2.4. Principal Layout of Backscattering Diffractometers

3.1.2. A bit of history: The first "Rückstreuspektrometer" at the FRM in Munich

The first backscattering spectrometer was built in Munich and became operational in 1969. [2.4].

Fig. 9. Schematic drawing of the 'Rückstreuspektrometer' at the FRM in Munich.

 The silicon monochromator was placed into the white beam coming from a glass neutron guide. For scanning the neutron energy the monochromator was mounted on a velocity drive . Already this instrument was operated with a conical nickel coated beam concentrator which enhanced the solid angle of the glass guide such that it 'fitted' a nickel coated guide. An increase of the flux by a factor 1.9 was thereby achieved. For the analyser, about 1900 silicon single crystal slices were cut with an accuracy of 0.1 degree parallel to the (111) planes, and glued on concave spherically curved aluminum shells. The Si crystals focused the scattered intensity into the detector. The energy resolution was 0.6 µeV (FWHM). Apart from severe intensity problems due to the small reactor flux and the lack of a cold source the performance of this spectrometer was very limited:

3.1.3 More history: The π-Spectrometer at the FRJ-2 in Jülich

After the first backscattering experiments in Munich it became clear that a reactor with higher flux and with a cold source was required for this kind of spectroscopy. Consequently, in 1969 construction of a new spectrometer was started at the FRJ-2 in Jülich with a 200 times higher cold  neutron flux [2.5]. As shown in Fig. 10,

Fig. 10. Schematic drawing of the π-spectrometer at the FRJ-2 in Jülich.

a white neutron beam is guided by a Ni coated glass guide to the silicon monochromator mounted on a velocity drive. The backscattered neutrons are deflected by a graphite crystal to the sample. The energy of the scattered neutrons is analyzed by the spherically curved silicon analyser crystals. The neutrons are detected by counters placed close to the sample. The analyser crystals, the sample, and the detectors are protected by a large shielding house. The deflecting graphite crystal has an area of 3 x 3 cm², compared to the guide cross section of 10 x 10 cm². In this way 9/10 of the 'right' neutrons first pass the deflecting crystal; from the back-reflected neutrons 100 % are available of the flux hitting the deflecting crystal. The analyser crystals were adjusted at different scattering angles and measurements at five different Q-values could be performed simultaneously. The instrument could be operated in two modes, with polished or with unpolished silicon crystals. The energy resolution was  0.3µeV (FWHM) and  0.7 µeV, respectively, with 2 to 3 times higher intensity in the latter case.

This instrument, shown in Fig 11

Fig. 11 the BS spectrometer IN10

is situated at the end of the curved nickel coated neutron guide H15 which views the vertical cold source of the ILL [2.24]. The cold neutron beam has a total flux of about 2.10⁹ neutrons/cm²s with a spectral distribution around 6 Å. The width of the neutron guide is 3 cm, the height 20 cm. Only the upper 5 cm of the beam is used for IN10. The beam travels along a straight neutron guide section with 3 x 5 cm² cross-section and 10 meters length. It is followed by a further section with the same width, but 8 cm high and 6.3 meters long. The neutrons are backscattered from the monochromator which is mounted on the piston of a crank shaft velocity drive. The Bragg angle is 89.8°.

About 40% of the backscattered monochromatic beam is deflected off a (002) oriented graphite crystal (situated just above the incoming primary beam) into a third neutron guide (branching-off tube) of 3 x  3 cm² and 4.25 m length. This guide has a supermirror coating in order to reduce transmission losses due to the increased divergency of the beam after the deflector. The deflector has an anisotropic mosaic distribution, i.e. η_vertical = 0.4°, η_horizontal = 1.2°.

The neutrons then pass a chopper and a monitor, enter the analyser container and hit the sample. The scattered neutrons are analyzed for momentum and energy changes by analyser crystals. The analysers consist of single crystal wafers that are glued in the (111) or (311) orientation to the surface of spherically curved aluminum plates. These spherical segments have a radius of curvature of 1.5 m and are aligned so that neutrons backreflected from each one are focussed onto a ³He-detector located near the sample. Initially the analyser spheres where covered by small hexagonal single crystals of 1 cm diameter and 0.7 mm thickness (see Figures below). All analysers of IN10 were a few years ago changed by adopting a new technique to deform large Si-crystals which had been developed on IN16 (images of different analysers / monochromators can be found here). Today the analysers consist of hexagonal silicon single crystal slices of 0.5 mm thickness and a diameter of 6 cm. Spectra with up to 8 different momentum transfers can be measured simultaneously. An additional set of seven circular analyzers centered around the forward transmitted beam covers the small angle region (0.07 < Q < 0.3 Å^-1). The chopper has a duty cycle of 50% and provides the trigger signal for the electronic gate. Neutrons, scattered into the detector directly from the sample, are not counted.

The graphite crystal, the branching-off-guide and the analyser container with the chopper can be rotated around a vertical axis, defined by the crossover of the mid line of the main guide and the branching-off-guide, when the wavelength is changed.

The Doppler velocity of the monochromator is measured with an induction coil and a magnetic core, which is rigidly connected to the monochromator. It provides a voltage signal directly proportional to the velocity of the monochromator. The output voltage is amplified and digitalised. Together with the detector code it defines the channel number of the core storage into which the neutrons are sorted. (more information on data acquisition here)

The analyser container can be filled with Argon gas to reduce neutron losses and the background.

For further details see IN10. The instrument was commissioned in 1974. Subsequently, it became very popular and could until recently be considered as the first work horse in neutron backscattering spectroscopy. The majority of publications until about 1995 (see references to chapter 4) is based on experiments performed on this machine.

Click on 2004-1993 or 1993-1984 or 1984-1974 to access a more or less complete list of  publications of experiments carried out on IN10.

IN10 Analyser Plates

Cold neutron backscattering has one drawback: the limitation in momentum transfer: Q < 2Å^-1 for 6 Å neutrons. Therefore the IN10 spectrometer has an additional set of Si(311) monochromator and analyzers crystals which permit to access Q values up to 3.8 Å^-1. This setup has been used with success to measure the elastic incoherent structure factor of adamantane [4.47]. The problem with this setup is however the low intensity. Therefore a dedicated backscattering spectrometer for thermal neutrons, IN13, was developed at the ILL and commissioned in 1980 [2.24].

IN13 is installed at the thermal neutron guide H24 of the ILL with a total flux of 5.10⁸ neutrons/cm²s (see Fig. 12).

Fig. 12. Schematic drawing of the backscattering spectrometer IN13 for thermal neutrons at the HFR of the ILL in Grenoble.

CaF₂ crystals with (422) orientation are used for the monochromator and analyzers yielding a final energy of 16.45 meV (λ = 2.23 Å) in  backscattering. The incident energy is scanned via thermal expansion of the monochromator crystals (13.5 cm high, 5 cm wide and 1 cm thick) which are mounted in a cryofurnace operating with liquid N₂ as coolant. The temperature of the monochromator can be scanned from 80 K to 720 K continuously with a stability of 0.5 K yielding an energy transfer range from -125 µeV to 300 µeV. The analyser crystals are held at room temperature. The monochromator Bragg angle can be varied between 89° and 81° with a corresponding energy resolution of the spectrometer between 8 µeV and 24 µeV (FWHM). A vertically curved composite graphite crystal (9 lamella of 5 x 1.5 x 0.4 cm³, mosaic spread 0.4°) focuses the  beam onto the sample. The scattered neutrons are analyzed for energy and momentum transfer by a set of seven spherically curved composite crystal analyzers (60 cm high, 30 cm wide, radius of curvature 1 meter) with individual flat crystals of CaF₂  of 2 x 2 x 0.15 cm³.

 An additional set of three circular analyzers centered around the forward transmitted beam covers the small angle region (0.15 < Q < 0.5 Å^-1). A disk chopper with 4 windows and a duty cycle of 33% between the deflector and the sample is used to suppress the background of directly scattered neutrons and higher order contaminations. The neutrons are counted with a cylindrically shaped polydetector consisting of 32 vertical ³He detectors in two staggered rows and three end window individual 3He counters for the small angle analyzers.

Click on 2004-1989 and 1989-1982 to detect a complete list of publications of experiments performed with IN13.

Recent progress on the improvement of IN13 has been publishes in a paper of Natali et al

'Artistic' view of one IN13 analyser

View of the inside of the IN13 secondary spectrometer

Gradient Furnace of IN13

The option of a gradient furnace for the monochromator has been used to adapt the energy resolution to the problem under study. (This option is no more available on IN13).

