Never put off till tomorrow what you can do the day after tomorrow. (Mark Twain)

The Backscattering Story: a personal view

Backscattering History: Talk at the ILL in December 2013

1.  Introduction

2.  Theory

2.1  Introduction

2.2  Backscattering from Perfect Crystals

2.3  Backscattering Geometry and Energy Variation

2.4  Principal Layout of Backscattering Diffractometers

2.5  Principal Layout of Backscattering Spectrometers

2.6  Resolution Considerations

2.7  Instrument Simulations


3.  Instruments

3.1  Backscattering Spectrometers

3.1.11 Fixed Window Scan

3.2   TOF Backscattering Spectrometers

3.3  Backscattering Diffractometers


4.  Applications of Neutron Backscattering Spectroscopy

4.1  Introduction

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

4.13  Quantum Liquids


5.  Special Applications

5.1  Precision Measurement of h/m

5.2  Backscattering and Polarisation

5.3  Neutron Magnetic Resonance Shift

5.4  A Perfect Crystal Storage Device

5.5 Antibunching of Neutrons


6.  Future Prospects and Outlook

6.1  BS Instruments under Construction

6.2  Technical Developments in BS


7.  Conclusion


8. Appendix

NSE

History of Neutron Guides



1. Introduction

Neutron backscattering spectroscopy (BS) has been proposed by Maier-Leibnitz 50 years ago. The basic idea is to use Bragg angles of near 90° with moderate collimation for beam monochromatisation and analysis in order to obtain very high energy resolution. The first BS experiments were carried out at the Munich research reactor FRM in 1969. This method improved the energy resolution of neutron spectrometers by about two orders of magnitude, pushing it into the µeV range. The prototype BS spectrometer built at the FRM yielded an energy resolution of 0.6 µeV (FWHM). For the first time the hyperfine splitting of the vanadium nucleus in magnetically ordered V2O3 could be measured with neutron scattering. Since these early experiments BS spectroscopy has substantially evolved. BS spectrometers have been developed and used in Jülich (Germany), at the ILL (France) and more recently at other neutron scattering centers at reactors and spallation sources.

In this review we shall explain the BS technique, describe BS instruments, give an overview of applications in the field of condensed matter research and more specialized fields and finally present an outlook into future developments in BS spectroscopy. Section 2 is devoted to some basic considerations about energy resolution and flux. In section 3 we present BS instruments including a short description of X-ray BS instruments. Some common applications of BS spectroscopy are outlined in section 4. Special applications and future developments are described in sections 5 and 6.

A short comparison between neutron backscattering and neutron spin echo spectroscopy which both are very high energy resolution techniques can be found in appendix A


2. Theory

2.1 Introduction

The main idea of backscattering consists in using a Bragg angle  close to 90° for the selection and for the analysis of the wavelength λ  of neutrons or X rays in a spectrometer or diffractometer [2.1 - 2.6]. Under this condition the Bragg reflected wavelength band Δλ becomes very narrow. This can be seen easily by differentiating the Bragg equation

                     (1)

        (2)

where d is the lattice spacing and ΔΘ  is the angular divergence of the beam.  represents the relative width of the wavelength band for an infinitely sharp collimated beam. is caused by lattice strains, primary and secondary extinction. It follows from Eq. 2 that for θ  = 90° the second term in Eq. 2 becomes zero, or in other words  becomes, in first order, independent of the beam divergence ΔΘ . In order to investigate the resolution in more detail near θ  = 90° it is convenient to look at the Bragg equation in reciprocal space ( see Fig.1).

 

Fig. 1. Backscattering geometry in reciprocal space near θ  = 90°.


  is a reciprocal lattice vector and

Defining Δk as the difference between the longest and shortest k vector caused by a divergent beam and  extinction we find:

          (3)

where ε = 90° - θ. For small values of we can expand Eq. 3 to  first order, so that
                      (4)

For exact backscattering we obtain

                                  (5)

Therefore in this special case the angular divergence contributes only in second order and we can rewrite Eq. 2 for the case of near backscattering

    (6)


2.2 Backscattering from perfect crystals

It has been shown by Darwin [2.7] and Ewald [2.8] that even for perfect crystals the term has a finite value due to primary extinction. They derived functions for the reflectivity R of perfect crystals which may be represented by [2.9]
           (Ewald) (7)

    (Darwin) (8)

Both curves are plotted in Fig.2. They have a central plateau Δy  where the reflectivity is 100 % and are slightly different for y >1 [2.63] . We will show now how this plateau is related to .


Fig. 2. Ewald and Darwin curves for perfect crystals.


The variable y is given by
                          (9)

where for neutrons
                                     (10)

                                                (11)

                                              (12)

Fτ is the structure factor associated with the reciprocal lattice vector Nc the number density of unit cells, E the kinetic energy of the neutrons and  is the angle between the incident beam and the inward normal on the surface of the crystal slab,  is the angle between the reflected beam and the inward normal. V(τ) is the Fourier transform of the crystal potential :

            (13)

It follows that V(0) is given by :

                             (14)

where Nc is the number density of atoms. α and therefore y are, depending on how R is measured, linear functions of θ, λ or k :

where                                 (18)
Therefore we obtain for the width Δy of the plateau of the reflection curves:

We note that only in the case in which one measures R as a function of k, the width of the reflection curve is independent of the Bragg angle ΘB and depends only on the crystal properties :

                                      (22)

One can call the "radial" mosaic distribution caused by primary extinction in analogy to the angular mosaic spread η in imperfect crystals. This is shown in Fig. 3 where we depicted the influence of primary extinction on the Bragg reflection plotted in reciprocal space.


Fig. 3. Effect of primary extinction on the Bragg reflection shown in reciprocal space


The neutron energy width ΔEext resulting from the term only neglecting correction factors ( FWHM or convolution factors: For more details click here.) is given by:
       (23)

Thus ΔEext is independent of τ and depends only on the structure factor Fτ and on Nc. Table 1 shows values of and ΔEext for a few crystal planes. Obviously the values of ΔEext are in the range below 1µeV and therefore one can neglect the influence of and ΔEext on the energy resolution of conventional crystal spectrometers using Bragg angles far away from 90° where the resolution  is dominated by the geometry term ctg ΘΔΘ.

Table 1

Crystal plane

ΔEext (μeV)

λ (Å)
for Θ = 90°

Si (111)

1.86.10-5

0.077

6.2708

Si (311)

0.51.10-5

0.077

3.2748

Ca F2 (111)

1.52.10-5

0.063

6.307

Ca F2 (422)

0.54.10-5

0.177

2.23

Ga As (400)

0.75.10-5

0.153

2.8269

Ga As (200)

0.157.10-5

0.008

5.6537

Graphite (002)

12.10.10-5

0.44

6.70

For Bragg angles near 90° however we cannot neglect  .

A few remarks to the longitudonal and transversal coherence lengths:

The longitudonal coherence length is given by:

Dx = 1/Dk = Dt/t *k = 5,4 mm for Si(111) (see table 1).

