Neutron Scattering from Magnetic Materials part 9 docx

Magnetic structures with zero propagation vector It has already been pointed out in Section 2.3 that in structures with τ = 0, nuclear andmagnetic scattering occur in the same Bragg refl

Spherical neutron polarimetry 237

Fig 10 The h0l section of reciprocal space for the incommensurate structure of CuO The fundamental (nuclear)

reflections are shown as open circles and the magnetic satellites as filled ones The scattering vectors for the

002+ τ and 000 − τ reflections are shown as dashed lines.

4.4 Magnetic structures with zero propagation vector

It has already been pointed out in Section 2.3 that in structures with τ = 0, nuclear andmagnetic scattering occur in the same Bragg reflections and interference between them canoccur In this case SNP can determine the ratio between magnetic and nuclear scattering,allowing the magnitude of magnetic moments to be established from SNP alone withoutrecourse to supplementary integrated intensity measurements

4.4.1 The magnetic structure of U 14 Au 51 The intermetallic compound U14Au51 tallises in the hexagonal Gd14Ag51 structure with space group P 6/m [11] The uranium atoms occupy three crystallographically distinct sites 6(k), 6(j ) and 2(e) labelled U1,

crys-U2 and U3, respectively Susceptibility, specific heat and resistivity indicate a magneticphase transition at 22 K This has been confirmed by neutron powder diffraction measure-ments: an antiferromagnetic structure with zero propagation vector and magnetic space

group P 6/m was proposed [12] The ordered magnetic moments aligned parallel to c

were 0.5 and 1.6µBon U1 and U2, respectively No moment was assigned to the U3 atoms

which have a particularly small separation, it was argued that direct f -electron wave

func-tion overlap prevents a magnetic response An SNP study of the magnetic structure wasundertaken because it proved impossible to reconcile the intensity of magnetic scattering

by single crystals of U14Au51 with the proposed magnetic structure The U14Au51 tal was mounted with its [01.0] axis vertical The polarisation matrices determined for the20.0, 20.1 and 10.1 reflections are given in Table 1 They enable severe constraints to beimposed on the possible magnetic structures

crys-1 For incident polarisation parallel to the scattering vector x the scattered beam is partly

depolarised, and reversed but not rotated The depolarisation must be due to diagonal terms (P xy,P xz) of opposite signs coming from 180◦domains This means

off-that the J terms in (14) must be nonzero showing that the magnetic scattering is

2 For the 20.0 reflection there are no significant off-diagonal terms andP zzis not

sig-nificantly different from 1 showing that for 20.0 M⊥must be parallel to z

(crystal-lographic[01.0]), and there are therefore no significant components of the magnetic

structure factor parallel to c.

3 For 20.1 there is some depolarisation for all three incident polarisation directions andoff-diagonal componentsP yzandP zy are observed This is consistent with moments

in the a–b plane The observation that the depolarisation for incident directions in

the y–z plane, that containing the magnetic interaction vector, is less than for the

x direction implies that all the depolarisation is due to the 180◦ domains and that

there is none due to orientation domains The magnetic structure therefore probablyhas the full symmetry of the crystallographic space group

There is just a single magnetic space group and basic model structure which is ble with all these constraints The U1 and U2 sites lie on the mirror planes perpendicular tothe hexad and from (2) and (3) their moments lie in it These mirror planes cannot thereforeinvert the moments and must be combined with time reversal To satisfy (1) which impliesthat centrosymmetrically related atoms have opposite moments the hexad must operate

compati-without time inversion The magnetic space group is therefore P 6/m and the magnetic

moments on the groups of 6 U1 (and U2) atoms related by the hexad have a star

struc-ture as illustrated in Figure 11 In this magnetic group any moment on the U3 atoms is

constrained to be parallel to c since these sites are on the hexagonal axes The SNP surements show that the c component of moment is small or zero so it can be concluded

mea-that the U3 moment is also small or zero To describe the structure completely it is

neces-sary to determine the magnitude and the orientation within the a–b plane of the moments

on the U1 and U2 atoms

The SNP data for the 10.1 reflection (Table 1) allow rough values of the moment

direc-tions within the a–b plane to be deduced The incident polarisation parallel to y is hardly changed on scattering so its magnetic interaction vector M⊥ is nearly parallel to y The magnitude of M⊥can be obtained from

P xx=1− γ2

γ gives, as before (equation (10)), the ratio of magnetic to nuclear scattering M⊥ for

Spherical neutron polarimetry 239

Fig 11 The hexagonal star arrangement of moments found in the U1 and U2 layers of U14Au51 The angle φ

used to fix the orientation of the moments is marked.

the 10.1 reflection is the sum of contributions from the U1 and U2 layers The y and z components of these were computed separately as a function of the angle φ in Figure 11

and are plotted in Figure 12 By moving one curve relative to the other it was found that

there is only a small range in which a pair of φ’s exist for which the M ⊥zfor U1 and U2

cancel whilst their M ⊥y reinforce one another It corresponds to φU1∼ 140◦, φU2∼ 90◦.