IN13 Secondary Spectrometer

The backscattering instrument BSS was built in the new guide hall ELLA at the Jülich research reactor [2.6]. The instrument is very similar to IN10. Although installed at a medium flux reactor with about 10¹⁴ neutrons/cm²s, BSS has a flux which is not much smaller than that of IN10. This improvement was achieved by the efficient use of the rather large area beam (7 x 10 cm²) from a Ni⁵⁸ coated neutron guide. This large beam is focused to 3 x 3 cm² by a vertically curved graphite deflector with an anisotropic mosaic spread in conjunction with a converging supermirror guide. The anisotropic mosaic spread (η_v = 0.4°, η_h = 1.2°) is obtained by mounting 3 graphite crystals behind each other with orientations differing in the horizontal plane by η_v. Another specialty of BSS1 is a hydraulic velocity drive which allows energy scans with velocity profiles at the choice of the experimentalist. A drawback of BSS is the use of a monochromator Bragg angle smaller than 90°  in a beam from a Ni⁵⁸ coated guide, yielding an energy resolution of not better than 1.0 µeV (FWHM).

photo album of IN16

IN16 was the first spectrometer of the second generation (see chapter 3.1) for which the energy resolution is decoupled from the guide divergence (others are HFBS and RSSM). The spectrometer is located at the H53 guide on the horizontal cold source of ILL. A coarse wavelength band is extracted from the primary guide by a double deflector setup and is offered in nearly exact backscattering to the Doppler monochromator.

The layout of this instrument is shown in Fig. 13

Fig. 13: schematic view of the backscattering spectrometer IN16 [2.24].

The first vertically focusing triple graphite deflector, assembled by 9 triple graphite lamella according the same principle as the BSS deflector, together with a conical supermirror guide focuses the 120 mm high and 60 mm wide beam of the neutron guide H53 onto a 27 x 27 mm² spot at the focusing guide exit. Higher order contamination is eliminated by a liquid N₂ cooled beryllium filter. A background chopper avoids that high intensity neutron pulses enter the spectrometer with the graphite chopper in open position.

At a distance of 200 mm from the focusing guide exit the beam hits a graphite deflector chopper with 90° sectors of alternating open or reflecting segments (radius 294±30 mm; rotating with 2430 rpm; crystal speed of about 74.8 ±7.6 m/s). The reflecting segments are built up by trapezoidal shaped single cassettes hosting three horizontally inclined PG(0 0 2) crystals of roughly 80′ mosaicity each, making up a total horizontal mosaicity of about 240′ for deflecting the beam from the focusing guide towards the backscattering monochromator. This graphite chopper sends periodically, with a duty cycle of 50%, the coarsely monochromatic neutrons to a spherically curved monochromator (Radius R = 2.1 m at a distance of 1.9 meters) in nearly exact backscattering (focus at the guide exit).

This spherical monochromator (5.5 kg on aluminium support) is mounted on a mechanical crankshaft velocity drive operating at a variable frequency of maximum 14 Hz and a constant amplitude of ±25 mm, which allows for ±15 μeV energy transfer. Click here to see a movie of the Doppler drive, which is composed of two symmetric machines operating in opposite direction in order to minimise vibrations. Doppler movie  .

In the secondary spectrometer, like on IN13, the analyzers cover the full angular range and the neutrons are counted with a polydetector. The distance between the sample and the analyzers is 2 meters. The analyzers have a height of 140 cm and are mounted on goniometers sitting on air pads. This allows for a fast change of the analyser configuration. Three different analyser configurations are available:
– polished, flat Si(111) analysers for high energy resolution (HWHM = 0.2 microeV; both, the monochromator and the analyzers consist then of many perfect (111) oriented silicon crystals with a size of 4 x 4 x 0.4 mm³ glued on spherically curved surfaces).
– deformed, unpolished Si(111) crystals of size 6cm and 0.75 mm thickness.
– (the total number of analyser crystals is of the order of 450 000 !

The advantages of this layout are:

IN16 has a neutron flux about five times higher than that of IN10.

A complete list of publications can found by clicking on 2004-1996 or 1996-1992.

IN16 Analyser Plates

3.1.8  The HFBS Backscattering Spectrometer at NIST

This first backscattering instrument that uses the principle of phase space transformation (PST) explained in section 3.1.1.  Therefore the  flux of this instrument is increased and is somewhat higher than that of IN16. Due to the wide wavelength band of the primary beam hitting the PST, the signal to background ratio is less favorable.
Energy scans in a rather large window of 60 µeV are possible. The NIST WEB site dedicated to this instrument contains very detailed information about all technical aspects and a complete list of publications of experiments carried out with this spectrometer. The instrument is described in full detail in a recent review article.

3.1.9  The Backscattering Spectrometer at the FRMII: RSSM

This instrument is under construction at the new research reactor FRMII close to Munich. Again it uses the principle of phase space transformation  in order to increase the flux. The monochromator is mounted on a new Doppler drive which uses  a powerful linear motor. Velocities up to 4.7 m/s are possible.  The spectrometer should be operational in the near future. Look at the very nice WEB site to get very detailed information on the construction of this new instrument and a list of publications.

Schematic view of the Doppler drive

Figs%20BS/rssm_anim.gif

click on the title above to get an animated view of the spectrometer

3.1.10 . The Three Axis BS Spectrometer at the HFBR Brookhaven

This chapter again belongs to the history. We mention it because it contains some interesting aspects.
The idea behind this spectrometer proposed by Axe, Moncton and Passell [2.25] in 1983 is to use backscattering as an option on a conventional three axis spectrometer in order to obtain very high energy resolution without loosing the advantages of a three axis spectrometer, i.e. the flexibility, the ease of selecting particular Q scans and the high momentum transfer resolution.

The instrument (see Fig. 8d) is a three axis spectrometer in which both the monochromator and the analyser are replaced by a special two-crystal setup. The first crystal is oriented with a fixed Bragg angle of 90° - ε, where ε ≥ ΔΘ+η. The second crystal serves as the deflector of the backscattered neutrons. Energy scans are performed via thermal expansion of the monochromator backscattering crystal like on IN13. Therefore the energy range is rather limited compared to a normal TAS spectrometer, however, the Q resolution is as good. The flux at the sample position is as low as expected for a monochromator in backscattering geometry.

One problem with this setup resides in the fact that the deflector crystals are crossed by the primary beam before the neutrons hit the backscattering crystals. On the monochromator side this implies background problems created by the white neutrons scattered by the deflector. Therefore severe 'selection rules' have to be employed in the choice of the deflector material :

For the analyser the problem is less critical because its deflector is hit only by monochromatic neutrons. CaF₂ crystals with (422) orientation were used as backscattering crystals. The monochromator deflector is a silicon crystal with (331) orientation and an anisotropic mosaic spread (η_v  = 5', η_h =  20 ′). The analyser deflector is pyrolytic graphite with  η =  20′. The energy resolution was 10 μeV (FWHM), the energy range ± 120 μeV and the flux about 10⁴ neutrons/cm²s. Applications were limited to high energy resolution studies around the elastic line such as critical scattering near phase transitions or truly elastic measurements, i.e. the separation of elastic from inelastic scattering with μeV resolution.

Larry Passell and one of the authors aligning the BS TAS monochromator at HFBR.

The Brookhaven BS TAS monochromator

3.1.11 Fixed Window Scan

click on the title above to get more information about this method, which has been introduced by B.Alefeld.

3.1.12. X-Ray Backscattering Spectrometer

For completeness we mention the case of Backscattering of X-rays which  is as old as backscattering with neutrons [2.26].

History
Bottom [2.26] was the first to propose X-ray backscattering as a means for high resolution crystal diffraction work. Bottoms' idea was taken up by Sykora and Peisl [2.27] for the construction of a 'first generation' X-ray instrument for high precision measurements of relative lattice parameter changes  better than 10^-6. Freund et al developed a similar instrument [2.28]. Diffraction phenomena of X-rays in the backscattering case were studied theoretically on the basis of the dynamical theory of diffraction in several papers [2.29 - 2.31]. Later the idea emerged to try high energy resolution X-ray spectroscopy using the backscattering method, probably triggered by the very successful application in the neutron case. First results were published in 1982 by Graeff and Materlik [2.32]. They used a double crystal setup with two separate silicon (111) crystals at a Bragg angle of 89,84°. The energy scan was performed using thermal expansion of one of the two crystals. With the (888) reflection they obtained a relative energy resolution  of 5.3 .10^-7 or in absolute units  8.3 meV. Similar results were obtained by Egger et al [2.34]. Dorner and Peisl continued to develop this method and proposed a real backscattering spectrometer for X-rays [2.33]. This instrument is shown in Fig. 14.

Fig. 14. Schematic drawing of a backscattering spectrometer for X-rays.

