap from the previous burst of the Fermi chopper. The scattering centered near 1.8 Å is the inelastic scattering of interest. Corrections for chopper transmission, counter wavelength sensitivity, air attenuation, and scattering were made. The most troublesome correction is for multiple scattering, where a neutron creates or destroys two phonons in one scattering event or the annihilation of one phonon followed by the creation or annihilation of a second phonon in a second scattering event. The calculation of the two-phonon scattering requires the integration of the double phonon cross-section over the observed spectrum [36,37]. Both effects give intensity over the whole region of the one-phonon scattering. The corrected vibrational spectrum for V is shown in Figure 10. The low-frequency part of the spectrum matched a Debye spectrum corresponding to a Debye temperature, θ

_{D}, of 338 K, as found from low-temperature specific heat measurements.

#### 3.4. Structure and Dynamics of Liquid He^{4}

^{4}were reported by Henshaw and Hurst in 1953 [38], thus beginning a field of endeavor that continued for over half a century. The results corrected for background and the change of effective volume with scattering angle were accurate to ±7%, while double scattering and the effect of resolution were noted to be smaller than the statistical error. The main finding was the position of the first peak in the structure factor at 2.15 ± 0.0.11 Å

^{−1}. This peak was attributed to a peak in the radial distribution function, g(r), at 3.6 Å.

^{4}cryostat with a monochromatic beam of neutrons of wavelength 1.04 Å from an NaCl monochromator at temperatures of 5.04, 4.24, 2.25, 2.15, 1.95, and 1.65 K and covered a range of density from 0.095 to 0.145 gmcm

^{−3}. The scattering was measured for angles between 4 and 80° (Q between 0.42 and 7.77 Å

^{−1}). The angular resolution was taken to be the width of the KBr {002} diffraction peak, which is near the maximum in the structure factor. The results were corrected for ambient background, effective scattering volume in plate geometry, and instrumental resolution. The multiple scattering was calculated to be negligible. At large Q, the scattering would be expected to decrease in a similar way to the scattering from free atoms, i.e., in an uncorrelated incoherent way. The experimental points were divided by the differential cross-section of a free He atom and the result was normalized to make the average quotient unity at large angles, since g(r) is unity at large r and S(Q) is unity at large Q. The assumption inherent in this normalization is that most of the scattering is incoherent and that the coherent scattering that reveals the structure of liquid He

^{4}is restricted to small angles and in the region of the peak in S(Q) around Q = 2.057 Å

^{−1}. The measured energy loss for scattering by a free He atom at the peak of the structure factor is 2 meV, which is small compared with the incident energy of 75.8 meV. Thus, the assumption that a diffraction measurement integrates over the inelastic spectrum is reasonable at 2.057 Å

^{−1}although not at 6.04 Å

^{−1}, where the energy loss is 17 meV.

^{4}at the λ-point. This is not to be expected, since there is a change in the velocity distribution at the λ-point but no change in the spatial distribution. With the assumption that the normalization procedure captures the coherent scattering, the radial distribution function rρ(r)/ρ

_{0}was calculated from S(Q) by Fourier transformation. While the radial distribution function shows spurious peaks below 2.5 Å

^{−1}due to the finite Q-range of the data, a shell of neighbors around 3.8 Å and also departures from uniform density around 7 Å are observed. The area under the main peak corresponds to about eight nearest neighbors, which would be the case for a body-centered cubic material.

^{4}were reported in [40]. Measurements were made at 1.27, 1.57, 2.08, and 4.21 K, below and above the λ-point of 2.2 K, with the rotating crystal spectrometer [41] providing incident neutrons of wavelength 4.14 Å (4.77 meV). A peak in the excitation spectrum was observed at a scattered wavelength of 4.5 Å (4.04 meV) corresponding to an energy transfer of 0.73 meV and wavevector transfer of 1.87 Å

^{−1}. Measurements were also made with the Chalk River filter chopper spectrometer [19], and the two sets of measurements were consistent. The measurements of the excitation spectrum, Figure 11, agreed with previous neutron time-of-flight measurements [42,43]. It was noted that as the temperature was raised toward and through the λ-point of 2.2 K, the excitation energies decrease and the widths of the peaks increases. At small momentum transfers, the spectrum tends toward a linear phonon relation and at higher wavevectors follows a form consistent with the Landau [44] roton curve. Under a pressure of 21.4 atmospheres, the scattered energy decreased by about 0.09 meV, which was also in agreement with the prediction of the Landau theory. These experiments marked the beginning of a series of experiments of the inelastic scattering from He

^{4}over a period of 50 years by Henshaw, Woods, Cowley, Svensson, and collaborators.

#### 3.5. Structure of Liquid Neon

^{4}cryostat. Measurements were made from Q = 0.5 to 6.3 Å

^{−1}.

#### 3.6. Dynamics of Water

_{2}O is 95% incoherent and D

_{2}O is about 80% coherent.

_{s}(

**r**,t), is “the probability that, given an atom at position 0 and time 0, the same atom is at

**r**at a later time t”. The pair correlation function G

_{p}(

**r**,t) is “the probability that, given an atom at position 0 and time 0, any atom is at position

**r**and time t”. The incoherent and coherent partial differential scattering cross-sections per atom are, following Squires [37],

**k**and

**k′**are the incident and scattered wavevectors, and

**Q**=

**k**−

**k′**is the scattering vector. Note that Squires [37] defines the scattering function S(

**Q**,ω) by

**r**along the outgoing direction

**k′**at its incident velocity. Thus, the validity of the static approximation reduces to the question of whether anything of interest happens to the system in time t

_{0}. Quoting from Brockhouse, “Crudely, the neutron experiments are a kind of microscope with a field whose spatial extent is about ${Q}_{0}^{-1}$ Å. If the range of neutrons accepted by the counter, within the resolution function, is limited to values between ±ω

_{1}, then

**Q**need not be included in the discussion in this case.

_{0}, about 17% of neutrons of wavelength λ

_{0}/2 are present, and these were corrected for. The results for a range of wavevectors near the maximum in the structure factor, 1.88 Å

^{−1}(which cannot be observed in H

_{2}O with neutrons because the scattering is primarily incoherent), and beyond are shown in Figure 13, over a limited range of neutron energy gain (11 meV) and loss (6 meV). Brockhouse considered there to be a qualitative separation of energy scales in Figure 13 corresponding to the broad distribution of scattering beyond ±3 meV and a nearly elastic contribution at lower energies. At wavevectors lower than 1.88 Å

^{−1}and greater than 2.9 Å

^{−1}, the distinction between the two distributions is less clear, although a fitting process can establish the relative amounts of each. The broad spectrum can be fitted to a calculated spectrum for a monatomic gas of mass (16 + 2 = 18) at all wavevectors, and at 8 Å

^{−1}, all the scattering can be attributed to a gas of mass 18. For H

_{2}O, the quasielastic scattering has a maximum at Q = 0 and has diminished to zero at 5 Å

^{−1}, as shown in Figure 14, and it follows a Debye–Waller like form with $\overline{u}$ about 0.4 Å. The width of this distribution increases with Q and the different kinds of diffusive motion, jump-type and continuous, which have different Q dependences. Brockhouse concluded that both contribute to this width. The fact that the broad inelastic contribution could be simulated by a gas of mass 18 indicated that the scattering observed originated from scattering by the whole molecule. Measurements with the filter difference spectrometer [19] in neutron energy gain at 300 K showed the presence of a peak in the scattering around 60 meV, which is in the same energy range, 30–110 meV, as the band of hindered rotations observed in Raman scattering by Cross et al. [48].

_{2}O at 1.88 and 2.33 Å

^{−1}, at which wavevectors there had been a clear separation between the inelastic and quasielastic contributions in H

_{2}O. The inelastic contribution is far smaller for D

_{2}O because of the ratio of the scattering cross-sections, 19/168. At a resolution of 3 meV, it was difficult to separate the two contributions. The broad part appeared wider at the higher wavevectors but is still narrower than expected on the basis of continuous diffusion. However, the correlation function that enters does include effects from the oxygen atoms, and so the scattering would be expected to be different.

_{0}= 4.964 meV and an energy resolution of 0.2 meV. The data were corrected for fast neutron background, air scattering, second-order wavelength contamination of the incident beam, counter efficiency, and multiple scattering. Twenty measurements at different scattering angles were used to create a grid of S(Q,ω) in steps of 0.2 Å

^{−1}and 0.05 meV, as seen in Figure 15, which was then Fourier transformed to obtain the self-correlation function G

_{s}(r,t). Only diffusive broadening of the quasi-elastic peak was observed over the whole range of ω (50 meV) and Q (2.0 Å

^{−1}). The energy width, W, of the diffusive peak, corrected for resolution, followed the relation

^{2}> for water derived from the data followed the diffusion law <r

^{2}> = 6Dt at times greater than 6 × 10

^{−12}s.

#### 3.7. Magnetic Structure of Chromium Oxide

_{2}O

_{3}(T

_{N}= 318 K) was determined by Brockhouse in 1953 [51]. Measurements were made at 80 and 295 K with 1.303 Å neutrons from a crystal spectrometer, and the reflections were indexed on the basis of the rhombohedral unit cell. Strong increases in the intensities of the {110}, {211}, and {200} reflections at 80 K were ascribed to magnetic Bragg scattering. Four Cr ions lie on the <111> diagonal of the unit cell, and three possible antiferromagnetic structures may occur, namely ↑↓↑↓, ↑↑↓↓, and ↑↓↓↑. Values of the magnetic structure factor were calculated and compared with experiment, and it was concluded that the ↑↓↑↓ arrangement was consistent with the strong intensity increases of the {110}, {211}, and {200} reflections and the absence of a magnetic contribution to the {210} reflection. With a polycrystalline sample, it was not possible to assign the crystallographic orientation of the Cr moments.