These initial values provided an adequate starting point for a least squares refinement ofthe structure using both SNP and integrated intensity data

4.4.2 Magnetoelectric crystals. The property of magnetoelectricity in centrosymmetriccrystals is restricted to those having antiferromagnetic structures with zero propagationvector in which the centre of symmetry is combined with time-reversal These are just

the requirements for J ni(equation (16)) to be finite giving rise to off-diagonal termsP xz,

P zx in the polarisation matrix (equation (14)) It is known that although the temperaturedependencies of magnetoelectric (ME) susceptibilities are unique to each material, theirmagnitudes and even their signs are specimen dependent This specimen dependence isdue to the existence of 180◦antiferromagnetic domains which have opposite ME effects.

The measured ME susceptibility χobsis related to the intrinsic susceptibility χ0by

240 P.J Brown

Fig 12 Curves showing the variation with φ of M ⊥y (full) and M ⊥z (dashed) components of the magnetic interaction vector of the 10.1 reflection of U14Au51 (a) is for the U1 atoms and (b) for the U2 atoms The origins

of the two figures are displaced so that on the vertical line marked ((a) φ= 140◦, (b) φ= 90◦) the z components

of U1 and U2 cancel whilst their y components reinforce one another.

Fig 13 The moment directions of Cr3+ions at the centres of the double octahedral coordination polyhedra

found in Cr 2 O 3 , after (a) cooling in parallel, (b) antiparallel electric and magnetic fields.

q y is+1 if M(Q) is parallel to y and −1 if it is antiparallel Measurement of the

polar-isation matrix therefore allows both η and γ to be determined The absolute directions

of rotation of the neutron spins when η = 0 determine the magnetic configuration of themore populous domain This in turn allows the effects of electric and magnetic fields onthe domain population to be studied The results shed light on the fundamental mecha-nisms leading to the ME effect Perhaps the best known ME material is Cr2O3in whichthe Cr3+ions are octahedrally coordinated by oxygen and the structure is made up of pairs

of octahedra, sharing a common face as illustrated in Figure 13, linked to other pairs bysharing the free vertices SNP has shown that electric and magnetic fields, applied parallel

to one another and to the c axis while cooling through the Néel transition, stabilise the

Spherical neutron polarimetry 241domain in which the moments point towards the shared face of their coordinating oxygenoctahedra [13].

5 Determination of antiferromagnetic form factors

As has been shown in Chapter 4 the classical polarised neutron diffraction technique [14]

is widely used to study the magnetisation distribution around magnetic atoms and ions inferromagnetic and paramagnetic materials It is very much more difficult to measure thisdistribution in antiferromagnetic systems because in antiferromagnets the cross-section isseldom polarisation dependent so the classical method is not applicable As a consequence,very few measurements of magnetisation distributions in antiferromagnetic materials havebeen made since usually they require very precise integrated intensity measurements ofrather weak reflections In the few cases where such measurements have been undertaken,they have given very interesting results [15–17] An antiferromagnetic magnetisation dis-tribution is more sensitive than a ferromagnetic or paramagnetic one to the effects of cova-lency because the overlap of positive and negative transferred spin on the ligand ions leads

to an actual loss of moment rather than just to a redistribution

Until recently no precise measurements had been made for the class of antiferromagneticstructures with zero propagation vector, in which magnetic atoms of opposite spin are re-lated by a centre of symmetry In such structures the magnetic and nuclear scattering aresuperimposed, making separation of the nuclear and magnetic parts difficult Additionallythe magnetic and nuclear structure factors are in phase quadrature so that is no interfer-ence between them to give a polarisation dependent cross-section However, it was shown

in the previous section that it is in exactly this case for which the magnetic and nuclear

scattering are in quadrature that the polarisation matrices depend sensitively on the ratio γ between the magnetic and nuclear structure factors when there is an imbalance (η = 0) inthe populations of the two 180◦domains The high precision with which the ratio γ can

be determined in favourable cases allows the magnetic structure factors to be determinedwith good accuracy and so gives access to the antiferromagnetic form-factors