It resembles very much a neutron backscattering spectrometer. There are however a number of characteristic differences:

            (37)

Therefore in the X-ray case the absolute energy resolution is inversely proportional to  τ, or in other words it improves by going to higher order reflections in contrast to the neutron case (see section 2.1, Eq. 23). The only limitation is the quality of the crystals, the absorption and not to forget the geometry term ctg ΘΔΘ. In order to reduce this latter to values smaller than 5.10^-7, one has to go very close to 90° and use very high collimation. But with the event of the development of dedicated synchrotron sources and associated equipment (wigglers etc) very intense X-ray beams with very high collimation are available.

 First measurements of this kind have been published [2.35- 2.42]. A  review on inelastic scattering of X-rays with very high energy resolution has been published by Burkel [2.70]. 

Recent developments in X ray BS spectroscopy
Two X ray BS spectrometers are in operation at the ESRF in Grenoble since a few years: ID16 and ID28. The diagram below shows the layout of these instruments.A full account of the new research field these instruments are giving access to is available at the ESRF. The interested reader is referred to this site. 

X-ray BS spectrometer at the ESRF

3.2. Time-of-Flight BS Spectrometers

The principle of this spectrometer is shown in Fig. 7 It is an inverted time-of-flight spectrometer with a very long primary flight path and a crystal analyser secondary spectrometer with a Bragg angle near 90°. This instrument was proposed by Scherm and Carlile in 1976.

3.2.1 The ISIS TOF-X Spectrometer IRIS

In the following we will describe the instrument IRIS, which is operational since 1987 at the spallation source ISIS of the Rutherford Laboratory [2.43, 2.44]. It views the ISIS liquid hydrogen cold source via a 34 meter long curved neutron guide which terminates with a 2.5 meter long supermirror coated converging guide. At 6Å the flux at the sample is enhanced by a factor of 2.8 due to this device. The flux at the sample position is 5.10⁶ neutrons/cm²s at 100 µA operation of the spallation source. At a distance of 6.4 meters from the moderator a variable aperture disc chopper serves to define the wavelength band to prevent frame overlap at the detector and to eliminate pulses for the case when a wider energy transfer range is required. The scattered neutrons are analyzed for energy and wavevector changes by an array of pyrolytic graphite crystals in near backscattering (θ  = 87.5°) and a position sensitive detector.

At full ISIS frequency the wavelength window at the sample is 2.0 Å. The energy transfer range spanned depends upon the phase of the chopper with respect to the ISIS pulse. If the wavelength window is centered on the elastic line an energy transfer range from +0.65 meV to -0.45 meV can be observed. In downscattering only to the elastic line an energy transfer range from +1.85 meV to 0.0 meV can be observed. Decreasing the chopper frequency to 25 Hz increases this measuring window to 11.1 meV. The maximum energy transfer range is limited by the short wavelength cut-off of the curved guide to a value of approximately 15 meV. At this energy transfer the resolution is ~80 µeV (FWHM) compared to 15 µeV (FWHM) at the elastic line.

It is a feature of inverted geometry spectrometers that the measuring range in neutron energy loss is very large compared with a direct geometry spectrometer as a result of the opposite handedness of the (Q, w) loci. Direct geometry machines achieve high resolutions by reducing the incident neutron energy such that in the limiting case a very narrow energy transfer range with only a small momentum transfer range is observable.

Recently a new analyser array consisting of mica crystals became available with an energy resolution of 4 µeV (FWHM) for 9.5 Å neutrons. The instrument has been and is  used successfully to study single particle motions of protons in a variety of materials in an energy range up to 1 meV [2.59-2.68]. A full list of publications can be found on clicking here.

An instrument similar to IRIS is  in operation at KENS in Japan [2.45] . A new instrument is in construction at the SNS in Oak Ridge. A TOF-X spectrometer called OSIRIS is in operation at ISIS. It operates with polarized neutrons.

3.2.2 The SNS TOF-X Spectrometer 

3.2.3 The ISIS TOF-X Spectrometer OSIRIS

3.2.4 The TOF-X Spectrometer MARS at PSI

This whole chapter belongs to the history . We describe these very specialized diffractometers just  for the sake of completeness.

The principle of the backscattering diffractometer was explained in section 2.4. As outlined in section 3.1.8, this type of instrument was first proposed by Bottom [2.26] for X-rays, then adapted to neutrons by Alefeld [2.17] and to X-rays by Sykora [2.27] and Freund [2.28]. We will limit the discussion to neutron diffraction.

3.3.1. Alefeld's Experiment

In 1965, Alefeld was the first to verify experimentally the ideas about neutron backscattering promoted by Maier-Leibnitz. A test setup was installed at a vertical neutron guide tube of the Munich research reactor FRM [2.1]. A sketch of the instrument is shown in Fig. 15.

Fig. 15. Schematic drawing of the first two-crystal backscattering diffractometer installed at the FRM in Munich by Alefeld.

Two silicon crystals were adjusted in a distance of 3.8 m such that the first crystal reflected the neutron beam with a Bragg angle of 89°. After a fall height of 3.8 m, the beam was reflected again by a second crystal with a Bragg angle of 89°. The neutrons were detected by a BF3 counter. The reflection curve (Fig. 16) was measured as a function of the temperature difference between the crystals.

Fig. 16. Reflection curve measured with two silicon crystals on the backscattering diffractometer of Alefeld.

It is clear from Fig. 16 that the curve is shifted from T₁-T₂ = 0. This shift was explained by the gravitational acceleration of the neutrons over 3.8 m which leads to a measurable energy shift of 0.38 µeV . This observation confirms within 20 % accuracy that the Hamiltonian for the neutron due to the gravitational force is m g h where g = 9.81 m sec^-2. With a dedicated experiment using the same technique an accuracy of 1 % could be obtained. This result was confirmed later with an accuracy of 1 % by measuring the phase shift of two coherent neutron beams in an interferometer, propagating in different gravitational potentials.

3.3.2. Double Crystal Neutron BS Diffractometers at the FRM in Munich

Alefeld [2.17] and later others [2.46 - 2.55] used the principle outlined in the preceding section to construct backscattering diffractometers dedicated to high precision measurements of relative lattice parameter changes.

The first neutron BS diffractometer ( thesis B.Alefeld) to measure the lattice parameter variation of SrTiO3

The principal layout of the backscattering diffractometer is shown in Fig. 5. Two versions can be distinguished, using either transmission or reflection geometry. In both cases the two crystals are aligned with the lattice planes parallel to each other and with a Bragg angle near or equal to 90°. The diffractometers developed in Munich were working in transmission geometry and used the Doppler effect for the scan. Alefeld [2.17] developed a rotating neutron counter to achieve exact backscattering. This system was abandoned on instruments of the next generation [2.48] because of technical problems. From that time on a simple neutron guide switch was used whereby Bragg angles of about 89.5° could be achieved.

The BS diffractometers in Munich have been used to study with high resolution relative lattice parameter changes of the following systems:

Similar experiments have been performed also at the ILL on the backscattering spectrometer IN10 [2.56, 2.57].

3.3.3. The Backscattering Diffractometer S21 at the HFR in Grenoble

For completeness, we mention the double crystal diffractometer S21 with reflection geometry and Bragg angles of 80°. Diffraction scans are performed either by thermal expansion or by rocking one of the two crystals [2.24].

3.3.4. A Four Beam Transmission BS Diffractometer

Alefeld and Springer improved the backscattering diffractometer in transmission geometry by using a so-called four beam method [2.58]. The principle is shown in Fig. 17.

Fig. 17. Schematic drawing of the four-beam backscattering diffractometer [2.50].

The neutrons are detected in four counter tubes arranged around the primary beam in a symmetric setup. It turns out that the accuracy of the relative adjustment of the sample and reference crystals is compensated in first order and that a second order correction can be performed easily. The method was applied to Ga As crystals. An accuracy of about ± 10^-7  was obtained with these crystals which have a Darwin width 1.6.10^-6 for the (200) reflection.

4. Neutron Backscattering Spectroscopy: Applications

The different fields of applications of high resolution neutron backscattering spectroscopy are listed in the table below:

4.2 Nuclear Spin Excitations

4.3 Critical Scattering near Phase Transitions

4.4 Dynamics of Spin Glasses and of the Glass Transition

4.5 Reorientational Motions in Molecular (Plastic) Crystals

4.6 Dynamics of Liquid Crystals and Molecular Liquids

4.7 Tunneling Spectroscopy

4.8 Dynamics in Polymers and Biological Model Systems

4.9 Diffusion Mechanisms in Metals, Alloys, Intercalation Compounds and of Hydrogen in Metals

4.10 Molecular Motions on Surfaces

4.11 Hydrogen Bonds

4.12 Fractals

N.B.This sequence of subjects is not ordered by inportance of the subjects. Historically the first applications of backscattering spectroscopy were studies of hyperfine interactions (V2O3) and of viscous liquids (Glycerol). They were followed by experiments in Jülich on liquid and molecular crystals and on diffusion mecanisms in metals. Tunneling spectroscopy with neutrons started also in Jülich.