#### 3.8. First Crystallographic Texture Measurements with Neutrons

#### 3.9. Inelastic Paramagnetic Scattering

_{CW}is the Curie–Weiss temperature determined by fitting the paramagnetic susceptibility to the Curie–Weiss Law

_{4}(Mn

^{2+}) and Mn

_{2}O

_{3}(Mn

^{3+}) with a fixed incident wavelength of 1.3 Å reflected from an Al mosaic monochromating crystal recently grown by Henshaw at Chalk River. The measurements were carried out at the NRX reactor. The energy analysis was carried out by stepping the scattered neutron wavelength through 1.3 Å by varying the angle of scattering from a second Al “analyzer” crystal. The intensity at zero energy transfer, corrected for ambient background, nuclear incoherent and multiple scattering, was expressed as a cross-section by calibration with the known incoherent cross-section of vanadium and shown to follow a 3d form factor with values in the forward direction of S = 5/2 and 2, respectively. That is, the Q-dependence of the scattering was shown to be magnetic and originating in the scattering from the known magnetic moments. Both materials showed energy distributions that were consistent with a Gaussian form. MnSO

_{4}showed an energy distribution no wider than the instrumental resolution, but Mn

_{2}O

_{3}showed a width greater than the instrumental resolution. The results were consistent with the low Curie–Weiss temperature, θ

_{CW}= −24 K and low Néel temperature, T

_{N}= 14 K, of MnSO

_{4}and the larger θ

_{CW}= −176 K and T

_{N}= 80 K of Mn

_{2}O

_{3}.

#### 3.10. Spin Waves in Magnetite

_{3}O

_{4}, were reported by Brockhouse in 1957 [54]. To put the paper in context, it was generally thought that the excitations out of the fully ordered magnetic state were wavelike by analogy with the vibrations of the crystal lattice. However, only integrations over the dispersion relations contribute to thermodynamic measurements such as magnetization and low-temperature specific heat. The theory of neutron scattering by spin waves had been worked in the previous decade (here, theory was ahead of experiments), and it was shown the neutron-spin-wave system should obey the same conservation laws in momentum and energy as the lattice vibrations, namely Equations (2) and (3).

^{3+}(S = 5/2) ions on A sites and 8 Fe

^{3+}and 8 Fe

^{2+}(S = 2) ions on B sites. The ions on A sites are antiparallel to those on B sites, leading to ferrimagnetism. At temperatures below 119 °K, the Verwey transition, the Fe

^{3+}and Fe

^{2+}are ordered on the B sites. There are six interpenetrating magnetic sublattices, and therefore, there is one “acoustic” mode that goes to zero energy at

**q**= 0 and five “optic” branches. The question arose: “Could the magnetic excitations be seen above the other contributions to the scattering, especially since Fe and O are strong coherent scatterers and give rise to intense lattice vibrations and there is a large diffuse contribution from the impurity content of the crystal?” Brockhouse estimated that the total incoherent scattering, both nuclear and magnetic, was between 5 and 8 bn per Fe

_{3}O

_{4}formula unit. Multiple scattering, in this case neutron inelastic scattering preceding or following Bragg scattering in the crystal, will augment the incoherent scattering. With characteristic thoroughness, measurements of the intensity integrated over energy in the form of a differential scattering cross-section, dσ/dΩ, calibrated with a standard V sample, were made and shown to be consistent with the estimates of intensity. In the experiment, measurements were made around the (111) reflection, which is primarily (97%) magnetic. Since the measurements were made at a relatively small angle, 8.1°, fast neutrons coming down the main beam would have been a problem, and tight collimation was used. Close to, but not at the (111) peak, dσ/dΩ, becomes very intense, as had been seen previously [55], and it was the investigation of this intensity that yielded the spin-wave behavior under energy analysis.

^{−1}. It was not possible to decide on the basis of the initial experiments whether the dispersion relation was linear or quadratic because of the scatter. However, the line through the data certainly did not correspond to the velocity of sound in magnetite and therefore was not a phonon dispersion relation seen through the magnetic cross-section (magnetovibrational scattering). For a linear dispersion relation, the dominant exchange interaction, J

_{AB}, would be 122 K (or 10.5 meV, 2.5 THz), while for a quadratic dispersion relation, J

_{AB}, it was 23.2 K (or 2.0 meV, 0.48 THz). On the basis of the latter number, the calculated Curie temperature in a molecular field model would be 1050 K, which is not too far from the actual value of 850 K. On this physical basis, it was concluded that a quadratic dispersion relation was more reasonable.

**q**= 0, the relative phases of the spin deviations would be the same as the static magnetic moments. On this basis, he guessed the structure factors and found them to agree within reason with the measurements of d

^{2}σ/dΩdE. Further work was promised on the temperature and field dependence of the intensities, which are unambiguous for magnetic materials.

## 4. Post Development of the Triple-Axis Crystal Spectrometer and Constant-Q, 1957–1965

^{14}neutrons·cm

^{−2}s

^{−1}, which is about a factor of 10 higher than NRX. A newly designed triple-axis spectrometer was being built at the NRU reactor, and immediately following the restart of the reactor after the NRU accident in the fall of 1958, it was rapidly deployed. In describing the instrument at the Conference on Neutron Scattering in Solids and Liquids in 1960 in Vienna, [41], Brockhouse said, “The C5 triple-axis crystal spectrometer at the C face of the NRU reactor was designed to be as flexible and generally useful as possible, allowing a wide range of energies, E and E′, and scattering angles, Φ, and crystal angles, Ψ. The resolution was readily changeable.” The neutron flux of all wavelengths at the source, which was 6 m from the spectrometer, was 2 × 10

^{14}neutrons·cm

^{−2}s

^{−1}.

_{1}, and a large moving platform with an accurate vernier scale, which carries the sample table, analyzer, and counter. Arguably, this is the most crucial part of the instrument, since it has to provide accurately moving parts and provide good shielding against the fast neutrons and gamma rays emerging from the beam tube. It was designed by W. McAlpin, who had worked on ship structures on the River Clyde in the UK during the war. The monochromator angle θ

_{M}followed the turning of the drum, the monochromator scattering angle, 2θ

_{M}, at half speed.

_{A}is connected to 2θ

_{A}by a half-angling mechanism, and both were provided with accurate angular scales. The BF

_{3}(96% B

^{10}) counter was 6.2 cm in diameter and 25 cm long and was made at Chalk River. A paper-tape control system was used, and the tape contained angular increments in 2θ

_{M}, Φ, Ψ, 2θ

_{A}prepared on the Chalk River mainframe computer corresponding to constant-

**Q**or constant-υ scans. The monochromator and analyzer crystals were mounted on permanently aligned mounts and cut from single-crystal ingots of Al with the [$1\overline{1}0$] direction vertical allowing access to the (111), (002), (220), (113), and (331) monochromator planes. Adjustable collimators were made of Cd-coated steel strips which slid into slots in the collimator boxes. Collimations around ½ to 1° were used at collimators C3 and C4 to define the beam directions and were relaxed at collimator C5. The collimation before the monochromator, C1, C2 is defined by the beam apertures and the distance to the source, and it is of order 1°. A photograph of the spectrometer taken in about 1965 is shown in Figure 19.

**Q**, ω) directly”.

**Q**method in this way:

**Q**method of observing phonons. A few weeks earlier, R.G. Stedman from Sweden had arrived at Chalk River to work for a year in P.A. Egelstaff’s United Kingdom Atomic Energy group on scattering from neutron moderators. Dr. Stedman had explained to us attempts made in Sweden to observe phonons in NaCl on the initial steep branch of the dispersion relation by moving at constant energy transfer across the curve. Bert brilliantly clued into this and realized that if you could control the angle of scattering and the sample crystal orientation along with the energy transfer, you could do a scan without changing the momentum transfer, hence constant-

**Q**”. The awkward Jacobian defined in Equation (13) becomes unity in the constant-

**Q**method.

#### 4.1. Lattice Dynamics of Crystals

#### 4.1.1. Lattice Dynamics of Lead

**k**, method and then with the constant-

**Q**method. The energy versus wavevector dispersion relation for Pb at 100 K is shown in Figure 20; it is far from being a simple sine curve and clearly displays higher-order harmonics. The neutron groups at higher temperatures are broadened but superposed on a strong sloping background, which was ascribed to two subsequent single phonon scattering events in the large crystal as well as multiphonon scattering. After absorption corrections and taking out the population factors and assuming that the cross-section is independent of temperature, the full-widths at half-maximum of the peaks at 425 K are shown in Figure 21.

**u**

_{l}is the displacement and Φ and Γ are the restoring and dissipative force constants between the l and l′ atoms. We assume the solution to be a damped wave of form

_{n}decreases faster as n increases. Brockhouse et al. postulated that damped phonons have an energy dependence that follows a Lorentzian form

**q**appears to follow a sine variation, as Figure 21 shows, and 1/γ may be interpreted as the lifetime of the phonon. Its value at the [00ζ] zone boundary is about 30% of the phonon frequency, showing that the lifetime is only about 50% of the period of the wave. The origin of the widths, whether due to electron–phonon or phonon–phonon coupling, remained unknown at that time.

**q,**around different reciprocal lattice vectors,

**τ**, proved that this was the case and constituted the first such direct observation of the interaction between phonons and electrons [61]. The experiments were carried out at the NRU reactor with the constant-

**Q**technique with a relative error from point to point of ±0.05 THz. For a spherical Fermi surface, the positions of the anomalies occur where the sphere cuts the wavevector,

**q,**are given by

**τ,**for example, (000), which gives a circle in the ($1\overline{1}0$) plane, or (200), which gives an arc in the ($1\overline{1}0$) plane. The most marked discontinuity is at $\frac{a}{2\pi}[0.4,0.4,0]$, and this position is in fair agreement with the dimension of the Fermi surface, assuming it to be spherical or as determined by de Haas–van Alphen measurements. The anomaly at $\frac{a}{2\pi}[1.5,1.5,1.5]$ is also expected to be strong but may have contributions from two different pieces of the Fermi surface. Actual discontinuities such as the Kohn anomalies in Pb underline the long-range nature of the electronic origin of the forces and are not sensibly amenable to the interplanar approach.