The transversal coherence length is given by:

 

We now consider the special case of a perfect crystal in exact backscattering at the end of a neutron guide. The theoretical beam divergence at the end of the guide is                                 (24)

Therefore from Eqs. 5 and 23
  (25)

The index g stands for neutron guide. In this special case the energy width DE is independent of the energy and depends only on the neutron guide material and on the crystal quantities. Using as an example the values for a nickel coated neutron guide and a perfect silicon crystal with the (111) planes in backscattering we get

ΔE= (0.24 + 0.08) µeV

In this case the beam divergence contributes to ΔE three times more than primary extinction.

In an optimized case one would match the contributions to ΔE from the divergence and from primary extinction i.e. (see Eq. 5)

       (26)

or

     (27)

From Eq. 27 we can calculate the optimum beam divergence in the case of exact backscattering as a function of the primary extinction . Fig. 4 represents Eq. 27 in reciprocal space and shows the dependence of  on .


Fig. 4


Finally let us calculate the flux of backscattered neutrons we would obtain for the case of a perfect silicon (111) crystal at the end of a nickel coated straight neutron guide assuming no losses.

One can show that the intensity of a neutron beam produced by a reactor with a moderator in thermal equilibrium is given by [2.10]         (28)
where Φ is the total thermal flux of the reactor and kT
is the "thermal" wave vector:
                                                  (29)
where T is the temperature of the moderator. We calculate I for two special cases :

1. For the first Munich research reactor FRM without a cold source we have:


2. For the high flux reactor HFR of the ILL with a cold source we have:

From these flux estimations it is obvious that it is extremely difficult to perform backscattering spectroscopy at the FRM because of the low flux. However at the HFR  inelastic high resolution experiments are certainly feasible.

For the sake of completeness we mention that backscattering from imperfect crystals has been considered by Hiismäki [2.11].


2.3. Backscattering Geometry and Energy Variation

The Bragg equation shows that the reflected wavelength can be varied either by a change of the Bragg angle Θ or by a change of the lattice spacing d. Obviously only the second method can be used in the backscattering case. The lattice spacing of a given crystal can be varied in several ways which will be explained in the following.

2.3.1. Doppler Effect

Neutron reflection from moving lattices has been treated in the past in several papers [2.12-2.16]. The energy and wavevector conservation laws allow us to calculate the relevant effects :

                            (30)

                                  (31)

where Ef and  (Ei and ) are the final (initial) neutron energy and velocity,  is the Doppler velocity. For the special case of backscattering from a crystal moving parallel or antiparallel to the neutron beam direction, i.e. parallel to we get in first order (VD << V)

    (32)

The energy change δE of backscattered neutrons is therefore in first order linear in the Doppler velocity. Doppler velocities of about 2.3 m/s, corresponding to energy changes of ±15 µeV for 6.3 Å neutrons were obtained easily on spectrometers like IN10. Today about 5 m/s are reached with mechanical and linear motor drives.

A chapter dedicated to the evolution of velocity drives in BS spectroscopy can be opened here.

2.3.2. Thermal Expansion 

Lattice parameter variation by temperature changes is another means to vary the reflected neutron energy :

                      (33)


with d(T) = do (1 + ßo T + ß1 T2 + ...)     (34)

Table 6 shows relative energy variations for several crystals for a temperature variation between 80K and 700K. Energy variations up to a few percent are possible for crystals with high thermal expansion.


 

 

Monochromator

 

 

Structure

 

ao(Å)

Lattice plane

(hkl)

 

Tm(K)

(meV)

(meV)

(meV)


%

KCl

NaCl

6.29294

200

1040

-15

80

0.113

2.71x10-5

4.60

CaF2

CaF2

5.462

111

1696

0

70

0.063

1.52x10-5

3.36

BaF2

CaF2

6.196

200

1628

-65

- 10

0.051

1.19x10-5

2.58

AgCl

NaCl

5.547

111

728

60

140

0.047

1.16x10-5

4.01

NaI

NaCl

6.4728

200

934

100

190

0.067

1.71x10-5

4.61

NaCl

NaCl

5.62799

111

1074

120

210

0.074

1.92x10-5

5.03

NaBr

NaCl

5.97324

200

1020

- 240

-140

0.100

2.18x10-5

4.36

AgBr

NaCl

5.7745

200

705

-400

-280

0.145

2.82x10-5

4.89

NaCl

NaCl

5.62799

200

1074

-530

-400

0.157

3.03x10-5

5.03

TIBr

CsCl

3.97

110

753

-545

-400

0.130

2.5x10-5

5.59

AgCl

NaCl

5.547

200

728

-610

-520

0.195

3.67x10-5

3.38

CaF2

CaF2

5.462

200

1696

-690

-600

0.078

1.42x10-5

3.36

TICl

CsCl

3.834

110

703

-730

-620

0.173

3.12x10-5

3.95

NaF

NaCl

4.62

111

1266

-815

-700

0.042

0.73x10-5

4.00

RbI

NaCl

7.342

220

9 20

-1050

-850

0.056

0.93x10-5

6.59

Si

Diamond

5.4306

111

1683

- 2

      4

0.077

1.86x10-5

0.29

Table 6. Possible monochromators for cold neutron BS spectrometers. Tm is the melting point,  and  the minimum and maximum energy transfer using a silicon (111) analyser at room temperature and a temperature scan from 80K to 700K for the monochromator.  is the corresponding relative energy change of neutrons. One of the best examples of the use of this technique on IN10 can be found by clicking here. A chapter dedicated to monochromator temperature scan devices can be opened by clicking here.



2.4. Principal Layout of Backscattering Diffractometers

It was recognized early on [2.17] that backscattering can be used for high accuracy lattice parameter measurements. The principle is explained in Fig. 5.


Fig. 5

A white neutron beam penetrates the sample crystal which is adjusted with the lattice planes perpendicular to the beam. For the wavelength λ = 2d an intensity dip in the white spectrum is generated. For perfect crystals the width of the dip is given by (see Eq. 22). The spectrum is analyzed by a reference crystal, again in back reflection. The reference crystal is moved back and forth by a velocity drive. If the backscattered intensity is plotted against the Doppler velocity, the intensity dip is observed for

            (35)

where dd is the difference between the lattice parameters of the sample crystal and the reference crystal,  the Doppler velocity for the intensity dip, b is the adjustment error of nonparallelity between the two crystals. The precision of this method is determined by the quality of the crystals and the accuracy of the adjustment.

2.5. Principal Layout of Backscattering Spectrometers

Two different principal layouts of backscattering spectrometers to measure quasielastic or inelastic neutron scattering can be distinguished. In the first type backscattering from crystals is used in both the primary (monochromator) and secondary (analyser) spectrometer. We call these instruments X-X-type backscattering spectrometers. In the second type, backscattering is used only for the analyser and the monochromatisation is determined by time of flight (TOF).  We call these instruments TOF-X-spectrometers.

2.5.1. X-X-Backscattering Spectrometers

The principal layout of this instrument which was proposed originally by Maier-Leibnitz is shown in Fig. 6.


Fig. 6. Principal layout of a X-X-backscattering spectrometer.