Two domains can be distinguished, inelastic scattering and quasielastic scattering. Studies of nuclear spin excitations and of tunneling motions belong to the first category, all the others to the second, except maybe the fractal experiments, where one measures the density of states of excitations. In most of the above mentioned applications one investigates single particle motions via incoherent scattering. There are however significant exceptions like critical scattering near phase transitions, spin glasses and in some cases studies of the glass transition where one looks at collective phenomena. An excellent review about high resolution quasielastic scattering was written by M.Bee.

A more or less complete list of publications obtained with BS spectrometers so far can be found here.

4.2. Nuclear Spin Excitations

Hyperfine interactions can be studied by high resolution inelastic spin-flip scattering of neutrons. The theory behind this field can be reviewed shortly as follows [4.1]: If neutrons are scattered spin-incoherently from nuclei, the probability that their spins will be flipped, is 2/3. Due to the conservation of angular momentum, the nucleus at which the neutron is scattered with spin-flip changes its magnetic quantum number M to M±1. If the nuclear ground state is split up into different energy levels EM, for example by a magnetic field or an electric quadrupole interaction, the spin-flip produces a change of the energy of the nuclear ground state i.e. a nuclear spin excitation:

                                                                                                          (38)

         (39)

where e-2w is the Debye-Waller factor and a' the spin incoherent scattering length. For DM we used a constant D independent of M. This is the case for a Zeeman split nuclear ground state due to a magnetic field H:

                           (40)

where µ is the number of nuclear magnetons µ_k, I the nuclear spin. Eq. 39 has been derived assuming no correlation between the nuclear spin system and the lattice as well as no nuclear spin-spin interactions. Neutron scattering cross-sections have been derived also for the more general case where part of these correlations and interactions have been taken into account [4.13]:

We can measure hyperfine interactions with neutrons provided that:

Concerning the first condition the lower limit of a detectable splitting is of the order of 0.03 µeV or 8 MHz with backscattering spectrometers. Table 4 shows a number of potentially usable nuclei for inelastic spin-flip scattering. D is calculated for a magnetic field of 100 kOe.

Isotope	Spin			A %
1H	1/2	79.9	0.33	99.98	1.77
2H	1	2.04	0.0005	0.015	0.27
7Li	3/2	0.68	0.045	92.5	0.69
23Na	3/2	1.59	0.53	100.0	0.47
25Mg	5/2	0.27	0.19	10.0	0.11
35Cl	3/2	4.6	44.1	75.7	0.17
39K	3/2	0.23	2.1	93.0	0.08
45Sc	7/2	4.67	27.2	100.0	0.43
47Ti	5/2	1.5	1.7	7.3	0.10
51V	7/2	5.07	4.9	100.0	0.46
53Cr	3/2	5.85	18.2	9.5	0.10
55Mn	5/2	0.40	13.3	100.0	0.43
59Co	7/2	4.83	37.2	100.0	0.42
65Cu	3/2	0.40	2.17	30.8	0.50
67Zn	5/2	0.28	6.8	4.1	0.11
105Pd	5/2	0.80	20.0	22.3	0.08
127I	5/2	0.31	6.2	100.0	0.35
133Cs	7/2	0.21	29.2	100.0	0.23
139La	7/2	1.15	8.9	99.2	0.25
165Ho	7/2	0.36	64.7	100.0	0.36
171Yb	1/2	3.9	48.6	14.3	0.31
173Yb	5/2	2.47	17.1	16.1	0.08
143Nd	7/2			12.17

Table 4. Potentially usable nuclei for the study of hyperfine interactions by high resolution inelastic spin-flip scattering of neutrons. A is the natural abundance of the isotope, sinc and sabs are the spin incoherent and absorption cross-sections. Delta is the Zeeman splitting calculated for a field of 100 K0e.

Fig. 18

Fig. 18 shows a characteristic example of a spectrum of neutrons scattered in ferromagnetic cobalt [4.4]. Until now only four nuclei have been used to measure hyperfine interactions with neutrons: Vanadium, Cobalt and Hydrogen and more recently Nd.

Fig. 19 shows a spectrum of neutrons scattered from protons in Tb Fe2 H4 at 4.2 K [4.8].

Fig. 20

Fig. 20 shows the temperature dependence of the internal magnetic field at the vanadium nucleus in V3 O7 [4.9]. In the following we compare the neutron method with Mössbauer and NMR techniques.

Mössbauer Effect. Most nuclei in Table 4 are not Mössbauer isotopes. Therefore a new class of substances is accessible with the neutron method. Although the energy width of ⁵⁷Fe is 9 neV compared to the 300 neV neutron energy resolution, the sensitivity to internal magnetic fields of a Mössbauer experiment in iron is roughly the same as that of a neutron experiment in cobalt due to the higher nuclear magnetic moment of cobalt. The neutron energy spectrum with its three lines is simpler than the general rather complex Mössbauer spectrum. This is advantageous for investigations of substances with more than one hyperfine field or with field distributions. In the neutron case the Debye Waller factor is close to unity even at room temperature. Therefore no intensity problems arise at elevated temperatures.

NMR has the advantage of a very high resolution. However, from this a number of difficulties arise, which do not exist in the neutron experiment. It is often rather tedious to find an unknown resonance. This is especially true in antiferromagnetic samples where no enhancement factor exists. Furthermore, short relaxation times make the signal to noise ratio small. Investigations of mixed systems in the high concentration range are often impossible and magnetically hard materials are not easily accessible. The sensitivity of the neutron experiment does not depend on relaxation times and enhancement factors. Therefore all nuclei in a substance are contributing equally to the signal and always the bulk properties are measured. NMR experiments are performed at a momentum transfer of Q~0 Å-1 whereas the neutron experiments can cover a range in Q. Only in the case of nuclear spin-spin correlations which have been neglected in the derivation of Eq. 39, would different results be obtained by NMR and neutron scattering.
More recent publications concerning Nd compounds can be found by clicking here. Spectra of NdCu2 obtained at various temperatures are shown below:

4.3. Critical Scattering near Phase Transitions

Neutron scattering has been used extensively to study phase transitions. Soft modes, central peaks and critical scattering are the key phenomena observed in these systems. Very high energy resolution is often required for these studies. Critical quasielastic scattering has been observed in systems with order -disorder phase transitions. It originates from the formation of clusters of the ordered structure within the disordered phase [4.18]. The energy width of this quasielastic component is proportional to the inverse of the cluster relaxation time t_c. The intensity and t_c diverge at the phase transition. This corresponds to a critical slowing down of fluctuations.

An investigation of this type of phenomenon has been performed on IN10 on the molecular crystal paraterphenyl [4.18]. The molecules consist of three phenyl rings which are in a non planar configuration. At Tc = 179.5 K paraterphenyl undergoes an antiferrodistortive phase transition. The purpose of the experiment was to study the temperature dependence of the cluster lifetime t_c at  .

Fig. 21. Critical scattering in para-terphenyl near its phase transition:

a) directly measured spectra .

b) Data after Fourier transformation and deconvolution

Fig. 21a shows spectra measured at three temperatures close to the phase transition. The same data after Fourier transformation and deconvolution is plotted in Fig. 21b on a logarithmic scale. Apparently S(q = 0, t) cannot be described by a single exponential. From the slope of ln S(0, t) at short times t one obtains the line width _^G = 1/t_c which is plotted in Fig. 22 as a function of _^DT = T - T_c.

Fig. 22. Quasielastic line width of the critical scattering in paraterphenyl as a function of   DT = T-T_c.

A linear behavior is observed. For this kind of investigation very high resolution in energy is crucial.

The phase transition in paraterphenyl has also been investigated by quasielastic incoherent neutron scattering [4.22, 4.23, 4.24]. Far above the transition point the measured quasielastic spectra revealed the dynamics of an almost random disorder of molecular orientation. In the neighborhood of T_c, however, the quasielastic incoherent neutron spectra were clearly affected by the appearance of orientational short-range order and critical fluctuations of the orientational order parameter. The results obtained by high resolution spectroscopy were analyzed in terms of a model involving orientationally short-range-ordered clusters. The single-molecule residence time t_R, the cluster lifetime t₀ and cluster concentration c, as defined in this model, were determined as a function of temperature.