#### 4.1.2. Lattice Dynamics of Sodium and Potassium

_{n}is a linear combination of the atomic force constants α

_{1}, β

_{1}, α

_{2}, etc. The contributions from the five nearest neighbors are given explicitly in Tables 1 and 5 of ref. [63]. Typically, the ratio of φ

_{3}to φ

_{1}is about five to ten times smaller for Na than for Pb. The single crystal of Na, having a length and diameter of 4.5 cm, was encased in a thin-walled Al can to prevent oxidation. Measurements were made on the C5 triple axis crystal spectrometer at the NRU reactor by the constant-

**Q**method. The majority of measurements were made at 90 K, and measurements at 215 and 296 K showed that the frequencies decreased by a few percentage points, and the phonon widths increased as the temperature approached the melting temperature of 370.6 K. The dispersion relation for Na at 90 K is shown in Figure 22. The Fourier analysis showed that while three terms, corresponding to five shells of neighbors, gave quite a good fit, a more precise fit required n = 5, corresponding to eight shells of neighbors, but with values which were only of order 1% of φ

_{1}. Na is non-superconducting, unlike Pb, and the electron–phonon interaction is very small. Considerable effort was spent to see whether there were any Kohn anomalies in Na where the Fermi surface is nearly spherical and well known. There were no discontinuities in the dispersion relation to within 1% in the locations where the free-electron Fermi surface cuts the dispersion relation in wavevector.

**Q**scan, the cross-section and hence the integrated intensity is given by

_{j}is the temperature factor and g

_{j}is the structure factor.

_{l}is the scattering length for the lth atom in the unit cell, and ${U}_{jl}$ is the polarization vector for the lth atom in the jth mode. The ratio of the integrated intensities of the neutron groups for the two modes, one longitudinal and one transverse, propagating in the $(1\overline{1}0)$ plane is, for neutron energy loss, for monatomic Na

**Q**, and the [001] axis and α is the angle between ${U}_{1}$ and the [001] axis, then

_{D}, which was in substantial agreement with the experiment.

_{α}, specifying the polarization of the mode, obeys certain orthogonality relations and Φ

_{αβ}(0,l) are the atomic force constants following the notation of ref. [65]. The equations are non-linear, and it is not possible to solve uniquely at a general point in the Brillouin zone unless both the frequency squared and the polarization vector are known. Since measurements are typically made in high-symmetry directions, such as [ζζζ], in general, the force constants derived from these measurements alone are not unique. This effect was also noted in [66] for Ni and also for Si by Dolling [30], where there are sometimes ambiguities in the fits such as non-physical parameters.

**Q**), are defined, which only depend on the distance r between the atoms. Then, D may be written

^{R}from short-range interactions, which are expected to be small, and D

^{C}from electrostatic forces assumed to be acting between the ions, which can be calculated from the Ewald θ-transformation [68]. Finally, D

^{E}describes the interactions between the ions and the conduction electrons. In the method of Cochran [67], the bare ion contribution, D

^{C}, is subtracted from the measured D

_{αβ}(

**q**), and the remainder is analyzed in terms of a local pseudopotential interaction between the ions and the conduction electrons, which is written in the form

^{E}was initially specified by a Fourier series with up to 30 terms up to a value Q

_{max}, beyond which it was assumed to be zero. The Fourier series was only used to obtain the first estimate of V

^{E}, and thereafter, it was determined by a table of values, the entries of which were parameters, in a non-linear least-squares fit. The values of the potential were found to be independent of the value of Q

_{max}. An alternative analysis, which developed a potential between atoms instead of ion cores with an ion-conduction electron interaction was found to be less satisfactory.

#### 4.1.3. Lattice Dynamics of Alkali Halides

^{+}, ions and the negative, I

^{−}, ions lie on interpenetrating fcc lattices. Kellerman [68] had computed the lattice dynamics for a series of point ions interacting with Coulomb forces and a central repulsive force between first neighbors, which became known as the rigid ion model. This approach neglects the polarizability of the ions and thus cannot explain their dielectric properties. In addition to the neutron measurements and calculations in the rigid ion model, the lattice dynamical theory, which takes account of the polarizability of the ions, known as the shell model, was also presented. Both were compared with experiment in forms with no adjustable parameters. The approach drew on the expertise of Prof. W. Cochran (who had pointed out to Crick and Watson that the X-ray patterns of DNA indicated a helical structure) and who was a visitor from Cambridge UK for a year at Chalk River in the late 1950s.

^{−}ions is much larger than the Na

^{+}ions, leading to being able to neglect the latter. Large single crystals were available, since they were in common use for γ-ray detectors. Unfortunately, the incoherent scattering cross-section for sodium is 1.6 bn, which produced inelastic incoherent scattering, as in V, in the vicinity of the optic modes. In this case, the determinantal equation which gives the phonon frequencies and polarization vectors, has the form

**I**is the unit 6 × 6 matrix. The shell model leads to two further variables corresponding to deformable shells on the two ions. Then, there are forces between the cores, between the shells, and between the core of one ion and the shell of the other. Then, the matrix,

**D**, is 12 × 12. However, the expressions for

**D**contain constants, which are all known from the elastic constants and from the high and low frequency dielectric constants, so the dispersion relation may be calculated with no adjustable parameters in the simple shell model and the rigid ion model.

**Q**technique. The frequencies of the optic modes in NaI at 100 K were established, and the complete dispersion curve is shown in Figure 26 together with the rigid ion and shell model calculations. The rigid ion model tends to overestimate the frequencies, especially near the [001] zone boundary. The TO modes are fairly well captured, but the LO mode is badly overestimated. The simple shell model is a considerable improvement. While not completely correct, particularly at the [111} zone boundary, it certainly captures the basic behavior with no adjustable parameters. One key feature mentioned in ref. [72] was that the LO mode appeared to be weak and quite broad compared with the TO mode. The measurements in KBr also showed anomalous behavior of the LO mode: the widths of the corresponding neutron groups increase strongly with temperature even at 400 K, which is well below the melting temperature of 1003 K, and the LO mode at

**q**= 0 is also broader than the TO mode, suggesting that the lifetime of the LO phonons is reduced. The calculated dispersion relation with the rigid ion and simple shell models are shown in Figure 27.

^{+}ions implies a positively charged shell, which was considered to be unrealistic. It was pointed out in this paper that the shell model was formally equivalent to introducing dipoles in addition to the cores on the ion sites and then considering the force constants between the cores, between the dipoles, and between cores and dipoles.

**q**= 0 are shown in Figure 28 for the LO and TO modes in NaI and KBr. The TO modes have a well-defined frequency but broaden somewhat as the temperature is raised. For the LO modes, the center of gravity of the frequency distribution decreases with temperature but also broadens markedly, especially to the high-frequency side. Roger Cowley remarked that, “At least a substantial part of the anomalous temperature dependence of the shape of the LO modes in alkali halides, as found by Woods et al. in ref. [72], can be explained in terms of phonon–phonon interaction”. For NaI and KBr, there are two and possibly three peaks in the LO response as the temperature increases arising from anharmonicity. Further calculations reported in the comprehensive review of phonons in perfect crystals by Cochran and Cowley [24] shows that the broadening increases as

**q**approaches zero in the [111] direction for KBr.

#### 4.1.4. Lattice Dynamics of Pyrolytic Graphite

**Q**. As a result of the 5± mosaic spread of the sample, the vertical divergence at small offsets from the (002) reflection gives rise to a range of phonon wavevectors, and this tends to give a spurious increased frequency at small wavevectors. The frequency versus wavevector dispersion relation is well fitted by a single sine wave, suggesting that there are longitudinal forces only between nearest planes. The transverse modes appear as broad distributions about 50% lower than the longitudinal frequencies and are in any case somewhat ambiguous.

#### 4.1.5. Lattice Dynamics of SrTiO_{3}

_{3}in 1962 [76,77]. Cochran [78] had recognized that the anomalously large dielectric constant is associated with a low-frequency transverse-optic mode of small phonon wavevector,

**q**. The lowest frequency [ζ00] TO branch was measured by neutron inelastic scattering on the C5 triple axis spectrometer at the NRU reactor as a function of temperature between 430 and 90 K and found to be strongly temperature-dependent at small wavevectors. Most of the measurements were made by constant-

**Q**method, but the constant-υ method was used where the branch dips steeply. The ζ = 0 mode obtained with neutrons agreed well with that determined by far infrared measurements. The temperature dependence interpreted on Cochran’s theory [78] should behave similar to υ

^{2}~(T-T

_{c})

^{−1}, where T

_{c}is the temperature of the phase change. The experimental confirmation is shown in Figure 29, which also shows the Curie-like variation of the reciprocal of the dielectric constant. The paper marked the beginning of a major field of experimental work on the connection between crystallographic phase transitions and the divergence of a generalized susceptibility. In the particular case of SrTiO

_{3}, a particular mode softens. When the quasi-elastic scattering at a particular wavevector increases strongly (as opposed to the frequency of the inelastic scattering decreasing), the mode is referred to as an over-damped mode, and this eventually transforms into a sharp Bragg peak.

#### 4.1.6. Lattice Dynamics of bcc Transition Metals

_{44}elastic constant in Nb). Ta in the same column of the periodic table shows the first anomaly noted for Nb, but not the second and third. It is as if the anomalies become less marked for the 5d as opposed to the 4d metals.

#### 4.1.7. Crystal Dynamics of β-Brass

**Q**or constant-υ methods. The dispersion relation at 296 K is shown in Figure 33.

**q**) would resemble a body-centered cubic material such as Na, where the zone-boundary in the [00ζ] and [ζζζ] directions is at ζ = 1.0. The presence of ordering in the CsCl structure halves the size of the Brillouin zone to ζ = 0.5, and the range ζ = 0.5 to 1.0 corresponds to optic modes. For a body-centered cubic material, the branches would be continuous at ζ = 0.5, but for the CsCl structure, gaps appear at ζ = 0.5 at about a frequency of about 4.5 Thz in β-brass. Gaps in the branches also appear at this frequency at [0.17, 0.17, 0.17] and at [0.22, 0.22, 0], where the LA and LO modes would otherwise cross in the reduced zone scheme for the body-centered cubic structure. In these cases, there is an absence of vibrations where these modes of the same symmetry cross and the eigenvectors are zero. There is also a sharp dip at this frequency in the phonon frequency distribution g(υ) for which Gilat and Dolling [89] had developed an accurate numerical method of calculating

#### 4.1.8. Lattice Dynamics of GaAs

**q**= 0 obeyed the Lyddane–Sachs–Teller [91] relation, namely

_{0}and ε

_{∞}are the static and high-frequency dielectric constants of GaAs and are also in agreement with infrared absorption and reflection measurements. The dispersion relations were generally similar, except for the absence of certain degeneracies, to the semiconductors Ge [27] and Si [30] such as the splitting of the LA and LO modes at [001], and the results were also interpreted in terms of the shell model. The model parameters required second-nearest neighbor short-range interactions, without which quite poor descriptions were obtained and lead to a small positive charge of about 0.04e on the Ga ion.

