Inelastic Neutron Scattering
3.3 Scattering Function
where the first and second delta functions correspond respectively to the creation and an- nihilation of one phonon of energy ω. The sum over modes s brings about the phonon density of states and after further simplification, one obtains the incoherent neutron scat- tering cross-section for a one-phonon creation process, originally derived by Placzek and van Hove [12]:
∂2σ
∂Ω∂E inc
+1 phonon
= σinc 4π
kf ki
3N
2Me−2W × Q2
3ωg(ω)n(ω) + 1, (3.14) where exp(−2W) = expU2is the Debye-Waller factor andg(ω) is the normalized phonon DOS. Since phonons are Bosons,n(ω) is the Bose-Einstein distribution.
tering experiment of figure 3.1. A schematic of a direct-geometry chopper spectrometer, as found most commonly at a spallation source, is shown in figure 3.3. A pulsed source delivers neutrons to a moderator, in which the neutrons thermalize through multiple col- lisions with the medium, exiting with a maxwellian velocity distribution representative of the moderator temperature. The spectrum typically also contains epithermal neutrons of higher energy that were not fully equilibrated. A port passes a beam of neutrons, which is monochromatized by a pair of rotating choppers (t0 and E0 rotors in figure 3.3), whose synchronized opening times only let through neutrons of a chosen energy. This energy selection rests on the energy-velocity relationship for the free neutrons, E = 12mv2. The neutrons are incident on a sample and some of them are scattered, transferring momentum and energy, and the scattered neutrons are collected by a set of detectors covering possibly a large angular range. Typically, the detectors are3He-filled tubes, which produce a charge after capture of the neutron and a low-energy nuclear reaction. This type of detector does not offer energy discrimination. Instead, knowing the time of the initial neutron pulse, the velocity of the scattered neutrons is determined from their arrival time on the detector, the time at which they were impinging on the sample, and knowledge of the sample-detector distance. This velocity is converted to energy with the energy-velocity relationship. In this time-of-flight approach, the final energies are most accurately measured when the sample- to-detector flight path is long. However, because of spatial and monetary constraints, the surface that can be covered with detectors is finite and there is thus a trade-off between the length of the flight path and the solid-angle coverage.
Another important parameter in neutron scattering experiment is the relatively low flux of neutron sources, especially when compared to x-ray sources. Because the density of neutrons in the beam is low (with particle densities on the order of a magnitude of a good vacuum), the dimensions of the beam tend to be large to compensate. This results in the need for large samples, with typical transverse sizes on the order of a few centimeters.
Figure 3.3: Direct-geometry time-of-flight neutron spectrometer.
Another aspect, resulting from this sample size, is that one needs to minimize multiple scattering events. Because the elastic cross-section is typically much larger than its inelastic counterpart (see examples below), multiple scattering events mostly involve several elastic scattering events or a combination of one elastic scattering and one inelastic scattering.
Since the neutrons involved in an extra elastic scattering event travel extra distances inside the sample, they will reach the detector with a delay, thus appearing to have an extra energy loss. Multiple scattering is difficult to correct for and as a result one tries to limit the number of such processes by having samples that are thin enough. A typical working value is to make samples that scatter 10% of the incident neutrons, limiting double elastic scattering events to less than 1%. This has the unfortunate consequence that most of the already scarce incident neutrons are wasted. Such limitations are intrinsic to the neutron time-of-flight approach, until detectors with sufficient energy resolution are devised.
q
of coherently scattering nuclei, the phonon dispersions can be accessed as well as the DOS.
The possibility of collecting data over a large swath of reciprocal space at once is in fact the main strength of this technique, whereas more detailed studies of excitations at specific points in the Brillouin zone are perhaps best undertaken with a triple-axis spectrometer.
From the incident and final neutron energies and the scattering angle Φ, the momentum transfer Q can be determined. In practice, one most often works with polycrystalline samples and only the magnitude of the momentum transfer is relevant,
Q = { 1
2.072(2Ei(1−cos Φ
1−ω/Ei)−ω)}1/2, (3.17) whereQis in ˚A−1 and the energies are in meV. TheQ(Φ, E) relation for the low-resolution medium-energy chopper spectrometer (LRMECS) instrument at the intense pulsed neutron source (IPNS) at Argonne National Laboratory is shown in figure 3.4, for different scattering angles covered by the detector bank.
From figure 3.4, one can see that the relevant range of phonon energies for a vanadium crystal is sampled with momentum transfer ranging from close to 0 ˚A−1 up to about 8 ˚A−1. A spherical sampling region with such range is compared to the size of the first Brillouin zone for the vanadium reciprocal lattice in figure 3.5. In this figure, the sphere denotes the volume sampled by the neutrons and the central thick square is the Brillouin zone of the fcc reciprocal lattice. One can see on this figure that many Brillouin zones are sampled.