Related studies by incoherent neutron scattering had been carried out already much earlier by Toepler et al [4.16] with single-crystalline NH4Cl near its order-disorder phase transition. The complementary study on ND4Cl single crystals on IN10 has also been published [4.21].

The central peak phenomenon of SrTiO3 has been investigated on IN10 by Toepler et al [4.15]. In a temperature range between T_c and
T_c + 12 K no line broadening was observed within an experimental upper limit of 0.08 µeV.

4.4 Dynamics of the Glass Transition in Spin Glasses and Molecular Glasses

The dynamics of disordered systems has been studied extensively in the past by inelastic and quasielastic neutron scattering. Initiated by the paper of Edwards and Anderson [4.31], high resolution neutron experiments were undertaken by Murani et al [4.25] to investigate the phenomenon of the phase transition (freezing) of spin glasses like Cux Mn1-x. These early measurements showed that the intermediate scattering function S(Q,t) extends over many decades in time. A distribution of relaxation times was observed. Therefore it was necessary to use several types of spectrometers (Backscattering, TOF and Neutron Spin Echo (NSE)) to explore the broad range of relaxation times. The quasielastic width G of the scattering function of the amorphous spin glass Al2 Mn3 Si3 O12 [4.26, 4.29] is shown in an Arrhenius plot in Fig. 23.

Fig. 23. 

The data of  _^G between 0.01 and 10 µeV were obtained with the backscattering spectrometer IN10.

The study of the glass transition in molecular glasses has become and still is a fashionable subject of research. Theoretical papers have triggered high resolution neutron scatterers to investigate this problem experimentally [4.32 - 4.45]. The physics behind the glass transition is related to that of the freezing in spin glasses.

Colmenero et al have published a paper in 2004 which reviews the field of the glass transition.

Molecular (plastic) crystals have been investigated with the aim of understanding the dynamic orientational disorder present in these systems. The most striking feature in the high resolution neutron spectra observed in such experiments is the appearance of incoherent elastic and quasielastic scattering. From the Q dependence of the former (the elastic incoherent structure factor EISF) we can get detailed information about the geometry of the orientational motions. From the width of the quasielastic peak we obtain the dynamical behavior of the orientational motions. High energy resolution is crucial in these experiments in order to be able to separate clearly elastic from quasielastic scattering. We will briefly present two examples from the numerous experiments already performed [4.46 - 4.68].

The measurement of the EISF of adamantane C10H16 in its plastic phase by Lechner et al [4.47] is a text book example on how to extract the information about the geometry of the orientational motion from neutron data (see Fig. 24).

Fig. 24 Elastic incoherent structure factor (EISF) of adamantane measured on IN10. - C4 Reorientations , - - - C3 Reorientations, ..... Rotational Diffusion

The energy resolution in this experimental study on IN10 was so high that the elastic incoherent intensity could be observed directly and the quasielastic scattering contributed to the spectra in form of a flat background. The difficulty in these experiments is related to the presence of coherent Bragg scattering,

which has to be avoided very carefully. The conclusion of this study was that the reorientational motion in adamantane at room temperature is probably dominated by C4 rotational jumps with a correlation time of 1.7.10^-11 sec.

The motions of 1-cyanoadamantane C10H15 CN were investigated by Bee et al [4.62] on IN10 in the glassy phase obtained by rapid quenching of the room-temperature orientationally disordered phase.

Fig. 25. Intermediate scattering function of 1-cyanoadamantane.

From a direct Fourier transform analysis (see Fig. 25) it was demonstrated that the experimental data could not be described in terms of any simple jump model involving a single correlation time. Instead the interpretation was based upon a distribution of correlation times, a phenomenon mentioned already in conjunction with the glass transition.

4.6 Dynamics of Liquid Crystals and Liquids

The knowledge of the self diffusion coefficients in liquid crystals and molecular liquids is important for the understanding of the flow properties of these systems High resolution neutron spectroscopy using the backscattering technique at small momentum transfers Q < 0.3 Å^-1 is a very efficient tool for these studies. Small momentum transfers are necessary in order to avoid the influence of rotational motions on the shape of the quasielastic signal. Indeed earlier neutron experiments with meV energy resolution and at higher momentum transfers have led to much too high diffusion constants via a misinterpretation of the results. Using the backscattering technique results could be obtained which were then in agreement with tracer experiments or NMR spin-echo measurements.

Fig. 26 Quasielastic line width of the liquid crystal TBBA at different temperatures as a function of Q².

Fig. 26 shows the quasielastic line width of the liquid crystal TBBA at different temperatures as a function of Q2 [4.71]. The diffusion constant is obtained from the slope of the straight lines.

Related studies have been performed by Salmon et al [4.86, 4.87, 4.88] on the dynamics of water-protons in aqueous ionic solutions.

Fig. 27 shows typical results obtained on an aqueous NiCl₂ solution. The continuous curves are 2-Lorentzian fits, the HWHM of both being plotted as a function of Q², resulting in two diffusion constants D₁ and D₂. D₁ is called the diffusion coefficient of protons bound to the cation for a time t₁, D₂ the diffusion coefficient of the remaining protons in solution.

4.7. Tunneling Spectroscopy

Since 1975 high resolution neutron spectroscopy has been used very successfully to study rotational tunneling motions of small molecules, molecular groups or ions like CH₄, CH₃ and in solids. It seemed at the beginning that only a few materials show the effect of tunneling in the energy range accessible to neutron spectrometers. The systematic research performed in the meantime has shown that many molecules are weakly hindered from rotation in the solid. Thus it was possible to study tunneling in various chemical surroundings (ionic, van der Waals, metallic systems), as a function of chemical substitution (isomorphous and/or deuterated compounds), bond length and unit cell dimensions (pressure), the coupling to phonons (temperature) and to other methyl groups, etc. The results gave a rather complete understanding of rotational tunneling itself and of the effects or materials under investigation. The method is very sensitive because the tunnel splitting is determined by the overlap of the wavefunctions describing the different orientations of a molecule which decays approximately exponentially with the potential height. It is, however, only applicable to molecules with large rotational constants (small momenta of inertia) e.g. protonated or deuterated small molecules.

A first review of the field was given in 1981 by W. Press [4.146]. His book describes the fundamental theory of neutron scattering from a quantum rotor, mainly concentrating on CH3 and XH4 molecules and contains an almost complete list of references of the work performed up to 1980. In the meantime considerable progress has been achieved [4.147]. A more or less complete list of the investigated materials containing published and unpublished results has been assembled by Prager et al [4.148]. A complete list of publications of results obtained by backscattering spectroscopy is given [4.89 - 4.145]. Typical spectra and results of a few selected cases are shown in Figs. 28, 29 and 30.

Fig. 28. Tunneling spectrum of NH4 ClO4 at 5 K measured on IN10.

Fig. 29. Tunneling spectrum of CD4 at 3 K measured on IN10.

Fig. 30. Tunneling spectra of a) CD₃  groups, b) CH₃ groups in lithium acetate measured on IN10 and IN13 respectively

First exploratory high resolution QNS experiments using the backscattering technique were performed in 1974 on poly-dimethyl-siloxane (PDMS) at small momentum transfers [4.149]. Later studies were made on polymers in melts and solutions and on monomer molecular motions in micellar aggregates [4.150 - 4.156]. Ferroelectric copolymers were investigated recently by Legrand [4.157, 4.160].

From about 1980 on the neutron spin echo technique became available with the NSE spectrometer IN11 at the ILL [2.24]. It turned out rapidly that this technique is ideally suited for the study of slow collective motions in polymers via coherent quasielastic neutron scattering in the time domain at small Q > 0.02 Å^-1. But spin incoherent QNS using backscattering spectrometers remains very useful and complementary tool for studying the single particle motions in these systems.

Going a step further in complexity, biological model systems have been investigated by high resolution neutron scattering [4.161-4.167]. The interpretation of the results is extremely difficult and the amount of information obtained very limited. However the hope is that in combining measurements with different high resolution neutron spectrometers on samples under various conditions of temperature, humidity, isotopic substitution, etc. and by comparison with molecular dynamics simulations one will be able to extract significant information on the dynamics in these systems.An review of this field has been published by Zaccai. An introduction into the field of biological systems studied by neutron scattering by Pynn can be found on the web.

Neutron scattering applications to biotechnology are discussed by Byron and Gilbert.