#### 4.1.9. Crystal Dynamics of UO_{2}

_{2}was the material of choice for the fuel elements in the CANDU reactor system developed at Chalk River in the decades between the late 1940s and the 1960s and was of great interest to Atomic Energy of Canada. The lattice vibrations in UO

_{2}, which has the calcium fluorite structure, were measured by Dolling, Cowley, and Woods [58] by triple-axis crystal spectrometry and interpreted by them in terms of the shell model, since it is an ionic material. Thus began a fascinating story which has only been clarified in the last few years, sixty years after it began. Since there are three atoms in the primitive unit cell, there are nine branches of the dispersion relation, three acoustic modes, and six optic modes. The measurements were made at 296 K on a crystal of about 3 cm

^{3}volume in the [00ζ], [ζζ0], and [ζζζ] directions. Structure factor calculations based on the rigid ion model were used to find the locations in

**Q**-space where the various branches are best observed. As a result of the low intensity of the optic modes around 18 THz, these were measured with a poor (5°) mosaic spread pyrolytic graphite analyzer with a Be filter in front of the detector to pass only 4 Å neutrons. The dispersion relations are shown in Figure 35. Dashed curves represent the rigid ion model, and solid curves represent the results of a shell model for which the ionic charge was allowed to vary. The latter model gave an accurate fit to the data, although the authors cautioned that the values of the parameters should not be taken as having any particular significance. The density of phonon states, g(υ), was calculated, and from this, the lattice specific heat up to 150 K for comparison with the experimental results, which include both lattice and magnetic contributions, since UO

_{2}is antiferromagnetic below 30 K A surprising feature of the comparison is that the magnetic-specific heat appears to persist far above the Néel temperature, indicating a strong short-range magnetic order or other contributions to the specific heat.

#### 4.1.10. Measurements of the Moderator Properties of Materials

_{2}, the fuel of choice for most reactor systems. Since the initial energy of neutrons released on fission is about 2 MeV and the final thermal energy is about 25 meV, most of the moderating collisions (about 30 for light water) are of the “billiard ball” variety, and only the final few collisions depend on collective properties such as the phonon spectrum. The approach to calculating the moderating properties is eventually to cast the problem into the average single nucleus response with an effective cross-section and effective mass dependent on the chemical formula. The interference effects due to the coherent scattering in the material are treated later as a perturbation. The case of UO

_{2}is considered here because both the polycrystal and single-crystal measurements were made at Chalk River. After the end of the Canada–United Kingdom collaboration in the late 1960s, the four-rotor spectrometer was not used for further experiments, as it required a great deal of technical effort and was rather less robust than the C5 triple-axis crystal spectrometer.

^{coh}(Q,ω), are subsequently treated as corrections, which may be large if the cross-section is principally coherent as for Be. A new function S(α, β) called the Scattering Law was introduced [94,95] with variables related to Q and ω by

^{2}, an energy, rather than Q. Considering the self-term in UO

_{2}with contributions from both U and O, the cross-section was written as

^{−β}

^{/2}was introduced to ensure that the principle of detailed balance is obeyed [94]. Since the scattering cross-section for U is about twice that for O, the last term in brackets in Equation (43) was approximated by ${\sigma}_{U}{S}_{U{O}_{2}}^{inc}$, where the response of U and O are bundled together and treated as the measured quantity. An example of the quantity $\frac{{S}_{U{O}_{2}}^{inc}(\alpha ,\beta )}{\alpha}$ for polycrystalline UO

_{2}at 296 K as a function of α is shown [93] in Figure 36 for three values of β. The rise at low α has contributions from coherent interference processes and multiple scattering, which mask the single nucleus response at small α. The dashed straight lines are fit to the part of the function for α > 5 corresponding to high Q (about 7 Ǻ

^{−1}), so the analysis removes the coherent scattering by extrapolating the single particle effects, which are expected at large α, to α = 0. The Fourier transform of the velocity correlation function p(β) is related to this extrapolated value [94,95] by

_{2}is shown in Figure 37 and is compared with g(υ) determined from the phonon dispersion curves for UO

_{2}[58]. The peak at β ≈ 0.5, 150 K, 3 Thz corresponds to the lower peak in g(υ) and the broader peak at β ≈ 2, 600 K matches the center of the broad distribution in g(ν) centered on 13 THz in g(υ), and the cut-off location is reasonable. The resolution of p(β) is good enough to proceed to the next stage of calculating S(α, β) over a wide range of α and β, as described by Egelstaff and Schofield [94,95]. However, from the point of view of getting at the lattice dynamics of UO

_{2}, these experiments are not very revealing.

_{2}O and D

_{2}O concentrations across the reactor, and in the AGR to account for the variations in temperature around graphite sleeves supporting the fuel elements.”

#### 4.2. Liquids

#### 4.2.1. The Dynamics of Liquid Lead

^{−1}) with the newly installed C5 triple-axis crystal spectrometer at the NRU reactor at Chalk River. This enabled measurements to be made up to Q of 7 Å

^{−1}. Measurements were also made with the rotating crystal spectrometer with E = 4.8 meV to cover the region of the first peak in the structure factor which is at 2.17 Å

^{−1}, where high-energy resolution is required. Measurements were made at many scattering angles down to the multiple scattering background [36,37] with a maximum energy transfer of ±20 meV.

^{−12}s are shown in Figure 38. A check on the time Fourier transform was made by making use of the second moment theorem of de Gennes [97], which relates I(Q,0) to the experimental second moment of the energy distribution by

^{−12}s and Q is in Å

^{−1}. At a wavevector of 8 Å

^{−1}, that is a large Q, for t = 0.1, I(8, 0.1) = 0.45, this matches the experimentally determined function in Figure 38. At t = 0.2, I(8, 0.2) = 0.04, and this also matches the experiment. At smaller wavevectors than Q = 8 Å

^{−1}, the structure of the liquid matters, and the perfect gas model is no longer appropriate. Now, the widths of G(0,t) and G(nn,t), where nn signifies the near-neighbor separation in the first coordination shell r = 3 Å, in Figure 38 resemble a perfect gas for t < 0.2 × 10

^{−12}s and match the expected value calculated from the macroscopic diffusion constant up to t = 1 × 10

^{−12}s, but they fall well below at larger t. Brockhouse and Pope remarked that this occurred also for H

_{2}O and surmised that this might be a general feature of liquids. The fact that the widths lay below the expected macroscopic diffusion and did not account for it suggested to them that there might be other diffusion processes occurring, such as jump diffusion, beside continuous diffusion.

#### 4.2.2. Structure and Dynamics of Liquid He^{4}

^{4}above and below the λ-point at the saturated vapor pressure and at various pressures [98,99] up to 51.3 atmospheres with the diffractometer at the NRX reactor, extending earlier measurements [38,39]. An incident neutron wavelength of 1.064 Å was used with scattering angles between 5 and 60± corresponding to a range of wavevectors between 0.5 and 6.0 Å

^{−1}. Since the measurements were made without an analyzer, the static approximation was made, although this was probably satisfactory in the Q-range covered, since the incident energy was 72.2 meV, and the single-particle excitation energy was only 0.96 meV at 2.1 Å

^{−1}. Measurements of the diffracted intensity as a function of angle were corrected for background contributions and for multiple scattering processes where the neutron scatters more than once in the sample and then used to construct the structure factor, S(Q), which is normalized to unity at the highest wavevectors. Values of S(Q = 0) for various pressures were determined from the relation, S(0) = ρk

_{B}Tχ

_{T}, where χ

_{T}is the isothermal compressibility.

^{−1}does not change, but the peak height decreases by about 5%. It is fair to say that the significance of this decrease as a measure of the superfluid fraction was not appreciated at the time. Later measurements confirmed this decrease. Apart from the region of the first peak, S(Q) was unchanged—for example, at the location of the weak second peak around Q = 4.3 Å

^{−1}. Following Squires [37], the relation between the radial distribution function and the structure factor is given by the Fourier transform of Equation (48), namely

^{4}at Chalk River was carried out by Henshaw and Woods [100] using a triple-axis spectrometer set up at the thermal column of the NRU reactor. The experiments were carried out with incident neutrons of wavelength 4.039 Å (5.01 meV) from the (111) planes of an Al single crystal and the (111) planes of a Pb single crystal as analyzer. The incident beam was filtered through Be and quartz single crystals. Measurements were made over a wavevector range from 0.27 to 2.68 Å

^{−1}and up to a maximum energy of 1.73 meV. The energy versus wavevector dispersion relation at 1.12 K is shown in Figure 40. At wavevectors below 0.6 Å

^{−1}, the dispersion relation is linear and matches the velocity of sound, 237 ms

^{−1}in He. Above 0.6 Å

^{−1}, the curve falls below the velocity of sound, reaches a maximum at 1.10 Å

^{−1}and 1.18 meV, falls to the roton minimum at 1.91 Å

^{−1}and 0.75 meV, and then rises again. The results agreed quantitatively with those of Yarnell et al. [101] but extended the range of measurements to much higher energies and wavevectors. In particular, it was shown that the phonon–roton curve begins to flatten off in energy at the highest wavevectors well below the energy for free particles in the region, and its intensity decreases rapidly.

^{−1}), and then decreases so as to be at least ×12 times less than the value at the roton minimum by Q = 2.68 Å

^{−1}. It was realized that the cross-section for the phonon–roton peak was far smaller than the cross-section integrated over energy, S(Q), measured in ref. [39] in 1955 for He at this temperature and wavevector. This was consistent with the hypothesis that most of the intensity is associated with multiphonon scattering. That is, the weight in the excitation spectrum shifts from the single particle excitations to multiple processes, which resemble the scattering from non-interacting He atoms. This prompted later detailed measurements of this inelastic multiphonon scattering.

^{−1}, the energy of the excitation increases rapidly with temperature below the λ-point, resembling an order parameter variation, for example, a Brillouin curve. Above the λ-point, the energy is practically constant with a value of 0.43 meV (5.2 K), as shown in Figure 42. The results confirmed that the excitations and their interactions are a strong function of temperature below the λ-point and therefore are a strong function of the superfluid fraction. On the other hand, the widths at the roton minimum above the λ-point are consistent with a gas of free He particles.