The spectrometer is similar to a three axis spectrometer with Bragg angles equal or near to 90°. The energy scan is performed by varying the incident energy via the lattice spacing of the monochromator as described above. The chopper is needed to discriminate between neutrons scattered by the sample directly into the detectors and those backscattered from the analyser. The chopper can be avoided only if one uses for the analyser a Bragg angle θ  significantly smaller than 90°, i.e. 90° - Θ >> ΔΘ .



2.5.2. TOF-X-Backscattering Spectrometers

The principal layout of this instrument is sketched in Fig. 7.


Fig. 7

Principal layout of a TOF-X-backscattering spectrometer.


A pulsed, white neutron beam travels down a very long neutron guide and hits the sample. The scattered neutrons are analyzed for energy (as a function of time of flight) and momentum change by a backscattering analyser crystals. The backscattered neutrons are detected with detectors near to the sample.



2.6. Resolution Considerations

A backscattering spectrometer is in principle a very special three axis spectrometer. Therefore one could think that energy resolution calculations which have been published for three axis spectrometers [2.18, 2.19] should also be valid in backscattering spectrometers. This is however in general not the case for several reasons :

1. The calculations are done for mosaic crystals and not for perfect crystals: The term is neglected.

2. The calculations are valid only for the non backscattering case and not applicable for exact backscattering.

However Pynn [2.20] has calculated the resolution function of a perfect-crystal three axis X-ray spectrometer. Popovici [2.21] and later Grimm [2.22] have included the term in extended Cooper-Nathans calculations for three axis spectrometers with imperfect crystals. The backscattering case has only be treated by Birr et al. [2.4] in a simplified way yielding expressions for the energy resolution as given by Eqs. 6 and 22. This approach is however justified by the following facts valid for backscattering spectrometers.

1. The variation of the incident energy is performed either by the Doppler effect or by lattice expansion of the monochromator crystal. This means that the geometry of the instrument remains unchanged during a measurement and therefore that the energy resolution is independent of the energy transfer.

2. Perfect crystal backscattering analyzers are normally spherically curved (see section 3.1) and geometrically arranged such that the center of the analyser sphere coincides with the sample and detector position. In this manner horizontal and vertical spatial focusing and energy focusing are achieved simultaneously. Therefore on the analyser side it is sufficient to calculate the energy resolution for one small perfect crystal, i.e. again using Eqs. 6 and 22 with ΔΘ properly calculated from the effects of crystal and sample size.

A special feature of backscattering is the fact that the horizontal and vertical collimation have the same influence on the energy resolution in contrast to three axis spectrometers. Therefore ΔΘ in Eq. 6 consists of two contributions: ΔΘ horizontal and ΔΘvertical, which have to be added together either linearly or quadratically for Lorentzian or Gaussian shaped resolution curves. For example :

(36)

instead of Eq. 6

 

2.7. Instrument Simulations

 

Monte Carlo methods can be used with advantage to simulate the performance of neutron spectrometers. This had been done by Kraxenberger for the backscattering spectrometer IN13 [2.23] 25 years ago. He calculated the distribution of incident and scattered neutrons in momentum space. Neutron beam defining elements like the neutron guide, the monochromator and analyser crystals (with specified values of  and mosaic spread h ), the deflector and entrance and exist slits were simulated by program subroutines. A close agreement was found between the calculations and the resolution measurements.

Various program packages for instrument simulations based on the Monte Carlo technique have been developed in recent years. We mention only two of them:

VITESS has been initiated by a group at HMI, McStas by a group at Risø. These programs have been used to simulate backscattering instruments which were developed in recent years and on older instruments to check if these instruments could be improved. More information can be found below were the instruments are described.


 

3. Backscattering Instruments

3.1. X-X Backscattering Spectrometers

In the following we will describe neutron X-X backscattering spectrometers including an X-ray spectrometer which had been constructed in the past or which are today operational or under construction. Fig. 8 shows a comparison of four different layouts used on these instruments.


Fig. 8. Four different possibilities of X-X-backscattering spectrometers for neutrons.


The first of these has already been mentioned in section 2.5.1. Its drawback resides in the fact that the sample and detector are both very close to the white primary neutron beam.

The second version avoids this problem by using a so-called deflector crystal which allows the separation of the highly monochromatic beam from the white beam. Its drawback is that it - as the first version - requires an end position and its primary energy resolution is dependent on the beam divergency offered by the neutron guide.

The third version which uses two deflector crystals, avoids the above mentioned problems :

1. No transparent monochromator is needed.

2. The energy resolution of the primary spectrometer is independent of the neutron guide divergence but depends on the divergence defined by the size of the beam at the second deflector and the distance between the latter and the backscattering monochromator which is spherically curved.

Its drawback is its higher complexity and additional loss of intensity.

The fourth version represents a backscattering option on a standard three axis spectrometer. The three axis monochromator and analyser are replaced each by a pair of crystals, one being the backscattering crystal, the other the deflector crystal. In this way µeV resolution is achieved on a three axis spectrometer (TAS) whilst maintaining the advantage of flexibility and the good Q-resolution of a TAS instrument. The disadvantages of this version compared to a normal TAS spectrometer are the low intensity (due to the high energy resolution) and the limited energy scan range.



3.1.1  Matching of the Q-Resolution on X-X BS Spectrometers by phase space transformation

A characteristic property of the backscattering spectrometers is the relatively poor Q-resolution which results from the necessity of using large solid angle analyzers in the secondary part of the instrument to obtain more intensity. However on the primary side of the instrument, there is considerably less solid angle subtended due to the small divergence of the neutron beam coming from a neutron guide. In principle a gain of two orders of magnitude could be obtained by matching the incident beam divergence to that in the secondary spectrometer. The new spectrometers IN16 and BSS1 are equipped with supermirror compressors with an increase of the primary solid angle by a factor of about four. A further increase using this technique is difficult. In the following a method invented by Schelten and Alefeld [6.4] is described which increases the primary solid angle at least in one direction, leading to an intensity gain of about 5. The principle, often called Phase Space Transformation (PST), is shown in Figs. 42 and 43.


Fig. 42. Bragg reflection of neutrons from a mosaic crystal moving with a velocity Vk perpendicular to the reciprocal lattice vector.


Fig. 43. Sketch of a BS spectrometer with a matched Q resolution.


The neutrons are reflected by a mosaic crystal which moves with a velocity Vk parallel to the reflecting lattice planes. In this way a cigar shaped momentum space element of small divergence δk and large k-vector distribution Δk is transformed into a concavely curved one with divergence Δk' and k-distrbution δk'. Such a phase space transformation can be characterized by the term  'from white' to 'wide'. This divergent beam then impinges under backreflection upon silicon monochromator crystals arranged on a spherical surface. The backscattered neutrons are focused on to the sample. In terms of intensity a gain of the order of Δk'/δk is achieved.

The phase space transformation can best be illustrated by a diagram (see Fig. 42). The slope of the momentum element after diffraction is given by

           (49)

in which Θ0 is the Bragg angle in the laboratory system and θ that in the crystal system which is determined by the crystal velocity.

              (50)

χ can be chosen such that the momentum space element after deflection is perpendicular to wave vector k', which means that:   .