Backscattering spectrometers have extended the measurement range of diffusion constants down to values of 10-8 cm2/s and of relaxation times up to a few 10-9 s. This fact made it possible to investigate diffusion and jump processes by QNS not only in liquid and liquid like systems, but also in solids, like metals below the melting point [4.171, 4.175, 4.192], fast ionic conductors [4.174], intercalation compounds [4.172, 4.176] and hydrogen and impurities in metals [4.198 - 4.224]. From experiments on single crystals at larger Q (2 to 4 Å-1) information on the single step of the diffusion is obtained, e.g. the jump vectors and the space distribution of the atoms over the interstitial sites.

So far only two pure metals, Na [4.175] and Ti [4.192] have been studied by high resolution QNS on IN10. In both cases single- and polycrystalline samples were used. Fig. 31 shows typical results obtained on single crystals of bcc b-titanium below its melting point at 1680° C:

Fig. 31. Quasielastic line broadening of neutrons scattered in bcc Ti single crystals as a function of the scattering vector.

The quasi-elastic line width is plotted as a function of the momentum transfer  for different orientations a, the angle in the scattering plane between the crystal <001> axis and the incident neutron wave vector. These results clearly reveal that self-diffusion in b-titanium is dominated by 1/2 [111] jumps into nearest neighbor vacancies. Very similar results were obtained on solid sodium. Concerning the studies of motions of hydrogen in metals, the fact that protons have a very high spin incoherent scattering cross section makes it possible to investigate rather low concentrations of protons down to 10^-3 for backscattering spectroscopy and even below 10^-5 for TOF spectroscopy. Other relatively non-absorbing nuclei which have rather high incoherent scattering cross sections are listed in Table 5. All of these nuclei are possible candidates for high resolution neutron studies of diffusional motions in condensed matter.

Element
1H	79.9	0.33
2H	2.04	0.00052
7Li	0.68	0.045
natN	0.52	1.9
23Na	1.59	0.53
natCl	5.20	33.5
nat Ar	0.22	0.68
natK	0.39	2.1
45Sc	4.5	27.2
nat Ti	2.87	6.09
51V	4.98	5.08
nat Cr	1.82	3.07
55Mn	0.6	13.3
nat Fe	0.39	2.56
59 Co	4.8	37.2
nat Ni	5.0	4.5
nat Cu	0.52	3.78
nat Se	0.35	11.7
nat Rb	0.30	0.38
nat Mo	0.28	2.55
nat Ag	0.56	63.3
nat La	1.51	8.97
nat Nd	11.0	50.5
nat Dy	70.0	940
natYb	4.0	35.5
nat Hf	2.6	104
natW	1.82	18.4
natPt	0.6	10.3
		Lambda=1.8 A

Table 5. Nuclei suited for high resolution neutron scattering studies of diffusional motions in condensed matter. sinc is the incoherent scattering cross section.

Spectroscopic high resolution neutron studies of single particle motions on surfaces are particularly difficult for of two reasons : the weak signal from the particles moving on the surface (which is limited in size) and the strong background signal from the substrate. Systems studied so far include motions of hydrogen atoms chemisorbed on catalysts like Raney-nickel [4.225] and platinum zeolite [4.227] and motions of small molecules like NH₃ and CH₄ adsorbed on graphite [4.228]. Both substrates mentioned above have the advantage that rather large surface areas can be obtained (of the order of a few m²) with samples with a volume of a few cm³. Here even submonolayer systems can give a sufficient neutron signal on backscattering spectrometers. One problem in the data treatment of these systems arises from the fact that the QNS scattering law cannot be described by a single Lorentzian but rather by a more complicated function which has a logarithmic singularity at = 0 and very long tails. This is due to the two-dimensional character of the diffusive motions.

4.11 Dynamics of Hydrogen Bonds

The transfer of hydrogen atoms along preexisting hydrogen bonds is one of the simplest chemical reactions and is important in many chemical and biological systems. In a condensed phase the phonon coupling leads to a structural rearrangement of the environment and modifies the reaction path. It also provides mechanisms of relaxation for the system. Many carboxylic acids form dimers linked by two hydrogen bonds, and the interconversion of the two tautomer forms by a concerted two-proton transfer is governed by a rather symmetric double-well potential. These materials are therefore good models for the study of intermolecular hydrogen bonding and proton transfer reactions. Amongst other techniques like NMR and Infrared, high resolution QNS can be used to investigate these systems. Rather high momentum transfers are needed because of the short jump distances of the order of 1 Å involved. The scattering law can be derived by a semiclassical treatment of the two-site jump model [4.59] and yields:

          (41)

                                                                           (42)

                                (43)

Where is the Debye-Waller factor, R the proton jump vector between the sites A and B, P_x is the equilibrium occupation probability of the site x, and t is the residence time. Stoeckli et al [4.59] have performed QNS experiments on IN13 to study several carboxylic acids in both single crystal and polycristalline samples. Fig. 32 shows the Q dependence of S₁ obtained for acetylene dicarboxylic acid.

Fig. 32 Q-dependence of the intensity of the quasielastic intensity of neutrons scattered from single crystals of acethylene dicarboxylic acid at 250 K.

From the data the jump-rate and jump-distance could be extracted, as well as the occupation probability. The authors emphasized that no other experimental technique is known to provide an equally detailed picture of the double proton exchange process than high resolution QNS at high momentum transfers on single crystals.

Aerogels are amorphous solids of high porosity, and, in some cases, are fractal, i.e. there exists self-similarity of the inhomogeneities. This can be concluded from small angle neutron scattering experiments. Porosity is not only a structural characteristic of a solid, it rather affects also its dynamic properties. It has been shown that above a crossover frequency wco, the modes are no longer propagating (phonons) but localized vibrations which have been called fractons. In this regime the vibrational density of states g( ) is predicted to vary as  , where d is the fractal dimension [4.229-4.231]. Inelastic neutron scattering provides a direct method to determine g( ). In the limiting case hw/k_BT << 1 which is nearly always the case on backscattering spectrometers, the intensity of inelastically scattered neutrons is proportional to  which is a constant for Debye solids, and which decreases with increasing for fractal systems like aerogels. Due to the fact that in silica aerogels the sound velocities c_s are lower by more than one order of magnitude compared to normal solids and that    , it is possible to measure the density of states of aerogels on backscattering spectrometers. Fig. 33 shows typical spectra measured with the backscattering spectrometer BSS1 at Jülich for two values of the momentum transfer Q and three aerogels of different mass density  ( g/cm3)  [4.230].

Fig. 33

The arrows indicate the derived values for the crossover energy. The spectra consist of an elastic line superimposed on an inelastic contribution. The shape of the latter depends on the mass density of the samples. The inelastic signal for the low density aerogel exhibits a strong energy transfer dependence which is equivalent to a deviation from a Debye density of states. The sample with the high density, on the contrary, produces an energy transfer independent inelastic contribution over the energy-range investigated, thus showing Debye behavior. In contrast to theoretical predictions, no indication for a hump in the crossover region has been found. Similar studies have been pursued by Vacher et al [4.229] and many questions are still open.

 4.13 Quantum Liquids

A typical application of an offset monochromator is the high resolution study of the roton minimum of He4 in the mK temperature range. Here the excitation which has an energy of about 740 meV has been studied on IN10 (option 'IN10B') using a NaF111 monochromator in conjunction with the usual Si111 analyser with 0.7 meV (FWHM) energy resolution. The shift and broadening was measured as a function of temperature, pressure and isotope composition ( ie admixtures of He3 in He4). The roton minimum has also been investigated in Aerogels. The figure below shows a comparison between the exitations measured in bulk He4 and of He4 in an aerogel.

with the courtesy of the authors: Andersen et al

5. Special Applications

5.1 Precision Measurement of h/mn

5.2 Backscattering and Polarisation

5.3 Neutron Magnetic Resonance Shift

5.1. Precision Measurement of h/mn

In 1968 Stedman proposed to use two synchronously vibrating crystals in backscattering geometry for the accurate measurement of the neutron velocity in order to set up a new standard of the lattice constant of silicon [5.1]. By combining _^l = 2d for backscattering and the de Broglie relation v_^l = h/m_n , we obtain d = h/2v m_n. The main problem of this method is to control the strains of the two crystals which are induced  by the strong accelerations up to 24 000 m/sec². In 1981 the lattice constant of silicon was measured by means of a combined scanning X-ray interferometer and a two beam optical interferometer [5.2]. A value ao = (543102.018±0.034) fm, which corresponds to an accuracy of 0.063 ppm, was obtained. Since d is now the best known quantity in de Broglie's relation the accurate measurement of v now serves to increase the accuracy of the ratio h/m_n.

At the ILL an experiment was set up by Krüger, Nistler and Weirauch [5.3] where also backreflection from a silicon crystal is used to define the wavelength . The principle of the method is shown in Fig. 34.

Fig. 34. Instrument setup at the ILL to measure the ratio h/m_n.