^{4}was predicted by Hohenberg and Martin [103] to follow the form

_{1}is the velocity of ordinary sound and ρ

_{s}and ρ are the densities of the superfluid fraction of He and the density including both the normal and superfluid fractions. Since ρ

_{s}is a strong function of temperature, the slope should be temperature dependent. Measurements of the phonons were made with the rotating crystal spectrometer at Chalk River by A.D.B. Woods [104] and found to be independent of temperature, which disproved the prediction. However, the widths of the phonon peaks do increase with temperature above the λ-point, and a sharp peak is not observed at 4.2 K. The importance of these observations is discussed below.

^{−1}corresponding to longitudinal density fluctuations through a maximum in the curve to the roton minimum at Δ

_{R}= 8.67 K and Q

_{R}= 1.936 ± 0.005 Å

^{−1}and its eventual disappearance around Q = 3.5 Å

^{−1}and 18.40 ± 1.4 K. This is equal to twice Δ

_{R}to within the experimental uncertainty and is generally taken to mean that at higher energies, the elementary excitations would always decay into two rotons. Critically, these measurements showed that the phonon–roton curve terminated at a finite wavevector, Q = 3.5 Å

^{−1}, rather than continuing up to higher Q and becoming the free particle curve, as had been assumed previously.

_{11}(Q, ω) as expressed by

_{11}(Q, ω)

^{−1}are displayed in Figure 43. The lower solid line shows the phonon–roton part. The mean position of the multiphonon part and the position in energy at the half-heights of the multiphonon distribution are also shown. While the existence of a well-defined phonon–roton mode at higher wavevectors in the roton region arises from the Bose–Einstein condensation at low temperatures, it was the width of the multiphonon part that gave the first estimate of the fraction, n

_{0}, of helium atoms in the zero-momentum state,

**k**= 0. Beyond Q = 3.5 Å

^{−1}, only S

_{11}(Q, ω) is non-zero, and its mean energy is approximately that expected for free helium atoms recoiling after being struck by a neutron, namely $\frac{{\u0127}^{2}{Q}^{2}}{2M}$, where M is the mass of a helium atom. The phonon–roton curve does not connect with the free-atom recoil curve but lies well below it, and it is the center of the broad distribution that merges with the free atom scattering. The integration over energy of S(Q, ω), the zeroth moment including both the Z(Q) and S

_{11}(Q, ω) terms, was shown to be equal, to within the experimental uncertainty, to S(Q) as measured by X-rays, which certainly integrates over the whole energy spectrum. That is, all the inelastic neutron scattering is accounted for at 1.1 K. Measurements at 4.2 K, above the superfluid transition temperature T

_{λ}, showed that the linear phonon part for Q less that 0.3 Å

^{−1}still existed and that the rest of the phonon–roton curve was absent, while the broad multiphonon part persisted. A crucial observation was that the full-width at half-height of the broad peak, corrected for experimental resolution, was systematically larger at 4.2 K than at 1.1 K, as shown in Figure 44. The oscillations in the width were ascribed to coherent effects. It was also found that the increase in width of S

_{11}(Q, ω) occurs very close to T

_{λ}. The results suggested to Cowley and Woods that there is an increase in the kinetic energy of the helium atoms above T

_{λ}, which was reasonably ascribed to the depletion of the zero-momentum state and could be used to give an estimate of n

_{0}. The results were analyzed with the wave functions and distribution of particle states for helium proposed by MacMillan [107] and for different integrated intensities and widths of a Lorentzian lineshape describing the superfluid state, and the best match with experiment gave an estimate of n

_{0}= (17 ± 10)%. This was the first direct numerical evidence for the existence of the Bose–Einstein condensation in liquid helium below the lambda point and was an important step forward in the field of quantum liquids.

#### 4.3. Magnetism

#### 4.3.1. The Magnetic Structure of Mn_{3}ZnC

_{3}ZnC was clarified by Brockhouse and Myers in a paper [108] describing neutron diffraction measurements at temperatures above and below the Curie temperature of 392 K. The saturation magnetization of Mn

_{3}ZnC initially follows a Brillouin curve down to 230 K but then shows a slight arrest at about 1.3 μ

_{B}and then decreases again. Low-temperature X-ray measurements showed that the cubic perovskite structure deforms to a tetragonal structure at 233 K. Neutron diffraction measurements at 433 K confirmed the non-magnetic perovskite structure and revealed paramagnetic scattering corresponding to the observed Curie–Weiss law. Measurements at 295 K indicated magnetic contributions to the intensity in {100} and {200} peaks. At 100 K, the {100} peak continues to increase, but additional peaks appear at reciprocal lattice positions such as {½½½}, and these follow a second Brillouin curve below 230 K.

_{B}. Between 392 and 231 K, these are aligned ferromagnetically and would correspond to an average Mn moment of 1.66 μ

_{B}at 0 K. If the initial Brillouin curve above 392 K is extrapolated to zero, an average moment of this magnitude is obtained. Below the anomaly around 231 K, the moments on the 3 μ

_{B}sites remain ferromagnetically ordered, but half the moments on the 2 μ

_{B}sites remain parallel to the initial direction, but half align antiparallel—that is, antiferromagnetically—and so do not contribute to the net magnetization at 0 °K. In this case, the net ferromagnetic moment per Mn atom is 1 μ

_{B}, as measured by the saturation magnetization. The model also accounted for the magnetic structure factors. Interestingly, the magnetic anomaly does not occur in the ferromagnetic isomorphous structure Mn

_{3}AlC, where the Al would contribute an extra electron to the conduction band. It was noted that Kasper and Roberts [109] also found it necessary to describe the antiferromagnetic state of α-Mn, with three kinds of Mn sites, one with zero moment.

#### 4.3.2. Spin Waves in Metallic Co

**Q**method with a steep dispersion relation gives very broad peaks, so the constant energy transfer approach was used, varying

**Q**in a step-wise fashion, keeping the energy transfer constant.

^{3}) of a Co

_{0.92}Fe

_{0.08}alloy, which had been grown as a polarizer to reflect neutrons of one spin direction in a magnetic field. Fe has to be added to Co to stabilize the face-centered cubic structure. Aluminium crystals were used as monochromator and analyzer, and the measurements were made in the [ζζζ] direction around the [111] reciprocal lattice point. The decrease of the spin-wave intensity in a vertical applied magnetic field sufficient to saturate the crystal proved the magnetic nature of the excitations. Interestingly, the phonon intensity increased in the field due to additional magnetic coherent terms adding to the nuclear scattering. The process of observing phonons through the magnetic coherent cross-section is called magnetovibrational scattering.

#### 4.3.3. Crystal-Field Excitations

_{3}Fe

_{5}O

_{12}, is a ferrimagnet with a Curie point around 550 K. The Yb

^{3+}ions reside on effectively two inequivalent sites on the C-sublattice surrounded by distorted cubes of O

^{2−}ions. The ground state doublet is separated from the first excited quartet by about 70 meV. There is strong exchange coupling between the Fe

^{3+}ions, while the coupling between Yb

^{3+}ions is weak. The ground state doublet is split by the molecular field provided by the Fe

^{3+}ions. Far infrared measurements by Sievers and Tinkham [114] revealed peaks corresponding to three modes of excitation at 1.7, 2.9, and 3.3 meV at

**q**= 0. The first measurements of excitations associated with crystal-field splitting of rare-earth ions by neutron inelastic scattering were made by Watanabe and Brockhouse [115]. The measurements were made in neutron energy gain from the populated first excited state with the rotating crystal spectrometer at the NRX reactor on a polycrystalline sample, and only one peak was observed at 3 meV at 80 K. This corresponded to the mean position of the peak observed in infrared absorption. Low-temperature specific-heat measurements [116] on Yb

_{3}Fe

_{5}O

_{12}were consistent with the 3 meV excitation. However, the specific heat measurements below 4.2 K also required the contribution from the 1.7 meV level. This mode, identified in [114] at

**Q**= 0 as an “exchange mode”, would have had a strong wavevector dependence and probably corresponds to the lowest spin-wave mode, which does not go to zero because of the crystal-field anisotropy of the system. However, it would not have been observed as a sharp peak in a polycrystalline sample because of the averaging over wavevector.

_{2}O

_{3,}Er

_{2}O

_{3,}Tb

_{2}O

_{3}were reported by Brockhouse et al. [117] using the Chalk River rotating crystal spectrometer. Unfortunately, in these oxides, there are two distinct magnetic sites, each with low symmetry, so there are many possible peaks visible especially at temperatures of order of the multiplet splittings. Several peaks were observed, but it was not possible to assign these to particular transitions and so permit progress in understanding the results in terms of crystal fields. The problem is even daunting with single-crystal samples in these insulating rare-earth oxides. Much greater progress was achieved at Chalk River 15 years later in experiments on metallic rare-earth compounds such as TbSb, where the crystal fields are cubic.

#### 4.3.4. Spin Waves in Magnetite

**Q**method and the triple-axis crystal spectrometer, the complete spin-wave dispersion relation for the lowest, acoustic, mode of magnetite, Fe

_{3}O

_{4}, was measured to the [00ζ] zone boundary by Brockhouse and Watanabe [79] and shown in Figure 46. The spin-wave energy at ζ = 1.0 was 75 ± 2 meV, and the energy of the optic mode at

**q**= 0 was 58 ± 2 meV. The experiment is difficult because of the high spin-wave energies requiring E to be large, but the flux of neutrons with high energies is limited in a thermal reactor spectrum. In addition,

**Q**has to be small to keep the magnetic form factor large. In order to check that the inelastic peaks were actually spin waves, a vertical magnetic field was applied in the [$10\overline{1}$] direction sufficient to align the magnetic domains, and the peak intensities decreased as required. Constant-υ scans were made in addition to the constant-

**Q**scans. The spin-wave dispersion was found to be independent, to within the uncertainty of the measurements, of the direction of

**q.**The dispersion relation was analyzed in terms of the exchange model of Kaplan [118], and the antiferromagnetic exchange between Fe moments on the A and B sites was found to be J

_{AB}= 2.3 ± 0.2 meV. The interaction between Fe moments on B sites was determined to be much smaller and probably ferromagnetic. The discrepancy noted previously between the value of the low-temperature spin-wave specific heat as measured and that calculated from J

_{AB}is very likely to be a systematic error in the specific-heat measurements. The thermal conductivity of Fe

_{3}O

_{4}is about 200 times smaller than Ni for example. Typically, heat is applied for a short time, say 60 s, and the corresponding temperature increase is noted on a nearby thermometer. However, with a low thermal conductivity, the temperature only increases in that part of the sample closest to the heater, so that the thermometer would then give an incorrect reading of the average temperature rise and hence an incorrect specific heat. The present author had made low-temperature specific-heat measurements of polycrystalline magnetite in the early sixties and had not been aware of the consequences of a low thermal conductivity. Neutrons never lie!