Then from Eq. 50 it follows that

tgΘ = - ctg(2Θ0)       (51)

and from (50) and (51) the appropriate crystal velocity can be calculated, which for a moving graphite(002) deflector combined with a Si(111) backscattering crystal is of the order of 300 m/s.
This configuration is used on the BS spectrometers HFBS at NIST, SPHERES at MLZ and for the IN16B at ILL.
A computer simulation of this phase space transformation can be seen by clicking here.

 



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). The flux at the sample position was very small: 30 neutrons/s cm**2. The beam size was 14*9 mm**2. 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:

1. Measurements could be performed only by integrating over a large Q-range with an average Q-value of 1.4 Å-1.

2. The energy range was limited to  ± 4 µeV.

the Rückstreu-Spektrometer at the FRM


The crystal analyzer of the instrument

Two experiments were carried out successfully in a completely new energy transfer window with this spectrometer :
    1. Quasi-elastic scattering  in viscous Glycerol.
    2. Nuclear spin excitations in V2O3.
 With this experimental breakthrough the term ' microeV'  was coined in neutron scattering.

 The Munich BS instrument was dismantled  shortly after these experiments. But based on the new experience gained, later  more powerful BS spectrometers on reactors with much higher cold neutron flux were built and used. Many papers have been published since using  BS spectrometers despite the fact that 35 years ago there had been a lot of skepticism about the usefulness of this new technique. But Maier-Leibnitz was right when he said  that  a new experimental technique which allows to gain more than one order of  magnitude on some factor will open access to new fields in science.

Temperature Control


Background Chopper


 

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.The flux was 3000 neutrons/s cm**2 with the unpolished Si monochromator.The beam size was 3*3 cm**2.



3.1.4. The BS Spectrometer IN10 at the HFR Grenoble

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. ηvertical = 0.4°, ηhorizontal = 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. 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.

The flux at the sample position was 2*10**4 neutrons/s cm**2 with the unpolished monochromator. Beam size: 3*3 cm**2.

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.


The secondary spectrometer of IN10 with A. Magerl



IN10 Analyser Plates


Option 'IN10B' of   'Off-set' Monochromators on IN10:


Instead of using the Doppler effect for the energy scan thermal expansion of the monochromator could also be used on IN10.
IN10 has been taken out of the user program in 2012 and is used for the developpement of new BS technics.




3.1.5  The BS Spectrometer IN13 at the HFR Grenoble

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 2016-2002, 2004-1989, 1989-1982 and 1984-1977 to find a complete list of publications of experiments performed with IN13.



'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




3.1.6  The BS Spectrometer BSS at the FRJ-2 Jülich

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 (ηv = 0.4°, ηh = 1.2°) is obtained by mounting 3 graphite crystals behind each other with orientations differing in the horizontal plane by ηv. 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). The instrument is decommissioned.



3.1.7  The BS Spectrometer IN16 at the HFR Grenoble

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 SHERES). 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  . Get QuickTime

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:

1. Exact backscattering at the monochromator,

2. No losses in flux caused by the geometry ,

3. Energy resolution of the primary spectrometer independent of the neutron guide beam divergence.

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

A rather complete list of publications can found by clicking on 2016- 2006, 2004-1996 and 1996-1992.


IN16 Analyser Plates

IN16 has been replaced by IN16B in 2012.

 

3.1.7b  The new BS Spectrometer IN16B at the HFR Grenoble

Inauguration of IN16B in 2013


AEROLAS Doppler drive of IN16B

Recent publications from IN16B


3.1.8  The HFBS Backscattering Spectrometer at NIST

HFBS is the first backscattering instrument that used 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.

HFBS layout and photo

Sketchnotes about the HFBS by Rob Dimeo

 

3.1.9  The Backscattering Spectrometer at the FRMII: SPHERES

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
of SPHERES




Resolution function of SPHERES


3.1.10 . The Three Axis (TAS) 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 J.Axe, D.Moncton and L.Passell [unpublished] 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. Larry Lassell called this instrument the 'poor man's backscattering spectrometer'.

During my (A.H.) 6 month's sabbatical in the institute of Gen Shirane in 1984/1985 at the HFBR I had to pleasure to work on this project with John Axe and especially with Larry Passell.

sketch of the TAS BS spectrometer

The instrument 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 or analyser backscattering crystal. 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 :

1. Small incoherent and absorption cross section.

2. Need of single crystals: Pyrolytic graphite cannot be used because of the disorder in the a-b plane

For the analyser the problem is less critical because its deflector is hit only by monochromatic neutrons. The use of neutron filters before the monochromator can also reduce the background problem.

To demonstrate the method we choose two differents sets of TAS BS monochromators:

1. Al2O3 (222 orientation) crystals as BS monochromators ( incident energy: 4.2027 meV) combined with a deformed ( by deuterium loading and unloading) Niobium (110 orientation) crystal as monochromator deflector and pyrolytic graphite as analyser deflector. A Be filter in front of the monochrotor was necessary to reduce the background produced by the Niobium deflector and the second order contamination from the Al2O3 crystal. The energy resolution achieved was of the orger of 1.5 microeV.

2. CaF2 crystals ( provided by the company Minhorst in Germany) with (422) orientation as backscattering crystals ( incident energy: 16.3 meV). For the monochromator deflector we choose a silicon crystal with (331) orientation ( no second order contamination) and an anisotropic mosaic spread (ηv  = 5', ηh =  20 ′). These crystals had been produced by the monochromator group of A.Freund at the ILL.The analyser deflector choosen was a pyrolytic graphite with  η =  20′. The energy resolution achieved was of the order of 5μeV (FWHM). The analyser BS crystal was mounted in a special nitrogen cryofurnace developped at BNL. The temperature of the CaF2 crystal could be scanned between 77 K and 500 K, yielding an energy transfer range of about ± 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. These spectrometer options were never used to perform a real scientific experiment, unfortunately. Reasons for this were twofold: 1. no priority had been given to the use of this TAS option. 2. The HFBR ended its operation in 1991.

 

TAS BS Cyofurnace

Larry Passell and one of the authors aligning the BS TAS monochromator at the HFBR on H7

The Brookhaven BS TAS monochromator






 

3.1.10a Backscattering Spectroscopy at Ansto: EMU

 

3.1.10b BATS and GaAs for IN16B


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:

1. Exact backscattering is not feasible with X-rays (having the velocity of light) in contrast to neutrons,

2. The absolute energy resolution in the case of X-ray backscattering from perfect crystals neglecting the geometry term is given by:

            (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: 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 BASIS




  

3.2.3 TOF-X Spectrometer OSIRIS

3.2.4 The TOF-X Spectrometer MARS at PSI

3.2.5 Project Miracles at ESS

 

3.2.6 Fires at ISIS

 

3.2.7 The Performance of TOF near Backscattering Spectrometer DNA in MLF, J-PARC

 

3.2.8 BATS for IN16B


3.3. Backscattering Diffractometers

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:

1. Second order phase transitions.

    Examples: SrTiO3, KMn F3,Ni[2.17, 2.48 2.51, 2.55].

2. First order phase transitions at the melting point.

    Examples: Na, K [2.50, 2.51, 2.54].

3. Order-disorder phase transition in ice [2.53].

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

4.1. Introduction

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.

A talk reviewing the applications of backscattering spectroscopy has been presented by Dieter Richter at the ILL in 2002.