The polarisation matrix of (28) allows two independent estimates of γ :

(a) P xz = −P zx = ηξ = ηq y γ

1+ γ2,

(29)(b) P xx = P zz = β =1− γ2

1+ γ2,

the former only being useful if there is an imbalance in the 180◦domains Assuming the

polarimeter (CRYOPAD) is free of aberrations the precision with which γ can be

deter-mined depends on the statistical error in the determination of the components of scatteredpolarisation It should be recalled that in this type of structure the cross-section is inde-pendent of the polarisation direction The counting rate summed over the two polarisationstates accepted by the detector is therefore constant, and independent of either incident

or scattered polarisation direction The polarisation measured by the analyser is given by

P = (I+− I−)/(I++ I−) where I+ and I− are the counting rates in the two detector

N++ 1

N−



where N+ and N− are the counts recorded in each channel The variance is minimised

by dividing the measuring time available in the ratio t+/t−= (1 − P )/(1 + P ) With this division, if the total number of neutrons counted is N equation (30) becomes

If η is small (nearly equal domains) or γ is close to unity, the best estimate of γ will

be obtained from (29)(a) whereas for very small or very large γ (29)(b) will give a better value so long as η is nonzero Figure 14 shows the regions of γ –η space in which one or the other equation gives the better estimate of γ

The first example of the use of this technique was to determine the Cr2+ form-factor

in Cr2O3[18] Samples were cooled in combined electric and magnetic fields to obtain

dif-ferent domain ratios η as indicated in the previous section The crystals were aligned with

Fig 14 Plot of γ –η space The shaded region is that in which equation (29)(a) gives a more precise estimation

of γ than (29)(b) The γ axis is plotted on a logarithmic scale.

Spherical neutron polarimetry 243

Fig 15 The experimental values of the magnetic form factor measured at the h0. Bragg reflections of Cr2O3 The smooth curve is the spin-only free-ion form factor for Cr2+normalised to the experimental value at the

lowest angle reflection (10.2).

a[¯12¯10] axis vertical so as to obtain reflections h0 ¯hl in the horizontal plane The moment

direction is [0001] so that with this orientation M⊥(Q) is parallel to polarisation y The

elements P xz,P zx of the polarisation matrices obtained with different domain ratios arevery different, but the magnetic structure factors deduced from them were found to agreewell This confirms the supposition that extinction effects are not a major problem sincethe measurements of polarisation are made with a constant cross-section The points onthe Cr3+form factor obtained from the measured structure factors are plotted in Figure 15

where they are compared with the Cr3+ free in form factor It can be seen that for most

reflections an extremely good precision was obtained Exceptions are the 20.¯2 and 10.¯10 reflections; for the former the nuclear structure factor is small so that γ $ 1 and in thelatter the geometric structure factor for the Cr atoms is small so the reflection is insensitive

to the Cr form factor

This first pioneering experiment has shown that SNP can be used to make high sion measurements of antiferromagnetic magnetisation distributions However is should

preci-be emphasised that such measurements are only possible for a restricted class of magnets, those in which magnetic and nuclear scattering occur in quadrature in the samereflections Additionally high sensitivity can only be obtained if the population of the 180◦

antiferro-domains can be unbalanced Nevertheless, this class of antiferromagnets includes the netoelectric classes, and for these electromagnetic annealing can unbalance the domains.SNP therefore provides an important new tool for probing the magnetisation distributionsassociated with magnetoelectricity

[4] M Blume, Phys Rev 130 1670 (1963).

[5] S.V Maleev, V.G Bar’yaktar and P.A Suris, Sov Phys Solid State 4 2533 (1963).

[6] P.J Brown, Physica B 297 198 (2001).

[7] J.B Forsyth, P.J Brown and B.M Wanklyn, J Phys C 21 2917 (1988).

[8] D Yablonski, Physica C 171 454 (1990).

[9] Yu.G Raydugin, V.E Naish and E.A Turov, J Magn Magn Mater 102 331 (1991).

[10] P.J Brown, T Chattopadhyay, J.B Forsyth, V Nunez and F Tasset, J Phys.: Condens Matter 3 4281

(1991).

[11] A Dommann and F Hullinger, J Less-Common Met 141 261 (1988).

[12] A Dommann et al., J Less-Common Met 160 171 (1990).

[13] P.J Brown, J.B Forsyth and F Tasset, J Phys.: Condens Matter 10 663 (1998).