#### 4.3.5. Spin Waves in UO_{2}

_{2}at 9 K. Peaks were observed in the magnetic Brillouin zones enclosing the [001] and [110] magnetic reciprocal lattice points, corresponding to excitations propagating in the AF spin structure as well as phonons. However, the phonon frequencies were strongly modified in the regions near the magnetic excitations, indicating a strong coupling between the magnons and the phonons. This was the first magnon–phonon interaction ever seen at a finite wavevector. Measurements of the magnetic form factor for U

^{4+}suggest that the electronic configuration of the 5f electrons is (5f

^{2}). In the L–S coupling approximation, the ground state is

^{3}H

_{4}, which is further split by the octahedral crystalline–electric field of the O

^{2−}ions. As a result of the orbital angular momentum, the distribution of 5f electrons is no longer spherical but extended in lobes between the O

^{2−}ions. The lowest level in the crystal field could have been a ${\mathsf{\Gamma}}_{1}$ singlet or a ${{\mathsf{\Gamma}}^{\prime}}_{25}$ triplet, but the evidence of the magnon branches and the magnon–phonon interaction confirms that it is the triplet. A molecular field in the antiferromagnetic state splits this into three levels so the magnetic excitations would correspond to transitions between the ground state and the two excited states, since there are matrix elements of (L + 2S) connecting these states. The transverse phonon in the [00ζ] direction modulates the crystal field, leading to a coupling between the magnon and the phonon where they would otherwise cross.

**q**, where the corresponding Brillouin zones cut the ($1\overline{1}0$) plane. The major barrier to complete understanding was that there appear to be three branches of magnetic excitation, not two as expected on the basis of the triplet ground state split by the molecular field. One other consequence of the magnon–phonon interaction is that, for example, the Δ

_{5}transverse phonon is seen at [0 0 1.7] only by virtue of the admixture of the magnon eigenvectors into its description. The phonon eigenvectors lie perpendicular to

**Q**, so the purely phonon contribution is zero at this location. The theory for the magnetic excitations including orbital angular momentum was developed by Cowley and Dolling [120], who also derived the theory of the magnon–phonon interaction from first principles. While the theory of the magnetic excitations included all the interactions presumed to exist at the time, it was recognized as not fully satisfactory. A more complete story has taken over half a century to resolve [121], and a very recent paper in a volume dedicated to the contributions of Roger Cowley [122] summarizes the now-known facts. There are quadrupole moments on the U

^{4+}ions, and there are also quadrupolar transitions between the three states in the molecular field. These too can interact with the phonons and magnons, and this leads to the three apparently magnetic peaks seen in the original 9 K measurements.

## 5. Epilogue

**Q**scans and deciding the scans for the day. The day’s discoveries or problems were discussed at coffee time, which brought the whole group of scientists and technicians together every day. Beam time was considered sacred, and as soon as one experiment finished, the next one was set up and started. Regular buses to Deep River, after the nominal working day was over, were invariably full of scientists.

^{4}was carried out as summarized by H. Glyde in his contribution to the commemorative volume for Roger Cowley [123]. The most accurate estimate of the superfluid fraction, n

_{0}= (7.25 ± 0.25)%, is within the error band originally estimated at Chalk River. Interestingly, at sufficiently high wavevectors, for example at Q = 28.5 Å

^{−1}as measured with spallation sources, the difference in S(Q, ω) between the superfluid and the normal states can be observed directly.

_{2}, KCoF

_{3}[124], and RbCoF

_{3}. Here, the major perturbation of the free ion in a cubic compound is the crystalline–electric field with the spin–orbit coupling and exchange interactions determining the energy levels and wave functions of the lowest states. In general, excitations from the ground state to several excited states are allowed, and this leads to several branches of the spin-wave spectrum. Extensive investigations into the way in which excitations are altered in the presence of defect ions were mounted. For example, the Co

^{2+}ion in MnF

_{2}leads to a mode localized on the Co

^{2+}site, since it lies above the MnF

_{2}spin-wave band. On the other hand, Zn

^{2+}in MnF

_{2}leads to a resonant perturbation of the host spin waves. The work was extended to measurements and theoretical descriptions of concentrated alloys and clusters [125]. Measurements began on the spin waves in the rare-earth metals Ho [126] and Er, which have respectively incommensurate spiral and conical magnetic structures at low temperatures.

_{2}, and PrAl

_{2}followed. In the rare-earth compounds, the crystal field and long-range exchange interactions determine the low-lying level energy levels and wave functions, and several branches of the spin-wave spectrum are seen. In 1986, measurements on the spin-1 linear chain antiferromagnet CsNiCl

_{3}[128] provided the first experimental evidence for the Haldane Gap [129] for which Haldane later shared the Nobel Prize. The success in the interpretation of the rare-earth compounds led to work on antiferromagnetic UN [130] and ferromagnetic US and USe, which are the respective pnictides and chalcogenides with the smallest U–U separations, and whose ionic state is unknown. The magnetic excitations in these cases are not conventional spin waves with a well-defined energy at a given wavevector

**q**but are always broadened in energy, even at small

**q**. A quantitative theory for these excitations still does not exist.

_{2}As

_{2}. The study of thin films and depth profiling in materials such as LaSrMnO

_{3}was also enabled by reflectometry. New materials showing exotic properties such as Mott phases in Sr

_{3}(Ir

_{1-x}Ru

_{x})

_{2}O

_{7}and high-temperature superconductors were investigated. Many remarkable applications of neutron diffraction to critical components for industry worldwide were carried out, including work on stresses in highly active welds, assay of unknown radioactive materials within historic encapsulation, and also the solidification of alloys. The cumulative income from ANDI fee-for-service research was six million Canadian dollars. An account of the CNBC from 1997 to 2017 was given by Root and Banks [132] including the rather troubling interactions between AECL, NRC, NSERC, and the Department of Industry (NRCAN) over the possibilities for a neutron source for pure and applied research to replace the NRU reactor. A synopsis of the first decade of neutron scattering at Chalk River was given in ref. [133]. Statistical data on the performance of the neutron group over the years are to be found in ref. [134]. This recorded that the number of papers describing research with neutrons at Chalk River between 1949 and 1965 was 131, between 1966 and 1980 it was 330, between 1980 and 2000 it was 857, and between 2000 and 2017 it was 745 for a total of 2081 scientific papers over the 68 years of existence of the neutron group.

## 6. The Scientists

_{3}. At the end of his student days in 1964, he married his wife, Sheila, and set off on the “Empress of Canada” to Montreal, en route to Chalk River as a member of staff of Atomic Energy of Canada. Roger had an extraordinarily productive career at Chalk River working on the theory of anharmonicity, magnetic excitations in systems with spin and orbital moments, doped magnetic systems to find local and resonant magnetic excitations, and with Dave Woods on excitations in normal and superfluid helium. Roger and Sheila made many friends in Deep River, and their two children were born there.

_{2}. With R.A. Cowley, he measured the spin-wave excitations in UO

_{2}and discovered the magnon–phonon interaction and gave a convincing explanation of its origin. He was instrumental in developing high-quality squeezed germanium and silicon monochromators for triple-axis spectrometry. He authored 120 research papers and five book chapters. He strongly supported research into industrial applications of neutron scattering, which was a field he helped pioneer in the early 1980s.

_{c}superconductors. He rose to the position of Director of the Solid State Physics and Spectroscopy Division at BARC and was elected to the Indian Academy of Sciences, Bangalore. K.R. Rao died in 2008.

_{2}, O

_{2}, and N

_{2}), amorphous solids (Ge, Si, ZnCl

_{2}) and borate and phosphate glasses. Roger Sinclair died in 2000.

_{2}by neutron diffraction.

^{4}, which were obtained over the years from 1960. With colleagues in Grenoble, he made the first measurements on the excitations in the Fermion system, He

^{3}. He was branch head of Neutron and Solid State Physics from 1971 to 1979 and then became a special advisor to the vice-president for strategic planning at AECL. He was elected to the Royal Society of Canada in 1982 for his contributions to neutron scattering. He loved music, literature, travel, tennis, and badminton, and he took great pride in his family and their history in Newfoundland. Dave Woods retired in 1989 and died in 2019.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The main floor of the National Research Experimental (NRX) reactor at the Chalk River Laboratory about 1950. The powder spectrometer constructed by Donald Hurst and colleagues is visible near the center of the photograph. Most of the other equipment is concerned with nuclear physics or with physics of the neutron itself. For reasons of restricted space around the reactor, each heavily shielded apparatus is located at the end of a long tube. This figure is taken from Figure 1 of ref. [2] and reproduced by courtesy of Atomic Energy of Canada Limited. Copyright by the American Physical Society.

**Figure 2.**The first crude version of the triple-axis spectrometer. Monoenergetic neutrons are selected by the large single-crystal monochromator (X

_{1}) and impinge on the specimen (S), which is located on a table whose orientation (ψ) with respect to the incident beam can be selected. This table can be moved along the incident beam as desired. The analyzing spectrometer, which employs crystal X

_{2}, is a diffractometer of especially large aperture that can be translated as a unit; the angle, Φ, through which the examined neutrons are scattered is determined by triangulation. This figure is taken from Figure 3 of ref. [2] and modified to label the collimators C

_{2}and C

_{3}, the monochromator X

_{1}and analyzer X

_{2}. Copyright by the American Physical Society.

**Figure 3.**The transmission through cadmium for an aluminum scatterer. The inset shows the theoretical energy distribution of initially monoenergetic incident neutrons, 350 meV, after scattering from a gas which gives a Gaussian, and an Einstein crystal with discrete lines separated by kθ

_{D}= 25.9 meV. In the main diagram, the experimental points are plotted together with the straight line transmission corresponding to no inelastic scattering and the energy distribution produced by inelastic scattering from the Einstein model. This figure is taken from Figure 5 of ref. [15]. Copyright by the American Physical Society.

