Neutron scattering instruments

modulation along thekdirection.

2.1.3 Magnetic scattering cross-sections

Magnetic neutron scattering originates from the dipolar interaction between the neutron spin and the electron’s magnetic moment. And the special form of the dipolar interaction, after the Fourier transformation, leads to the fact that only the magnetic moment perpendicular to the scattering vectorkcan contribute to the scattering cross section. For a Bravais crystal, the magnetic cross section is directly related to the magnetic correlation function〈S₀^α(0)S^β_l(t)〉:

d²σ the electron,gis the Landé factor,F(k) is the magnetic form factor due to the distribution of the electrons around the nucleus,ldenotes the position of the nucleus, andS^α_l is theα component of magnetic moment atl.

For elastic magnetic scattering, there is only a contribution from the static magnetic moments:

dσ

Magnetic excitations,e.g.the spin wave (magnon) excitations, modulate the magnetic moment distribution in the crystals and thus can be detected by neutron scattering. The cross-section for one-magnon excitations can be expressed as:

d²σ whereqandħωqare the momentum and energy of the magnon excitation.

2.2 Neutron scattering instruments

As can be seen from the previous discussions, both the momentum and energy of the incident and scattered neutrons should be measured to study the excitations in a sample. However, neutron detectors usually rely on nuclear reactions like³He + n→³H + p + 0.8 MeV, which only count the number of detected neutrons and tell nothing about their energies. To get

more information of the detected neutrons, different instrumentation schemes have been em-ployed. In this section, we will briefly introduce the three most important neutron scattering spectrometers: the triple-axis spectrometer (TAS), the time-of-flight spectrometer (TOF), and the spin-echo spectrometer. The basics of neutron diffractometers are also discussed.

2.2.1 Triple-axis spectrometer

The triple-axis spectrometer uses the Bragg reflections of the monochromator and analyzer crystals to select neutrons with a specific wavelength before and after the sample. Fig. 2.2 presents a schematic of a typical triple-axis spectrometer. The monochromator and analyzer are typically co-aligned crystals of pyrolytic graphite (PG), silicon, or copper, and the selection of which depends on the specific wavelength and the required energy resolutions. Higher-order neutrons with wavelengthλ/ncan be removed by a low-energy-pass filter, which is normally composed of polycrystalline material of PG or Beryllium. For a Beryllium filter, it only scatters neutrons with wavelength shorter thanλcutoff=2dmax, wheredmax is the largest d-spacing of the planes in the crystal.

white neutron beam

monochromator

analyzer

detector sample

k_i

k_f

Figure 2.2 – The triple axis spectrometer employs the Bragg law of the monochromator and analyzer crystals to select neutrons with specific wavelength.

Triple-axis spectrometers employed for this thesis include TASP and EIGER at SINQ of Paul Scherrer Institut (PSI) for the study of the magnetic excitations in MgCr₂O₄, PANDA at Heinz Maier-Leibnitz Zentrum (MLZ) and ThALES at Institut Laue-Langevin (ILL) for the magnetic excitations in MnSc2S4.

2.2.2 Time-of-flight spectrometer

In the time-of-flight spectrometer (TOF), energies of neutrons are determined by measuring the flight time of the detected neutrons. Fig. 2.3 presents a schematic of TOF spectrometers at a continuous neutron source. The chopper monochromizes the incoming neutrons (sometimes in combination with a monochromator) and creates a pulsed neutron beam. The detector banks are time-sensitive, which allows the calculation of the velocities and energies for the detected neutrons.

Time-of-flight spectrometers employed for this thesis include the IN4 and IN6 spectrometers at ILL for the investigation of the crystal electric field transitions in the spin ice compounds of

2.2. Neutron scattering instruments

CdEr₂Se₄and CdEr₂S₄, the FOCUS spectrometer at SINQ of PSI for the magnetic excitations in MnSc2S4.

white continuous neutron beam

chopper

sample

detectors k_i

k_f

Figure 2.3 – The time-of-flight spectrometer measures the time it takes for a neutron to reach the detector. The chopper creates monochromatic neutron pulses and acts as the time initializer for the neutron beam.

2.2.3 Spin-echo spectrometer

In the spin-echo spectrometer, the energy (velocity) change of neutrons is measured by comparing the Larmor precession of the neutron spin before and after the scattering. Fig. 2.4 presents a schematic of the spin-echo spectrometer. Its main components are two precession coils with magnetic fields that obey the relation ofL₁H₁=L₂H₂, whereLis the length of the coil andHis the field strength. Incoming neutrons with velocityvare initially polarized with spin parallel to the velocity direction. Theπ/2 flipper flips the spin to thexdirection that is perpendicular to the velocity, making neutrons to precess when entering the coil. With theπ-flipper that reverses the spin direction, the overall precession angle in the two coils should beφ=φ1+φ2=γL1H1(1/v1−1/v2), whereγis the gyromagnetic ratio that equals 2.916 kHz/Oe. This overall precession angle determines the final spin direction of the neutrons and thexspin componentσx=cos(φ) can be detected using a supermirror analyzer.

neutron

sample

π-flipper

detector π/2-flipper π/2-flipper

precession coil

precession coil H

H analyzer

Figure 2.4 – The spin-echo spectrometer utilizes the precession of the neutron spin to detect the energy transfer in the scattering process.

Different from the other spectrometers that measure the energy dependence of the dynamic structure factorS(Q,ω), the spin-echo spectrometer instead measures the time-dependent S(Q,t), which is the Fourier transfer ofS(Q,ω). Here we present a simplified argument. For quasi-elastic scattering ofδv=v₂−v₁¿v₁, the overall precession angleφis proportional to

the energy transfer ofħω: φ=¡2ħωzL

mv₁³

¢ω≡ωτNSE, (2.6)

whereωz is the Larmor frequency. Thus the average ofx-spin componentσxfor all different scattering processes can be calculated as:

〈σx〉 = 〈cosφ〉 =Z

dωS(q,ω)cos(ωτN SE), (2.7)

which is just the Fourier transform for the dynamical correlation function ofS(q,ω).

In this thesis, we use a spin-echo spectrometer IN11 at ILL to study the high-temperature monopole dynamics in CdEr₂X₄(X= Se, S).

2.2.4 Neutron diffractometer

A neutron diffractometer has the major components of a spectrometer skipping the analyzer, but is usually equipped with an Eulerian cradle and/or lifting arm detector to facilitate collec-tion of datasets. This means a neutron diffractometer does not discriminate the energy of the scattered neutrons and is used for the investigation of magnetic and lattice structures.

In this thesis, we used the diffractometers TriCS (now updated to Zebra), DMC, and HRPT at SINQ and D9 at ILL to study the nuclear and magnetic structures of MgCr₂O₄and MnSc₂S₄. Also, we used DMC and the polarized neutron spectrometers DNS at MLZ and D7 at ILL in their diffraction modes to study the quasi-static spin correlations in MnSc₂S₄and CdEr₂X₄(X

= Se, S). The triple-axis spectrometer TASP was used in the diffraction mode to investigate the phase diagram of MnSc2S4.