[14] R Nathans, C.G Shull, G Shirane and A Andresen, J Phys Chem Solids 10 138 (1959).

[15] H.A Alperin, Phys Rev Lett 6 520 (1961).

[16] J.W Lynn, G Shirane and M Blume, Phys Rev Lett 37 154 (1976).

[17] X.L Wang, C Stassis, D.C Johnstone, T.C Leung, J Ye, B.N Harmon, G.H Lander, A.J Shultz,

C.-K Loong and J.M Honig, J Appl Phys 69 4860 (1991).

[18] P.J Brown, J.B Forsyth, E Lelièvre-Berna and F Tasset, J Phys.: Condens Matter 14 1957 (2002).

CHAPTER 6

Magnetic Excitations

Tapan Chatterji

Institut Laue–Langevin, B.P 156X, 38042 Grenoble cedex, France

E-mail: chatt@ill.fr

Contents

1 Introduction 247

2 Experimental methods 247

2.1 Triple-axis spectrometer 247

2.2 Intensity and resolution function of TAS 251

2.3 Size and shape of the resolution function 254

2.4 TAS multiplexing 256

2.5 Time-of-flight spectrometers 256

3 Spin waves in localized electron systems 259

3.1 Spin waves in Heisenberg ferromagnets 259

3.2 Thermal evolution of spin waves in Heisenberg ferromagnets 263

3.3 Spin wave damping in Heisenberg ferromagnets 265

3.4 Spin waves in Heisenberg antiferromagnets 268

3.5 Two-magnon interaction in Heisenberg antiferromagnets 274

3.6 Spin waves in Heisenberg ferrimagnets 275

4 Spin waves in itinerant magnetic systems 280

4.1 Generalized susceptibility and neutron scattering cross-section 281

4.2 Spin dynamics of ferromagnetic Fe 284

4.3 Spin dynamics of ferromagnetic Ni 291

4.4 Spin dynamics of weak itinerant ferromagnet MnSi 297

5 Spin waves in CMR manganites 300

5.1 Spin waves A 1−xBxMnO 3 , A = La, Pr, Nd; B = Ca, Sr, Ba 300

5.2 Thermal evolution of spin dynamics of A 1−xBxMnO 3 313

5.3 Spin waves in bilayer manganite La 2−2xSr 1+2xMn 2 O 7 315

6 Concluding remarks 327

Acknowledgments 328

References 328

NEUTRON SCATTERING FROM MAGNETIC MATERIALS

Edited by Tapan Chatterji

© 2006 Elsevier B.V All rights reserved

245

exper-by inelastic neutron scattering yielding eventually the sign and magnitudes of exchange teractions Before we explain how this may be achieved in particular cases we describe theessential experimental techniques of inelastic neutron scattering.

in-2 Experimental methods

As in the case of the determination of the magnetic structure, the neutron scattering nique is unique for experimental investigation of spin waves and other excitations in mag-netic crystals The spin wave energies in magnetic solids are normally in the meV rangeand therefore scattering of thermal neutrons is suitable for their investigation Sometimesthe spin wave energy is of the order of 0.1 meV for which cold neutron scattering is moreappropriate In some cases like transition metal ferromagnets like Fe, Ni, Co, etc., the spinwave energies lie at 0–200 meV for which it is necessary to use scattering of hot neutrons

tech-in addition to that of thermal neutrons

2.1 Triple-axis spectrometer

Spin wave dispersion is usually determined with a neutron triple-axis spectrometer (TAS).Magnetic and structural excitations have been investigated with triple-axis spectrometersever since Brockhouse [1] developed such a spectrometer at Chalk River in Canada Thetechniques of triple-axis spectrometers have been discussed in details by Shirane, Shapiroand Tranquada [2] and Currat [3] Here we will give an outline of this technique

Figure 1 shows schematically typical triple-axis spectrometer The three axes correspond

to the rotation axes of the monochromator, the sample table and the analyzer The axis spectrometer is the instrument of choice whenever it is necessary to have precise

triple-control on the positions in (Q, ω) space at which one wishes to measure the scattered neutron intensity The intensity at a single position in (Q, ω) space is measured in a step

by step manner where each spectrometer configuration corresponds to a well-defined value

of kiand kf, the incident and the scattered wave vector Q and ¯hω satisfy momentum and

energy conservation laws given by

¯h2k2i

2mn − ¯h2k2f

to ki by the rotation of the sample table (A3= ωs) and the double goniometer (tilt

an-gles) or Eulerian cradle The modulus of scattered wave vector kfis selected by the Braggdiffraction from the analyzer crystal (A5= ωa and A6= 2θa) and its orientation in thereciprocal space of the sample is determined by the scattering angle (A4= 2θs) at thesample position Figure 2 shows the reciprocal space diagram corresponding to the spec-trometer configuration of Figure 1 The magnitudes of the initial and final wave vectorare not equal since we are interested in measuring a finite energy transfer ¯hω The total

momentum transfer Q is decomposed into a reciprocal vector τ hkl and a wave vector q.