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)

This energy change is transferred to the scattered neutron. The double differential scattering cross-section in the zero phonon approximation is given by:

         (39)

where e-2w is the Debye-Waller factor and a' the spin incoherent scattering length.

a+ and a- are the scattering lengths for parallel and antiparallel orientation between the neutron and the nuclear spins I.

A detailed derivation of eq. 39 is given in the appendix A. Eq. 39 is valid only for the case of unpolarised neutrons and target nuclei. The latter cases are treated in appendix A.

The inelastic spin flip scattering cross section can be written in more detail assuming only one kind of nuclei:

Here the inelastic structure factor, which determines the inelstic peak intensity is given by:

The elastic structure factor is:

Examples:

1. I = 1/2:

 

M 1/2 -1/2
1/2 1 1
-1/2 1 1

 

 

 

2. I = 3/2:

M 3/2 1/2 -1/2 -3/2
3/2 9 3 0 0
1/2 3 1 4 0
-1/2 0 4 1 3
-3/2 0 0 3 9

 

 

 

 

3. I = 5/2:

M 5/2 3/2 1/2 -1/2 -3/2 -5/2
5/2 25 5 0 0 0 0
3/2 5 9 8 0 0 0
1/2 0 8 1 9 0 0
-1/2 0 0 9 1 8 0
-3/2 0 0 0 8 9 5
-5/2 0 0 0 0 5 25

 

 

 

 

 

 

1. Case: Nuclear Zeeman splitting

For ΔM we used a constant Δ 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 µk.. It is assumed that kT is much larger than the Zeeman splitting.

 Note from (39) that the inelastic and the elastic lines have the same intensity. A third of the spin-incoherent scattering cross section contributes to one inelastic line. A simulated spectrum is shown below using Gaussian shaped resolutions functions.

Fig.

Hyperfine splitting in ferromagnetic cobalt at room temperature[4.4] .

We can measure hyperfine interactions with neutrons provided that:

1. The energy resolution is equal to or narrower than the hyperfine splitting.

2. The spin dependent scattering length a' is sufficiently large to get a signal.

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. Δ is calculated for a magnetic field of 1 MOe.


 