The neutrons are polarized by Bragg reflection from a Heusler crystal H1. The polarization is modulated periodically with a meander coil M. After a path of about 10 meters the neutrons are backreflected from a Silicon single crystal Si with (331) orientation which defines the wavelength. The neutrons subsequently pass the modulator M a second time. The velocity of the neutrons is given by  v = 2Dln where Dl is the modulation length which is the distance between two intensity minima. n is the modulation frequency. In the experiment Dl is only 1mm. Therefore a large multiple of Dl (about 10⁴) is measured in order to increase the accuracy of the experiment. The most accurate value for h/m_n obtained by this method is h/m_n= (3.9560344±0.0000016) 10^-7m² sec^-1 [5.4].

5.2 Backscattering and Polarization

The Hamiltonian of a neutron in a magnetic field is

                   (44)

where is the neutron momentum operator, the neutron magnetic moment. If the magnetic field is time independent, the total energy is a constant of the motion and the change of the neutron momentum is dependent on the orientation of the field gradients relative to the neutron propagation direction. If the field gradient is oriented normal to the propagation vector, we deal with the Stern-Gerlach effect which describes the transversal separation of the two spin states. If the field gradient is parallel to the neutron propagation vector, it produces a longitudinal change of the neutron momentum which is connected with a change of the kinetic energy of the two spin states. This always happens when neutrons pass from a field free region into a space region with a magnetic field  where the change of the potential energy     inevitably implies an equivalent inverse change of the kinetic energy of the neutron for its total energy to remain constant (Zeeman splitting).

In 1976 Funahashi proposed to combine the high resolution of backreflection and the Zeeman splitting of neutrons in a magnetic field to produce a highly monochromatic polarized neutron beam [5.5]. A schematic drawing of his high resolution neutron polarizer is shown in Fig.35.

Fig. 35 Funahashi backscattering polarizer.

A perfect crystal in backscattering orientation extracts a highly monochromatic unpolarized neutron beam which penetrates into a strong magnetic field with the field gradient parallel to the propagation vector. The two spin states of the neutrons split into the energies Eo ± mH. An analyser crystal situated in this field , whose lattice constant is tuned (for example by temperature change) to one of these energies, reflects neutrons only of one polarization state. A slightly different experiment was performed [5.6] to verify this proposal. Instead of tuning the second crystal, the first crystal was Doppler moved, in this way both polarization states were measured simultaneously. The upper part of Fig. 36 shows an intensity spectrum dependent on the Doppler velocity with zero magnetic field.

Fig. 36. Longitudonal Stern Gerlach effect measured with a backscattering setup of silicon (111) crystals.

This spectrum represents the resolution of the experimental set- up. In the lower part of Fig. 36 a field of about 2 Telsa was applied at the second crystal. A clear indication of two split lines which correspond to different polarizations is visible.

For the production and use of an energy modulated polarized beam for a backscattering spectrometer it is easier to use two crystals with an offset of mH and to perform the energy scan by a simultaneous temperature change of both crystals.

It should be mentioned that Zeyen et al [5.7] also obtained a clear separation of the Zeeman-lines by using the double crystal diffractometer S21 with conventional Bragg angles. With these experiments however it is not possible to obtain the good wavelength resolution needed for a backscattering instrument.

Backreflection and transmission of the two Zeeman levels of ultracold neutrons by a monocrystalline magnetized iron foil has been used by Herdin et al [5.8] to demonstrate the efficient polarization of ultracold neutrons.

5.3 Neutron Magnetic Resonance Energy Shift (nMR) and Backscattering

Neutron magnetic spin resonance is mainly applied for the search and determination of fundamental properties of the neutron such as its magnetic moment [5.9] or its electric dipole moment (EDM) [5.10]. Kendrick et al [5.11] have studied the possibility of using nMR for a pulsed-neutron-polarization-inverter. As an active component of neutron optics, nMR was first discussed by Drabkin and Zhitnikov [5.12] and later in more detail by Krüger [5.13] and by Badurek et al [5.14]. This component shifts the kinetic energy E_kin of the two polarization states by ± 2 µHo. An unpolarized neutron beam entering a magnetic field Ho is separated into two polarized sub-beams. Energy conservation demands that the kinetic energy of the two sub-beams is changed by an amount corresponding to the potential energy change ± µHo. When leaving the magnetic field, the inverse change of the potential energy and correspondingly the kinetic energy occurs. Therefore the neutron kinetic energy (and the potential energy) has not changed after passing through a magnetic field. However if the polarization direction of the two sub-beams is inverted within the magnetic field Ho by a time-dependent magnetic field H1, an effective potential energy change of ± 2 µHo occurs. Thus the kinetic energy change of the two sub-beams when entering and leaving the magnetic field Ho is additive. As a result the two sublevels, after the neutrons have passed this neutron magnetic resonance device, are energetically separated by ^_DE = 4 µHo. Energy conservation is assured by photon creation and annihilation, respectively. The flipping probability  of a particle with spin 1/2 in a magnetic field Ho and a time-dependent, rotating field H1 perpendicular to Ho was calculated by Alvarez and Bloch [5.15], later by Rabi et al [5.16] and by Kendrick et al [5.11] and is given by

                             (45)

with

                                    (46)

                                     (47)

                              (48)

and t is the time during which the time-dependent rotating field H1 acts on the neutron.

In order to obtain = 1, the value for t has to be  . If this pulse-time for a p-turn is replaced by the time of flight To = lo/vo of the neutrons through the hf-coil, one obtains

                                       (49)

If the neutron velocity vo is fixed, then either the length of the hf-coil or the amplitude H1 of the rotating field has to be matched in order to obtain a p-turn.

An experiment to measure the nMR energy shift was performed at the p spectrometer in Jülich [2.5, 5.17]. Fig. 37 shows a sketch of the experimental set up.

Fig. 37. Experimental backscattering setup to measure the neutron magnetic resonance shift.

A monochromatic unpolarized neutron beam was backscattered from the crystal Si1 and deflected to the electromagnet containing an hf-coil in the flight direction of the neutrons and perpendicular to Ho. After passing the hf-coil, the energy of the neutrons was analyzed by the silicon analyser Si2 in backscattering. The energy scan was achieved by Doppler moving the monochromator Si1. After the nMR had been established, the voltage of the hf-coil was turned off and by Doppler moving Si1 a resolution measurement was performed which is shown in the upper part of Fig. 38. After switching on the hf-coil, the nMR energy shift was determined, which is shown in the lower part of Fig. 38.

Fig. 38. HF induced change of energy distribution of backscattered neutrons measured with the modified -spectrometer at the FRJ in Jülich.

Two energetically separated intensity peaks at  ± 0.24 meV were observed in good agreement with the calculated values of ± 0.237 meV. This result demonstrates the inelastic interaction of a neutron with the time-dependent magnetic field of a nMR system.

Several applications of nMR are conceivable:

The storage and study of ultracold neutrons in a bottle is exciting in itself and is used mainly in the study of fundamental properties of the neutron [5.18, 5.19]. Recently Schuster et al have tested a new type of storage device for cold neutrons which may act as a basic element for novel beam tailoring [5.20]. As shown in Fig. 39, two (111) oriented silicon crystals at both ends of a 1 m long monolithic perfect silicon crystal serve both as total reflecting walls for 6.27 Å neutrons in backscattering and as entrance and exit doors which can be activated magnetically.

Fig. 39. Sketch of the perfect crystal backscattering storage device installed at the ISIS neutron source.

To reduce lateral losses a glass guide tube is adjusted between the two crystal plates. The device was tested at the pulsed neutron source ISIS at the Rutherford Laboratory. The system was filled with neutrons by a synchronized magnetic pulse of 1.3 T applied to the first crystal. Due to this pulse, the kinetic energy of the two polarized sub-beams is shifted by ???? which corresponds to the width of the plateau of the Darwin curve of Si (111). After the neutrons have passed the magnetic field the energy has been shifted back and those neutrons which are reflected by the second crystal are also reflected by the first, if the field H was switched off before the neutrons come back to the first crystal. Fig. 40 shows a measurement where the first crystal plate was magnetically opened every two seconds and the second crystal plate was opened after six back and forth reflections.

Fig. 40. Response of the active perfect crystal storage system after opening of the second crystal at the sixth back and forth reflection.

The intensity observed after opening the second crystal is about 1600 times higher than in the case without magnetic switching (see Fig. 41).

Fig. 41. Response of the passive perfect crystal storage system.