**Figure 4.**Typical energy distributions of neutrons inelastically scattered by an aluminum single crystal with the energy resolutions indicated by triangles. The count rate is plotted against the scattered energy in eV and the angular phonon frequency in 10

^{13}radians/s. The incident energy E is indicated by the arrows. This figure is taken from Figure 1 of ref. [20]. Copyright by the American Physical Society.

**Figure 5.**The relation between the phonon frequency, ω (10

^{13}radians/s), and the phonon wavevector,

**q**, (10

^{8}cm

^{−1}), near the [111] direction in aluminum. Modes that have been positively identified as transverse or longitudinal are indicated by solid and open circles. Phonons of unknown polarization are indicated by dots. The dashed curve is a sine curve whose initial slope corresponds to a velocity of sound of 3080 ms

^{−1}for Al. This figure is taken from Figure 3 of ref. [20]. Copyright by the American Physical Society.

**Figure 6.**Schematic drawing of the filter-chopper spectrometer at Chalk River. The incident neutron beam passes alternatively through cold Be or cold Be and Pb filters. The scattering angle, Φ, at the sample is 90° and the scattered energy is determined in time of flight over a path of 4.5 m with a Fermi chopper rotating at 7200 rpm. The neutrons are counted in well-shielded BF

_{3}counters. The inset shows the Maxwellian spectrum and the position of the Pb and Be filter Bragg cut-offs. This figure is taken from Figure 3 of ref. [19]. Copyright by the American Physical Society.

**Figure 7.**The phonon dispersion relations in the [ζ00] and [ζζζ] directions for germanium at room temperature. The legend at the top of the diagram indicates the polarization and character of the vibrations. The signs indicate neutron energy loss (−) or gain (+). The heavy lines near (

**q**/2π) = 0 indicate the slope of the appropriate velocities of sound as calculated from the elastic constants. This figure is taken from Figure 5 of ref. [27]. Copyright by the American Physical Society.

**Figure 8.**The phonon dispersion relations in silicon for the [ζ00], [0ζζ], and [ζζζ] directions at 296 K. The curves are calculated from shell model IIC. The vertical dashed line at ζ = 0.75 in the [0ζζ] direction marks the position of the zone boundary. Open circles represent type I [0ζζ] modes, open triangles represent longitudinal modes, and solid points have undetermined polarization. This figure is taken from Figure 1 of ref. [30] by courtesy of the International Atomic Energy Authority (IAEA).

**Figure 9.**The Be-Pb difference patterns as measured with the filter difference spectrometer for vanadium at 300 °K and at 105 °K. The region of the spectrum around 1.2 Å corresponds to multiphonon scattering. Elastic scattering from the previous frame in time of flight is seen around 3.6 Å. This figure is taken from Figure 3 of ref. [35]. Copyright by the American Physical Society.

**Figure 10.**The vibrational spectrum, g(ν), of vanadium with the instrumental energy resolution shown as Gaussian curves. The full line is drawn through the combined histograms. The light dashed curve shows a possible spectrum with the resolution removed. The heavy dashed line at low frequencies shows part of the Debye spectrum for a Debye temperature θ

_{D}= 338 °K as determined by low-temperature specific heat measurements. This figure is taken from Figure 6 of ref. [35]. Copyright by the American Physical Society.

**Figure 11.**The excitation curve in superfluid He

^{4}. The points represent the measured change in energy and momentum of the scattered neutrons. The smooth curve drawn through the points is a guide to the eye and gives the expected form of the excitation curve. The minimum is around 1.93 Å

^{−1}and 8.1 K, which corresponds to the minimum of the Landau roton curve. The parabolic curve represents the excitation of free He

^{4}atoms, while the dotted curve gives the calculated phonon excitation curve for a velocity of sound of 237 ms

^{−1}. This figure is taken from Figure 1 of ref. [40]. Copyright by the American Physical Society.

**Figure 12.**The atomic distribution function 4πrρ(r) for liquid neon at 26 K. The arrow at 3.13 Å represents the measured separation between two atoms in the solid at 4.2 °K. The nearest distance of approach of two atoms in the liquid is 2.45 Å, while the number of neighbors around each atom is about 8.8. The oscillations below 2 Å are spurious and have their origin in the limited wavevector range, Q, of the experiments. This figure is taken from Figure 5 of ref. [45]. Copyright by the American Physical Society.

**Figure 13.**Measured energy distributions of 1.62 Å neutrons scattered by specimens of water, H

_{2}O, at room temperature for wavevectors

_{,}Q, near the maximum in S(Q). The results are plotted as a function of the angle 2θ

_{A}, of the analyzing spectrometer, the scattered neutron wavelength, and its energy. The resolution function as measured with vanadium is shown. The results indicate the separation of the quasi-elastic scattering from the inelastic scattering. The latter is consistent with the scattering from a gas of H

_{2}O molecules in this region of Q. This figure is taken from Figure 4 of ref. [46] courtesy of Nuovo Cimento.

**Figure 14.**The form factor of the thermal cloud for the protons in H

_{2}O at room temperature. The filled circles are obtained from patterns for which there is a fairly clear separation of “nearly-elastic “and inelastic” components and the open circles where the curves had to be fitted. The upper solid curve represents a Debye-Waller factor with <u>, the average amplitude of oscillation, of about 0.26 Å applicable to ice at room temperature while the lower solid curve is a fit of a Debye-Waller factor to the measurements for which <u> is about 0.4 Å. The shaded region is taken from a separate experiment where only the nearly elastic scattering was measured. This figure is taken from Figure 13 of ref. [46] courtesy of Nuovo Cimento.

**Figure 15.**The scattering function S(Q,ω) for H

_{2}O at 25 and 75 C for small values of energy transfer ħω as measured with an incident energy of 4.96 meV and an energy resolution of 0.2 meV obtained with the rotating crystal spectrometer. The resolution functions are shown as dashed curves. The multiple scattering A(ħω) is also shown. This figure is taken from ref. Figure 1 of [50]. © (1962) The Physical Society of Japan.

**Figure 16.**The energy transfer ΔE (eV) as a function of q/2π, the spin-wave wavevector, for acoustic spin waves in magnetite, Fe

_{3}O

_{4}, at room temperature propagating in the [00ζ] direction. The straight line corresponds to a velocity of greater than 10

^{4}ms

^{−1}, which is considerably greater than the mean velocity of sound for magnetite, and therefore, the results do not correspond to a phonon dispersion relation. The parabolic curve corresponds to the expected dispersion relation, E = Dq

^{2}, for spin waves at small wavevectors. The coefficient D, the spin wave stiffness, leads to estimates of the exchange interaction J

_{AB}of 2 meV and the Curie temperature, in molecular field theory, of 1050 K similar to the actual Curie temperature of 850 K. This figure is taken from Figure 5 of ref. [54]. Copyright by the American Physical Society.

**Figure 17.**Schematic diagram of the C5 triple-axis crystal spectrometer showing the monochromator scattering angle, 2θ

_{m}, sample angle, Ψ, scattering angle, Φ, and analyzer angle, 2θ

_{a}. The analyzer angle is generally held fixed, and the incident energy varied. A monitor in the incident monochromatic beam with a 1/v response compensates for the k′/k factor in the cross-section, so the measurement almost gives S(

**Q**, ω) directly. The collimator C

_{2}is placed within the drum before the monochromator, C

_{3}lies between the monochromator and the sample, and C

_{4}is placed between the sample and the analyzer. This figure is taken from Figure 4 of ref. [41] courtesy of the IAEA.

**Figure 18.**Cutaway drawing of the C5 triple-axis crystal spectrometer at the C-face of the National Research Universal (NRU) reactor. This figure is taken from Figure 2a of ref. [58] courtesy of Canadian Science Publishing and its licensors (DOI 10.1139/p65-135).

**Figure 19.**Photograph of the C5 spectrometer at the NRU reactor in the early 1960s. Margaret Elcombe, Brian Powell, and Dave Woods are setting up a low-temperature experiment. (Courtesy of Atomic Energy of Canada Limited.).

**Figure 20.**The phonon dispersion relations in lead at 100 K for high-symmetry directions. The open circles were obtained by the constant-

**k**method in the ($1\overline{1}0$) plane and the closed circles were obtained by the constant-

**Q**method in the {100} plane. The straight lines through the origin give the initial slopes of the curves as calculated from the elastic constants. This figure is taken from Figure 5 of ref. [60]. Copyright by the American Physical Society.

**Figure 21.**The full-width at half-maximum of the neutron groups in lead corrected for resolution as a function of the wavevector for the [ζ,0,0] T and L branches at 425 °K. The widths reveal a marked broadening of the phonons at high temperatures. The solid lines show the best fits to expressions of the form of Equation (28) with n = N = 1. The scatter gives a measure of the accuracy. This figure is taken from Figure 7 of ref. [59] by courtesy of the IAEA.

**Figure 22.**The phonon dispersion relation in body-centered cubic sodium at 90 K. The solid lines from

**q**= 0 have been calculated from the best available values of the elastic constants at 90 K. The abscissa for the [ζζζ] direction has been stretched to emphasize the equivalence of the points [½½½] and [111] with the corresponding points for [½½ζ] and [00ζ] shown above. This figure is taken from Figure 2 of ref. [63]. Copyright by the American Physical Society.

**Figure 23.**The approximate polarization directions for the high-frequency mode in the ($1\overline{1}0$) plane for sodium. These were deduced from symmetry considerations, intensity measurements, dispersion curve continuity, and the values of the force constants for the fifth neighbor model. This figure is taken from Figure 3 of ref. [64] courtesy of the IAEA.

**Figure 24.**The phonon dispersion relations in body-centered cubic potassium at 9 K. ζ, expressed in units of 2π/a, is a reduced wavevector coordinate. In the [00ζ], [ζζ0], and [ζζζ] directions, transverse (longitudinal) modes of vibration are denoted by circles (triangles). The solid curves represent the best fit to the results on the basis of a Born–von Kármán model with axially symmetric forces extending to fifth nearest-neighbor atoms. This figure is taken from Figure 1 of ref. [65]. Copyright by the American Physical Society.