Isotope Spin Nat Abundancy (%) b+ - b- (fm) sspin inc (barn) Methode for sspin inc sabs (barn) at 6.27 A µ(µk) Δ (µeV) for H=1 MOe quadrupole moment Q(barn) sspin inc (millibarn)
1H 1/2 99.99 58.24+/-0.02 79.9173 +/- 0.04 Difference 1.92 2.79 17.57  
79917+/-40
2H 1 0.01 8.55+/-0.07 2.0413 +/- 0.03 Difference 0.0027 0.85 2.7 0.00273
2041+/-30
3He 1/2 0.001 -3.9+/-0.5 0.4 +/-0.27 s 32000 -2.127 13.4  
400+/-270
7Li 3/2 92.6 -4.8+/-0.1 0.68+/-0.03 Pseudo-magnetism 0.26 3.26 6.86 -0.03
680+/-30
9Be 3/2 100 0.24+/-0.07 0.0014+/-0.0003 Pseudo-magnetism 0.044 -1.177 2.43 0.059
1.4+/-0.3
11B 3/2 80.4 -2.7+/-0.5 0.21+/-0.08 s 0.032 2.688 5.7 0.036
210+/-80
13C 1/2 1.1 -1.2+/-0.2 0.03+/-0.01 Pseudo-magnetism 0.008 0.7 4.4  
30+/-10
14N 1 99.6 4.2+/-0.4 0.5+/-0.1 s 11.0 0.4 1.3 0.016
500+/-100
17O 5/2 0.04 0.35+/-0.12 0.004+/-0.003 Pseudo-magnetism 1.4 -1.9 2.4 -0.0026
4+/-3
19F 1/2 100 -0.19+/-0.02 0.0009+/-0.0002 Pseudo-magnetism 0.06 2.63 16.6  
0.9+/-0.2
23Na 3/2 100 7.35+/-0.15 1.59+/-0.07 Pseudo-magnetism 3.1 2.22 4.7 0.14
1590+/-70
25Mg 5/2 10.13 3.0+/-0.2 0.27+/-0.04 Pseudo-magnetism 1.1 -0.85 1.1  
270+/-40
27 Al 5/2 100 0.55+/-0.02 0.009+/-0.0007 Pseudo-magnetism 1.3 3.63 4.6 0.149
9+/-0.7
29Si 1/2 4.7   0.15+/-0.06 Sears 0.6 -0.554 3.4  
31P 1/2 100   0.006+/-0.004 Sears 1.0 1.13 7.1  
33S 3/2 0.76   0.28+/-0.28 Sears 3.1 0.64 1.3 -0.064
35Cl 3/2 75.5 12.5+/-0.4 4.6+/-0.3 Pseudo-magnetism 255 0.82 1.7 -0.079
4600+/-300
37Cl 3/2 24.5 0.4 0.0001+/-0.0002 Sears 2.5 0.68 1.4 -0.062
39K 3/2 93.1 2.8 0.23+/60.1 Pseudo-magnetism 12.2 0.39 0.86 0.11
41K 3/2 6.7       8.5 0.21 0.43  
43Ca 7/2 0.15   1+/-1 Sears 36 1.31 1.1  
45Sc 7/2 100 -12.3 4.7+/-0.5 Difference Method 47 4.74 4.2 -0.22
47Ti 5/2 7.3 -7.0 1.5+/-0.16 s 10.0 -0.78 1.0  
49Ti 7/2 5.5   3.3+/-0.3 Sears 13.0 -1.1 1.0  
51V 7/2 99.8 12.81 5.08+/-0.01 Difference Method 28.4 5.14 4.57 -0.04
53Cr 3/2 9.6 14.1 5.86+/-0.2 s 106 -0.47 1.0
5860+/-200
55Mn 5/2 100 3.6 0.40+/-0.02 s 77 3.44 4.3 0.55
400+/-20
57Fe 1/2 2.2   0.5+/-? Sears 1.45 0.09 0.57  
59Co 5/2 100 -12.5 4.83+/-0.3 Pseudo-magnetism 220.0 4.61 4.1 0.4
61Ni 3/2 1.2   2.0+/-0.3 Sears 1.45 0.74 1.57  
63Cu 3/2 69 0.45 0.006+/-0.001 Pseudo-magnetism 26 2.29 4.85 -0.16
65Cu 3/2 31 3.7 0.40+/-0.04 Pseudo-magnetism 12.8 2.37 5.0 -0.15
400+/-40
67Zn 5/2 4.1 -3.05 0.28+/-0.03 Pseudo-magnetism 39 0.87 1.1 0.15
69Ga 3/2 60.4 -1.75 0.09+/-0.01 Pseudo-magnetism 12 2.01 4.3 0.178
71Ga 3/2 39.6 -1.69 0.084+/-0.02 Pseudo-magnetism 30 2.55 5.4 0.11
73Ge 9/2 7.8       81 -0.87 0.57 -0.2
75As 3/2 100 -1.43 0.06+/-0.01 Pseudo-magnetism 25 1.43 3 0.3
60+/-10
77Se 1/2 7.6       244 0.53 3.3  
79Br 3/2 50.5 -2.2 0.143+/-0.052 Pseudo-magnetism 64 2.1 4.4 0.33
143+/-52
81Br 3/2 49.5 1.2 0.042+/-0.021 Pseudo-magnetism 16 2.26 4.7 0.28
83Kr 9/2 11.5       1045 -0.97 0.71 0.15
85Rb 5/2 72.1   0.28 Sears 1.6 1.35 1.7 0.27
87Rb 3/2 27.8   0.28 Sears 0.7 2.74 5.7 0.13
87Sr 9/2 7.0       93 -1.089 0.7 0.2
89Y 1/2 100 2.5 0.147+/-0.012 Pseudo-magnetism 7.6 -0.137 0.85
91Zr 5/2 11.23 -2.2 0.148+/- Pseudo-magnetism 9.1 -1.303 1.7  
93Nb 9/2 100 -0.28 0.0024 Pseudo-magnetism 6.7 6.1435 4.3 -0.2
95Mo 5/2 15.7       81 0.9097 1.1 0.12
99Ru 3/2 12.7       41 -0.284 0.57  
101Ru 5/2 17.1       20 -0.69 0.86  
103Rh 1/2 100       865 -0.088 0.57  
105Pd 5/2 22.2 -5.2 0.83 Pseudo-magnetism 116 -0.639 0.86  
107Ag 1/2 51.8 2.3 0.125 Pseudo-magnetism 180 -0.113 0.71  
109Ag 1/2 48.2 -3.7 0.323 Pseudo-magnetism 505 -0.129 0.86  
111Cd 1/2 12.75       140 -0.592 3.7  
113Cd 1/2 12.3       116000 -0.62 3.9  
113In 9/2 4.3       70 5.49 3.9 1.14
115In 9/2 95.7 -4.3 0.575 Sears 1200 5.507 3.9  
115Sn 1/2 0.35       174 -0.913 5.7  
117Sn 1/2 7.6       13.3 -0.995 6.3  
119Sn 1/2 8.6       12.7 -1.041 6.6  
121Sb 5/2 57.2       34 3.3415 4.3 -0.5
123Sb 7/2 42.8       24 2.5334 2.3 -0.7
123Te 1/2 0.87       2400 -0.732 4.6  
125Te 1/2 7.0       9.0 -0.882 5.6  
127I 5/2 100 3.1+/-0.4 0.31+/-0.08 21 2.7937 145 -0.69
310+/310
129Xe 1/2 26.4       122 -0.772 4.9  
131Xe 3/2 21.2       500 0.687 1.4 -0.12
133Cs 7/2 100 2.6 0.209 Pseudo-magnetism 100 2.5642 2.3 -0.003
135Ba 3/2 6.6       34 0.8323 1.7 0.25
137Ba 3/2 11.3       30 0.9311 2.0 0.2
139La 7/2 99.9 6.1+/-0.4 1.15+/-0.15 Pseudo-magnetism 31 2.7615 2.4 0.21
1150+/-150
141Pr 5/2 100 -0.72 0.0158   40 4.09 5.1 -0.059
143Nd 7/2 12.2 56+/-7 Sears 1880 -1.063 1.0 -0.48
56000+/-7000
145Nd 7/2 8.3   5+/-5 Sears 348 -0.654 0.57 -0.25
147Sm 7/2 15   1+/-1 Sears 505 -0.807 0.71 -0.208
149Sm 7/2 14   98+/-5 Sears 240000 -0.643 0.57 0.06
147Pm 7/2     1.3+/-1.2 Sears 1000   0.7
151Eu 5/2 48   1.8+/-0.5 Sears 53000 3.463 4.4 1.16
153Eu 5/2 52   1.0+/-0.2 Sears 1870 1.529 2.0 2.9
155Gd 3/2 15       350000 -0.32 0.71 1.6
157Gd 3/2 16   85+/-20 Sears 1500000 -0.4 0.86 2.0
159Tb 3/2 100 -0.35 0.0036+/-0.0025 s 150 1.9 4.0 1.3
161Dy 5/2 19   3.3+/-0.4 Sears 3300 -0.46 0.57 1.4
163Dy 5/2 25   0.2+/-0.1 Sears 75 0.64 0.86 1.6
165Ho 7/2 100 -3.42 0.36+/-0.1 Difference Method 375 4.01 3.5 2.82
167Er 7/2 23   0.5+/-0.5 Sears 3800 -0.565 0.57 2.83
169Tm 1/2 100 2 0.094 Pseudo-magnetism 600 -0.231 1.4  
171Yb 1/2 14.3   0.5+/-0.5 s 280 0.4919 3.1  
173Yb 5/2 16   0.5+/-0.5 Sears 100 -0.678 0.86 2.8
175Lu 7/2 97       140 2.23 2.0 5.68
176Lu ? 3   1.1+/-0.3 Sears 12000 3.1 1.4 8.0
177Hf 7/2 19   0.12+/-0.12 Sears 2200 0.61 0.57 3.0
179Hf 9/2 14   0.14+/-0.05 Sears 240 -0.47 0.28 3.0
181Ta 7/2 99.99 -0.59 0.011+/-0.002 Pseudo-magnetism 75 2.34 2.1 3.0
183W 1/2 14       60 0.1126 0.86  
185Re 5/2 37       600 3.1437 4.0  
187Re 5/2 63       380 3.1759 4.0  
187Os 1/2 1.6   0.8+/- Sears 1850 0.0643 0.42  
189Os 3/2 16       140 0.65 1.4 0.8
191Ir 3/2 37       5500 0.144 0.28 1.5
193Ir 3/2 63       640 0.1568 0.28  
195Pt 1/2 34 -2.3 0.125   160 0.6004 3.9  
197Au 3/2 100 -3.8 0.43 Pseudo-magnetism 570 0.4349 0.85 0.59
199Hg 1/2 16.8   30+/-3 Sears 12000 0.4979 3.1  
201Hg 3/2 13.2       45 -0.553 1.1 0.5
203Tl 1/2 29.5 2.45+/-0.32 0.1414+/-0.036 Pseudo-magnetism 7 1.596 10
141+/-40
205Tl 1/2 70.5 -0.56 0.0074 Pseudo-magnetism 5 1.6116 10
207Pb 1/2 22.6 0.33 0.0026 Pseudo-magnetism 4 0.5842 3.7  
209Bi 9/2 100 0.52 0.0084 Pseudo-magnetism 0.12 4.0389 2.9 -0.4
Isotope Spin Nat Abundancy (%) b+ - b- (fm) sspin inc (barn) Methode for sspin inc sabs (barn) at 6.27 A µ(µk) Δ (µeV) for H=1 MOe

quadrupole moment Q(barn)

sspin inc (millibarn)

Table 4

Table 4 shows nuclei with with a spin unequal to zero. A is the natural abundance of the isotope, sinc and sabs are the spin incoherent and absorption cross-sections.

is the Zeeman splitting calculated for a field of 1 M0e:

Until now only five nuclei have been used to measure hyperfine interactions with neutrons: Vanadium, Cobalt, Hydrogen, Neodymium and Holmium ( blue color).