Small satellite peaks are observed for the third, fourth, fifth and seventh back and forth reflection which are caused by a finite transmittance of the crystal in the closed state. Three further experiments have been proposed by the authors to study this storage device in more detail. The device could be filled with about 25 times more neutrons by storing one burst after the other. Multiplexing the entrance and exit crystals is a possibility to enlarge the wavelength band of the stored neutrons. To store polarized neutrons a periodic spin flipper between the crystals could be used. Further experiments have been performed recently [5.21].

6 -Future Prospects and Outlook

a) BS Instruments under Construction

In previous sections we have described new BS instruments which are under construction at the FRMII, at PSI and at SNS. The latter one should provide a boost in intensity of one order of magnitude at a similar resolution of existing instruments or a boost in energy resolution of one order of magnitude with an intensity similar to existing instruments. This will be a big step forward and will open new fields. This progress is coming from the fact that the SNS source will have a peak flux at least one order of magnitude higher than existing pulsed sources. Remember that neutron scattering is a flux limited technique!

The instruments at the FRMII and at PSI will try to compete with existing instruments. Their problem is the source one which they are installed. The flux of these sources will not exceed those of existing comparable sources.

BS spectrometers on the ESS would have been the obvious choice. But the ESS project has been pushed away into some uncertain future. What a pity!

There are plans at the ILL to rebuild IN16 with the aim to gain a factor of 5 in flux by putting the instrument onto a better neutron beam and by using phase space transformations. Remember the the ILL reactor still has and will have in the foreseeable future the highest continuous cold flux in the world!

b) Technical Developments in BS

6.1 Gradient Crystal Monochromators

One intrinsic problem of BS spectrometers is the low incident flux as a result of the narrow band width of the monochromator and analyser. Sometimes it would be very useful to have a set of monochromators for a BS spectrometer only differing in their values, large yielding high intensity at the expense of energy resolution. In real crystals can be varied in several ways like by (i) the application of a temperature gradient parallel to the neutron wave vector or (ii) by a concentration gradient in mixed crystals. The first method has been investigated by Alefeld [6.1] and it was later used by Heidemann et al [4.128] on the BS spectrometer IN13. The second method can in principle be applied to mixed Si-Ge crystals. Si and Ge have the same crystal structure with lattice parameters differing by 4 % and are miscible over the full concentration range. The difficulty however arises in the growth of single crystals for which one method has been proposed by Magerl et al [6.2]. In principle it should also be possible to produce Si-Ge crystals with a lattice parameter gradient. However a number of technical problems have still to be overcome before a production of these crystals will be possible.

A third method to vary of a monochromator is the application of ultrasonic vibrations to the crystal. Two effects come into play :

It turns out that for cold neutrons in contrast to X-rays the second effect dominates over the first one. It has been demonstrated experimentally [6.13] on the backscattering spectrometer IN10, that an increase up to a factor of 16 of the intensity backscattered from a perfect silicon (111) oriented crystal as monochromator can be achieved easily in this way, accompanied by a corresponding change of energy resolution from 0.3 µeV to about 5 µeV (FWHM). The advantage of this method compared to static gradients in mixed crystals is that one can tune the energy resolution and intensity within a rather large range just by varying the power of the ultrasound generator.

A X-X BS spectrometer is an instrument matched to a continuous neutron source, a TOF-X BS spectrometer one matched to a pulsed neutron source with short pulses. Schelten [6.3] proposed a new type of X-X BS spectrometer called 'Musical' matched to the time structure of a pulsed neutron source with long pulses.

In order to make full use of the peak flux of a pulsed source, the monochromator consists of several crystals with the same orientation but slightly different lattice spacings mounted behind each other at a distance s ('multiplexing monochromator'). Each crystal reflects a slightly different wavelength. Due to the path length differences the differently 'coloured' neutrons arrive at different times at the sample.

The gain factor G of a 'Musical' backscattering spectrometer compared to a conventional one on a pulsed source is equal to the number n of crystals of the multimonochromator. In order to achieve a clear separation in time the distance s between two adjacent monochromator crystals has to be  where v is the neutron velocity and  the pulse length of the source. The optimum number of monochromator crystals is given by n_opt = T/2 . T is the period of the pulsed source. The factor 2 occurs because of the necessity of discrimination between the directly scattered neutrons and those scattered indirectly from the analyser. In order that the neutrons backscattered from the different monochromators do not overlap energetically, the energy difference between neutrons reflected from two nearest neighbor monochromators should be larger than the intrinsic full width at half maximum  of the Bragg reflection of one crystal. This energy difference can be produced either by the use of thermal expansion or by doping, thereby achieving different lattice parameters. For perfect Si (111) crystals the temperature difference between two adjacent crystals would have to be equal or larger than 9°C ( DEext = 77 neV, see table 1). The total energy band width covered by the multi-monochromator  nDEext is much smaller than the energy range of a BS spectrometer. Therefore the center of gravity of this energy band has to be scanned either via the Doppler effect or via thermal expansion.

6.3 Ultrahigh Resolution BS Monochromators

As we have seen in chapter 2, the energy resolution of a BS spectrometer is determined by the relation

                (50)

which is valid for both the primary and secondary spectrometer. Both contribute to the total resolution via a convolution. In an optimized intensity/resolution layout the first and second term of Eq. 50 are matched. For Si (111) crystals the primary extinction term yields DEext = 77 neV( see table 1). The geometry term of the spectrometer IN16 is 110 neV assuming a beam and sample size of 3 x 3 cm².

Alefeld and Springer recently investigated GaAs crystals on a BS diffractometer [2.58]. The intrinsic theoretical line width of the (200) reflection (2d₂₀₀ = 5.86 Å) is  8 neV, see table 1), i.e. 10 times narrower than that of Si (111). The measured deconvoluted linewidth obtained by Alefeld et al [2.58] was about 25 neV. Very similar experiments on GaAs performed by Liss et al [6.5] yielded a deconvoluted line width of about 15 neV, only a factor two larger than the theoretical value. The additional experimentally observed line width may be caused by several factors such as the crystal alignment errors, the temperature instability or the crystal imperfection.

It has been proposed [6.8] to use GaAs (200) crystals as monochromator and analyzers for a BS spectrometer with a geometry similar to that of IN10C. However in order to match the geometry-related line width to the one caused by primary extinction, a small sample size of only 1 x 1 cm2 (instead of 3 x 3 cm2 for Si (111)) can be used. The small sample size together with an intrinsic line width of GaAs 10 times narrower than that of Si would result in an intensity loss factor of about two orders of magnitude! There are however possibilities to overcome this problem. We mention three ideas:

Fig. 44a Correlation between direction and magnitude of the wave vector of neutrons backscattered from a perfect crystal.

Using this trick one could in principle obtain an energy resolution in the best possible case of DEext times a convolution factor f. This latter is 2 for Lorentzians, 1.4 for Gaussians and about 1.2 for Ewald or Drawin curves. Therefore one could hope to achieve a resolution of about 100neV with Si111 and 10 neV with GaAs200. However one must remember that those values can only be reached by using perfect crystals in a perfect focussing geometry. Tests on IN16 with Si111 reached so far an energy resolution of about 200 neV in the best case.

7. Conclusions

    We have explained the principle and applications of backscattering of neutrons (and X-rays) from crystals. We have described the different kinds of backscattering diffractometers and spectrometers. We have seen that neutron backscattering has opened a new field, the µeV- spectroscopy, in inelastic neutron scattering with many different applications ranging from hyperfine interactions to rotational tunneling.

The neutron backscattering technique is now very well established after more than 30 years of experience and experiments. It is a pleasure for those who were lucky to start to work in a totally unexplored field of neutron scattering in the sixties to see how this field has expanded since in the technique and in the applications. The jung thesis students at the FRMI are now retired. They look with amazement at the avalanche the idea of Maier-Leibnitz which they put into realization has created.

APPENDIX A

A short comparison between neutron backscattering and neutron spin echo spectroscopy

The figure below shows the time-distance scales probed by many different techniques. It includes the domain of neutron scattering and shows the areas covered by BS and NSE spectroscopy. The latter techniques overlap but NSE is penetrating well into the nanoeV region, whereas the BS technique works mainly in the meV range. They are also complementary to each other: The domain of NSE is mainly quasielastic coherent scattering, the domain of BS is mainly quasielastic and inelastic incoherent scattering (see second diagram below ( also obtained from R.Pynn)). For more information about NSE please visit the ILL web sites:

http://wwwold.ill.fr/YellowBook/IN11/

http://wwwold.ill.fr/YellowBook/IN15/

Reference: R.Pynn, private communication

Reference: R.Pynn, private communication

The figure above shows a typical tunneling spectrum of CH3 groups measured on IN10 and its Fourier transformation into the time domain. Obviously a time domain up to a maximum of 10 nanoseconds is available on IN10 in its highest resolution mode.