**Figure 25.**The screened pseudopotential for the conduction electron–ion interaction V

_{p}(Q)/ε(Q) in potassium at 9 K. A and B are theoretical curves based on the Heine–Abarenkov model, while C and D are fitted to the phonon data. Curve D resembles case A, as determined by Bortolani. This figure is taken Figure 5 of ref. [65]. Copyright by the American Physical Society.

**Figure 26.**The phonon dispersion relations in sodium iodide measured near 100 K. The measurements are compared with calculations based on the rigid ion (dashed) and simple shell (continuous) models. The broken vertical line indicates the [110] zone boundary. This figure is taken from Figure 1 of ref. [72]. Copyright by the American Physical Society.

**Figure 27.**The phonon dispersion relations in potassium bromide measured at 90 K, compared with calculated curves based on the rigid ion (dashed curve) and simple shell models (solid curves). This figure is taken from Figure 2 of ref. [72]. Copyright by the American Physical Society.

**Figure 28.**The calculated shapes of the inelastic scattering peaks from the

**q**= 0 longitudinal and transverse optical modes in potassium bromide and sodium iodide at several temperatures. The intensity is in arbitrary units. The figure is taken from Figure 25 of ref. [74] on anharmonicity in the alkali halides courtesy of Taylor and Francis.

**Figure 29.**A plot of the square of the frequency of the

**q**= 0 transverse optic mode against temperature in SrTiO

_{3}. The solid line is a linear regression fit through the points and gives a Curie temperature of 32 ± 5 K. The dashed line represents the reciprocal of the dielectric constant. This figure is taken from Figure 3 of ref. [76]. Copyright by the American Physical Society.

**Figure 30.**The two upper figures are the phonon dispersion relations in niobium at 296 K for lattice waves traveling in the [00ζ], [ζζ0], [ζζζ], [ζζ1], and [½½ζ] symmetry directions. The solid curves are the fits to an eighth-neighbor Born–von Kármán force constant model. The lower curves are similar measurements for Ta at 296 K. Lines through

**q**= 0 are calculated from the elastic constants. The upper two figures are taken from Figure 1 of ref. [80] and the lower figure is taken from Figure 1 of ref. [82]. Copyright by the American Physical Society.

**Figure 31.**The upper figure shows the phonon dispersion relations in molybdenum at 296 K. The straight lines through

**q**= 0 were determined from the measured elastic constants. The solid curves are the results of the best fit to the third neighbor axially symmetric force model. The model was not fitted to the region of the broken curves near the position H in the Brillouin zone, which is a Kohn anomaly. The lower figure shows similar measurements on tungsten at 296 K, which indicate no obvious Kohn anomalies. The upper figure is taken from Figure 1 of ref. [81] and the lower is taken from Figure 1 of ref. [83]. Copyright by the American Physical Society.

**Figure 32.**Interplanar force constants (10

^{4}dyncm

^{−1}) for the body-centered cubic transition metals niobium, molybdenum, tantalum, and tungsten derived from the [ζζζ] T modes. The data for this figure were taken from Table 1 of ref. [84] courtesy of the IAEA.

**Figure 33.**The phonon dispersion relations in β-brass at 296 K for high-symmetry directions. Triangular points denote L and Λ modes and circles denote T and Π modes. Solid points denote uncertain polarization. The solid curve represents the fit to a Born–von Kármán model to fourth nearest neighbors. The dashed curve is the fit obtained with an oscillatory pseudopotential model. This figure is taken from Figure 2 of ref. [87]. Copyright by the American Physical Society.

**Figure 34.**The phonon dispersion relations in gallium arsenide at 296 K. The solid points denote undetermined polarization. The vertical dashed line in the [0ζζ] direction represents the zone boundary. In this direction, points labeled I and II refer to modes whose polarization vectors lie within the ($01\overline{1}$) mirror plane. Other modes are strictly longitudinal (L) or transverse (T). The dotted and solid curves represent calculations based on two modifications of the dipole approximation or shell model. This figure is taken from Figure 1 of ref. [90]. Copyright by the American Physical Society.

**Figure 35.**The phonon dispersion relations for UO

_{2}at 296 K in three directions of high symmetry. The dashed (solid) curves show the best least squares fit to a rigid ion (shell mode). In the Δ and Λ directions, the open circles (triangles) denote transverse (longitudinal) modes. Solid points denote modes of uncertain polarization. This figure is taken from Figure 4 of ref. [58] and is reproduced courtesy of Canadian Science Publishing and its licensors.

**Figure 36.**A representation of the scattering law S(α, β) for polycrystalline UO

_{2}at 293 K determined by time-of-flight measurements at Chalk River at three values of β. This figure is taken from Figure 2 of ref. [93] courtesy of the IAEA.

**Figure 37.**The upper figure is the frequency distribution, g(υ), calculated from the best fit shell model for UO

_{2}[58]. The statistical fluctuations in the curve, which is a histogram plot, are too small to be shown in the figure. The upper figure is taken from Figure 5 of ref. [58] courtesy of Canadian Science Publishing and its licensors. p(β) is derived from time-of-flight polycrystalline measurements on UO

_{2}via the scattering law and is taken from Figure 5 of ref. [93] courtesy of the IAEA.

**Figure 38.**A selection of smoothed curves for liquid lead of I(Q, t) plotted against Q, boxes (

**a**–

**c**), and also G(r,t) boxes (

**d**–

**f**) against r for values of time t of 0, 0.1, 0.2, 0.6, and 2 × 10

^{−12}s. The closed circles in Figure (

**a**) show the experimental points for the integrated intensities I(Q,0) = S(Q). The open circles are values of I(Q,0) calculated from the second moment of the energy distribution. The left hand scales in boxes (

**b**) (

**d**) and (

**f**) apply to G(0,t). This figure is taken from Figure 2 of ref. [96]. Copyright by the American Physical Society.

**Figure 39.**The liquid structure factor, S(Q), for liquid helium under its normal vapor pressure at 2.29 °K (above the λ-point) shown by closed circles and 1.06 °K (below the λ-point) shown by open circles. The effect of the λ-transition is a lowering and a slight broadening of the main maximum. This figure is taken from Figure 2 of ref. [98]. Copyright by the American Physical Society.

**Figure 40.**The energy versus wavevector dispersion curve for superfluid liquid helium at 1.12 °K at its normal vapor pressure. The parabolic curve rising from the origin represents the calculated dispersion for free helium atoms at absolute zero. The circles correspond to the energy and wavevector of the measured excitations, and a smooth curve has been drawn through the points. The broken curve rising linearly from the origin is the theoretical phonon branch calculated with a velocity of sound of 237 ms

^{−1}. The dotted curve drawn through the point at Q = 2.27 Å

^{−1}has been drawn with a slope equal to the velocity of sound. This figure is taken from Figure 4 of ref. [100]. Copyright by the American Physical Society.

**Figure 41.**The temperature variation of the full-width at half-height of the phonon–roton excitation peak in liquid helium close to the roton minimum. The solid curve has been drawn through the points as a guide to the eye. The broken curve gives the calculated widths for a gas with an effective mass close to 4 above the λ-point. The width decreases to around 2.5 at the lowest temperature. The dotted curve represents the theoretical widths on the basis of the Landau–Khalatnikov theory. This figure is taken from Figure 8 of ref. [100]. Copyright by the American Physical Society.

**Figure 42.**The temperature variation of the mean energy change of 4.04 Å neutrons scattered through 80° from liquid helium. This corresponds closely to the energy of the elementary excitation at the Landau roton minimum. The smooth curve is drawn through the points as a guide to the eye. This figure is taken from Figure 6 of ref. [100]. Copyright by the American Physical Society.

**Figure 43.**The energy versus wavevector dependence of the scattering at 1.1 K in liquid helium. Shown are the one-phonon dispersion curve, the upper and lower energies corresponding to half peak intensity, and the mean energy of the multiphonon peak. The results were obtained using both the rotating crystal spectrometer and the C5 triple-axis crystal spectrometer. This figure is taken from Figure 6 of ref. [105] courtesy of Canadian Science Publishing and its licensors.

**Figure 44.**The width of the broad multiphonon peak in the scattering at large. Q in liquid helium as given by the width at half height at 1.1 and 4.2 K. This figure is taken from Figure 22 of ref. [105] courtesy of Canadian Science Publishing and its licensors.

**Figure 45.**The energy transfer ΔE as a function of the reduced wavevector for spin waves in a Co

_{0}.

_{92}Fe

_{0}.

_{08}alloy. The error bars are about half the full-width at half-maximum of the neutron groups. The solid line is the best fit to the functional form of a spin-wave dispersion plus a small constant anisotropy term, namely ΔE = A + Dq

^{2}. This figure is taken from Figure 2 of ref. [110]. Copyright by the American Physical Society.

**Figure 46.**Experimental dispersion curves for magnetite, Fe

_{3}O

_{4}, at 297 K. The dashed curve gives the results prior to correcting for vertical divergence. Closed circles show results from constant-

**Q**measurements, and open circles denote constant-υ measurements. This figure is taken from Figure 1 of ref. [79] courtesy of the IAEA.

**Figure 47.**The measured dispersion relations for magnetic excitations in UO

_{2}at 9 K, propagating along four directions in the crystal. The dashed curves show the phonon dispersion relations appropriate to 296 K. The solid curves are the result of fitting to a model including exchange interactions between uranium moments and a single ion anisotropy term of the form ${K}_{1}{\displaystyle \sum _{i}{({S}_{i}^{z})}^{2}}$. This figure is taken from Figure 4 of ref. [120]. Copyright by the American Physical Society.

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**MDPI and ACS Style**

Holden, T.M.
The First Two Decades of Neutron Scattering at the Chalk River Laboratories. *Quantum Beam Sci.* **2021**, *5*, 3.
https://doi.org/10.3390/qubs5010003

**AMA Style**

Holden TM.
The First Two Decades of Neutron Scattering at the Chalk River Laboratories. *Quantum Beam Science*. 2021; 5(1):3.
https://doi.org/10.3390/qubs5010003

**Chicago/Turabian Style**

Holden, Thomas M.
2021. "The First Two Decades of Neutron Scattering at the Chalk River Laboratories" *Quantum Beam Science* 5, no. 1: 3.
https://doi.org/10.3390/qubs5010003