Possible candidates for the study of hyperfine interactions by high resolution inelastic spin-flip scattering of neutrons are marked in red color (Zeeman splitting) or green color (quadrupol splitting)..

 

Mn nuclei could be interesting candidates for the measurement of internal magnetic fields (in Heusler alloys, Mn2Au, Mn2Y, CuMnAs,...).

The experimental value of sinc is quite uncertain.

------------------------------------------------------------------------------------------------------------------------------------------------------------------

Ref.:

Revealing the properties of Mn2Au for antiferromagnetic spintronics

V.M.T.S. Barthem, C.V. Colin, H. Mayaffre, M.-H. Julien & D. Givord

Nature Communications 4, Article number: 2892 (2013)

The Zeeman splitting of the Mn nuclei is about 0.8 µeV in this compound

---------------------------------------------------------------------------------------------------------------------------

.The same is true for Cr nuclei.In addition the nuclear magnetic moment of Cr is rather small. Thus the Zeeman splitting will be also small (well below 1 µeV).

NB: Reliable values for small spin dependent neutron scattering lengths were obtained only by the method of nuclear pseudomagnetism

References for neutron scattering amplitudes and cross sections:

L.Koester, H.Rauch, E.Seymann: neutron scattering lengths

THE INTEREST OF SPIN DEPENDENT NEUTRON
NUCLEAR SCATTERING AMPLITUDES

A. Abragam, G. Bacchella, J. Coustham, H. Glattli, M. Fourmond, A.
Malinovski, P. Meriel, M. Pinot, P. Roubeau

Neutron scattering lengths and cross sections

Varley F. Sears
AECL Research, Chalk River Laboratories
Chalk River, Ontario, Canada KOJ l JO

THERMAL-NEUTRON SCATTERING LENGTHS AND CROSS
SECTIONS FOR CONDENSED-MATTER RESEARCH

Varley F. Sears

Some experimental results:

Fig. 18 shows a characteristic example of a spectrum of neutrons scattered in ferromagnetic cobalt [4.4]. Apparently the large spin dependent cross section of the Co nuclei dominates the spectrum as seen from the intensity rations of the observed peaks close to 1:1:1.

Fig. 18

 

Fig.18a: Energy spectra of neutrons scattered in amorphous CoP alloys

 

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



Fig. 19

Fig. 20 shows the temperature dependence of the internal magnetic field at the vanadium nucleus in V3 O7 [4.9].


Fig. 20

 

More recent publications concerning Nd compounds can be found by clicking here. Spectra of NdCu2 obtained at various temperatures are shown below:

 

V2O3 on IN16B ( B.Frick, 2016, unpublished)


2. Case: Nuclear quadrupole splitting

If the nuclei have a non-zero quadrupole moment Q and are on sites with an axially symmetric electric field gradient Vzz, then the nuclear ground state is split into a number of levels given by:

e is the electron charge

M = I, (I-1)...-I

The energy levels are not equidistant, the degeneracy is only partly lifted. The distance between the levels is given by:

The level scheme is more complicated than in the magnetic case except for the cases I=1 and I=3/2. For nuclei with half numbered spins I>3/2 the neutron spin flip spectrum will consist of (2I-1)/2

inelastic peaks on both sides. For nuclei with I full numbered there will be I satellites on each side. The energy distance between the inelastic lines is independent of M and constant:

For a given value of eQVzz the splitting gets smaller with increasing I.

The intensities of the inelastic lines depend on . Following the equation (39a) the intensities of neutron energy loss and gain transitions are given by:

We consider now the special case of a nucleus with I=3/2 and a quadrupole moment Q on a site with a electric field gradient Vzz and a magnetic field H. The corresponding level scheme is shown in fig.

Applying eqs.40d and 40e one obtains the following neutron energy spectrum assuming that kT is much larger than the nuclear ground state splitting ( the y-axis scale is in arbitrary units):

M 3/2 1/2 -1/2 -3/2
3/2 9/4 3 0 0
1/2 3 1/4 4 0
-1/2 0 4 1/4 3
-3/2 0 0 3 9/4

 

 

 

 

Fig.

We assumed a quadrupole splitting 4 time as large as the Zeeman splitting.The Intensity ratio is: 3:3:4:10:4:3:3 from left to right of the INS spectrum.

For the case of a pure quadrupole splitting with H=0 one obtains for the intensity ratio between the elastic line and one of the inelastic lines the value 3:1 to be compared with the value of 1:1 for the case of a pure magnetic splitting.

NB.: The intensity of one inelastic peak is driven by only 1/5 of the spin incoherent cross section.

NB.: We consider here the case of pure spin incoherent scattering from nuclei with spin I.

The same calculation for I=5/2 ( see table below) yields for the pure quadrupole split neutron spectrum ( 2 inelastic lines on both sides)the following intensity ratio:

M 5/2 3/2 1/2 -1/2 -3/2 -5/2
5/2 25/4 5 0 0 0 0
3/2 5 9/4 8 0 0 0
1/2 0 8 1/4 9 0 0
-1/2 0 0 9 1/4 8 0
-3/2 0 0 0 8 9/4 5
-5/2 0 0 0 0 5 25/4

 

 

 

 

Fig.

where the index 'inel2' refers to the outer inelastic peak. NB.: The intensity of the outer peak is driven by only 2/21 ~ 1/10of the spin incoherent cross section.

Experiments

A number of experiments have been performed in the past to try to measure the quadrupole splitting with inelastic spin flip scattering of neutrons. These experiments have not been published for the simple reason that they were not free of doubts. In the following we will describe some of them. Three nuclei have been investigated: 75As, 79Br and 127I.

Isotope Spin Nat Abundancy (%) b+ - b- (fm) sspin inc (mbarn) Methode for sspin inc sabs (barn) at 6.27 A Q(barn) Dquadrupole (µeV) Compound Method
75As 3/2 100 -1.43 60 Pseudo-magnetism 24 0.21 0.4825 As2O3 NQR
79Br 3/2 50.5 -2.2 143 Pseudo-magnetism 64 0.33 1.59 solid Br2 NQR
81Br 3/2 49.5 1.2 42 Pseudo-magnetism 16 0.28     NQR
127I 5/2 100 3.2 313 or 0 Sears 21 -0.69 1.38 and 2.68 solid I2 NQR
                     
                     

Table shows the properties of those and other possible candidates for investigation.

Intensity estimations for experiments on IN16B:

 

1. Experiment to mesure the quadrupole split INS spectrum of As2O3 (A.Heidemann, B.Frick, 2001, unpublished)

2. Quadrupole splitting in solid I2 and Br2

3.Solid Iodine on IN16B July 2016

 

4.2.3 Nuclear spin waves

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. One case is the coupling of nuclear spins in small molecules like H2 or CH4, molecular groupes like CH3 or ions like NH4+ via the Pauli principle. This is the domain of neutron tunneling spectroscopy which is described in chapter 4.7. Another case is that of nuclear spin wave excitations: The theory of inelastic neutron scattering from such a system described by the Suhl-Nakamura Hamiltonian is given in the paper by R.Word et al:: On the Detection of Nuclear Spin Waves by Inelastic Neutron Scattering :R.Word, A.Heidemann, D.Richter, Z.Physik B 28,23-30 (1977). A first experimental observation of this phenomenum has been published by Chatterji et al: Direct evidence of nuclear spin waves in Nd2CuO4 by high-resolution neutron-spin-echo spectroscopy: Tapan Chatterji, Olaf Holderer and Harald Schneider, J. Phys Condensed Matter 2013 Nov 27;27 (47):476002:

Dispersion of the nuclear spin waves in
Nd2CuO4 along [001]. The red curve shows the least-squares
fit of the dispersion data with the nuclear spin wave model
explained in the text.

------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Direct observation of low energy nuclear spin excitations in HoCrO3 by high
resolution neutron spectroscopy

T. Chatterji, N. Jalarvo, C.M.N. Kumar, Y. Xiao and Th. Bruckel

We have investigated low energy nuclear spin excitations in strongly correlated electron compound
HoCrO3. We observe clear inelastic peaks at E = 22:180:04 eV in both energy loss and gain sides.
The energy of the inelastic peaks remains constant in the temperature range 1.5 - 40 K at which they
are observed. The intensity of the inelastic peak increases at rst with increasing temperature and
then decreases at higher temperatures. The temperature dependence of the energy and intensity of
the inelastic peaks is very unusual compared to that observed in other Nd, Co and V compounds.
Huge quasielastic scattering appears at higher temperatures presumably due to the
uctuating
electronic moments of the Ho ions that get increasingly disordered at higher temperatures.

Inelastic peaks of HoCrO3 at T = 1.5, 5, 10, 20, 30
and 40 K at both energy loss and gain sides tted by convo-
luting instrumental resolution function determined from vana-
dium with two delta functions for the two inelastic peaks, one
delta function for the elastic peak and a Lorentzian function
for the quasielastic scattering. The di erent components are
shown by continuous curves of different colors.

4.2.3a Relaxation Phenomena (project)

4.2.4 Comparison with other techniques

Mössbauer Effect.A number of nuclei in Table 4 are not Mössbauer isotopes. Therefore a new class of substances is accessible with the neutron method in principle. 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/NQR 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.

Conclusion.

The methods cited above are very powerful and it is very difficult to compete with inelastis spin flip scattering of neutrons.

Still it would be interesting to detect other nuclei than H, V, Co, Nd and Ho suitable for high resolution inelastic neutron scattering.

 



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 tc. The intensity and tc 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 tc 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/tc which is plotted in Fig. 22 as a function of DT = T - Tc.


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


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 Tc, 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 tR, 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 Tc and
Tc + 12 K no line broadening was observed within an experimental upper limit of 0.08 µeV.

SrTiO3: experimental setup



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.

W.Petry has presented a talk about the glass transition at the ILL in 2003.


 

4.5 Reorientational Motions in Molecular (Plastic) Crystals

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


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 quantum rotors, 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.

The study of tunneling motions of molecules by INS is based on the use of the spin dependent neutron scattering cross section. The latter is descibed in some detail in chapter 4.2 ( Nuclear Spin Excitations).

*The theory of the study of tunneling motions of three-dimensional molecules like CH4 by INS was first published by A.Hueller: Phys. Rev. B16, 1844 (1977). The equivalent calculation for methyl groups CH3 was published by A.Heidemann et al.: Z. Phys. B-Condensed Matter-49, 123 (1982).

 


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

A compilation of high resolution inelastic neutron scattering in the domain of tunneling spectroscopy has been published by Prager et al.:
Rotational tunneling and neutron spectroscopy: A compilation, Chemical Reviews Vol. 97 pp. 2933-2966, 1997

Further Reading:

A walk through the tunneling landscape with some emphasis to instrumentation by Michael Prager

The Development of the Dilution Insert and some Tunneling Experiments by Karl Neumaier

Almost 50 years of rotational tunneling by Werner Press



4.8 Dynamics in Polymers and Biological Model Systems

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.

 

 



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

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.

W.Petry has presented a talk about the diffusion in metals and alloys at the ILL in 2003.



4.10 Atomic and Molecular Motions on Surfaces

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, Px 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.

 


 

4.12 Aerogels and Fractons

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/kBT << 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 cs 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

Introduction

We have seen that the backscattering technique with perfect cristals produces a highly monochromatic neutron beam with dl/l = 2*10-5. The corresponding longitudonal coherence length

is of the order of a few microns for cold neutrons:

Dx = 1/Dk = Dt/t *k = 5,4 mm for Si(111) (see table 1).

The beam divergence is large of the order of 1° for cold neutrons.

Special applications of the neutron backscattering technique will use this high monochromacy.

5.1 Precision Measurement of h/mn

5.2 Backscattering and Polarisation

5.3 Neutron Magnetic Resonance Shift

5.4 A Perfect Crystal Storage Device

5.5 Antibunching of Neutrons


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 vl = h/mn , we obtain d = h/2v mn. 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/mn.

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/mn.


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/mn obtained by this method is h/mn= (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 Ekin 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:

a) The observed effect can be multiplied in a multiple stage arrangement.

b) The energy splitting system may be an important component in the realization of dynamical neutron polarization as proposed by Badurek et al [5.14].

c) Cold neutrons can be slowed down to energies of the order of meV.



5.4 A perfect Crystal Storage Device

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].

5.5 Antibunching of Neutrons



6 -Future Prospects and Outlook

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 :

1. The true deformation ( )def (strains) induced by longitudinal lattice vibrations.

2. The Doppler effect : Different volume elements of the crystal move with different velocities at the same time.

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.


6.2 Multiplexing Monochromators

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 nopt = 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 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:

1. Better geometrical focusing devices allowing a beam concentration by one order of magnitude in both, x and y directions. At present such a device does not exist since it does require very high quality supermirrors.

2. Devices for phase space transformations could be used.

3. Use of the correlation between the direction and magnitude of the wave vector of neutrons backscattered from perfect crystals (see Fig. 44a) : This correlation can be maintained in a BS spectrometer if (i) between the BS monochromator and the sample there are no beam tailoring devices and if (ii) a flat sample is placed in the bisecting orientation between the incident and scattered beam. In this case the energy resolution becomes independent of the beam divergence and is only determined by primary extinction. This concept is available in principle  on IN16.


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 100 neV 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.

GaAs for IN16B


March 2017: A Record in Backscattering Energy Resolution: First Tests on IN16B with GaAs Monochromator and Analyser Crystals:



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.

 

8. Acknowledgement

We would like to thank C.Carlile for encouraging us to prepare this web site and for his valuable critical comments.

We are very grateful to the late B.Alefeld, wo has very actively participated in the preparation of the internal scientific ILL report which was the basis of this web site. In fact the web site is essentially a modern and up to date version of this older internal ILL report:

REVIEW OF THE APPLICATIONS OF BACKSCATTERING TECHNIQUES IN NEUTRON AND X-RAY SCATTERING

A. Heidemann, B. Alefeld, Internal Scientific ILL Report, N° 91 HE 22 G, 2nd edition (April 1992)


APPENDIX

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:

IN11

IN15

one chapter of

Neutron Spin Echo, Lecture Notes in Physics 128, 122 (1979), Springer Verlag:

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.