One can measure a collective excitation with a dispersion ω(q) with such a spectrometer

configuration The measurement is done at the[hkl] Brillouin zone of the reciprocal space.

In principle the dispersion can be measured in any Brillouin zone but the intensity will

be different In magnetic samples the intensity is severely reduced as we go further awayfrom the center of the reciprocal lattice due to the magnetic form factor One can access

the same (Q, ω) point using infinite number of alternative combinations of kiand kf Thishas been illustrated by the dotted and dashed lines in Figure 2 However the intensity andresolution characteristics are different for these alternative configurations and therefore a

proper choice of ki and kfis important for the measurement

Two types of scan methods are normally employed: (1) constant-Q and (2) constant-E.

In the constant-Q method the spectrometer is set to a particular Q, which is kept fixed

Magnetic excitations 249

Fig 2 The solid lines represent the reciprocal space representation of the inelastic measurement with a TAS corresponding to Figure 1 The dotted and dashed lines represent alternative configurations leading to the

same (Q, ω) (from Currat [3]).

during the scan (hence called constant-Q) and the energy transfer is varied Also these scans are usually performed in two modes: (1) constant-ki or (2) constant-kfmode, the

later being more frequently used Figure 3 illustrates these two ways of constant-Q scans,

keeping ki= |ki| constant in the first case and kf= |kf| in the second case

The constant-Q scan is the most common mode of scan because the data collected in

this mode can be directly related to the dynamical susceptibility of the magnetic sampleinvestigated The model calculations to which one likes to compare the experimental data

are usually given in terms of the dynamical susceptibility χ (Q, ω) at high symmetry points

in the reciprocal space Also the integrated intensity of a constant-Q scan can give a direct

measure of S(Q, ω), multiplied by resolution volume associated with the analyzer arm of

the spectrometer If the scattered neutron wave vector kfis held fixed (constant-kfmode),

so that the energy transfer is varied by varying ki, then the phase space volume remains

constant during the scan In constant-E scans the spectrometer is set to detect a particular

energy transfer corresponding to the energy of the spin wave and Q is varied in a particular

direction in reciprocal space The choice between these two scans is dictated by the slope

of the spin wave dispersion and the form of the resolution ellipsoid of the spectrometer,which will be discussed in the next section

Apart from the neutron source which is usually a reactor, the monochromator crystal(or assembly of crystals) is the most important component of the triple-axis spectrometerthat determines the neutron intensity incident on the sample The monochromator crys-tal selects a specific neutron wavelength from the incident polychromatic neutron beamfrom the reactor by Bragg diffraction from a given set of lattice planes of the crystal Thechoice of the monochromator crystal is mainly dictated by the maximum reflectivity andless higher-order wavelength contamination For maximum reflectivity it is desirable tohave crystals with small unit cell volumes, large neutron scattering lengths and low ab-sorption coefficients To keep the background low it is desirable to have monochromatorcrystals with large Debye temperatures (rigid lattice) and small incoherent scattering cross-sections Phonons and incoherent scattering from the monochromator or analyzer crystals

250 T Chatterji

Fig 3 Upper panel: Illustration of a constant-ki constant-Q scan Lower panel: Illustration of a constant-kf

constant-Q scan (from Currat [3]).

give signals at unwanted and unintended wavelengths Certain crystal lattices are suitablefor suppressing higher-order wavelength contamination For example, the 111 reflection

from the diamond type structure (shared by silicon and germanium) will have no λ/2

con-tamination because of the forbidden 222 reflection of this type of crystal structure Anotherimportant characteristic of the monochromator is the mosaic width The horizontal mosaicwidth should be consistent with horizontal collimation, typically from 20 to 40, while

the vertical mosaic width should be as narrow as possible An ideal monochromator is

pyrolytic (or oriented) graphite (PG) which has highly preferred orientation of the (00l) planes, but all other (hkl) planes are oriented at random giving rise to powder peaks (De- bye rings) PG(002) is also often used as an analyzer crystal PG is also used as a filter for

higher-order wavelength contamination Other typical monochromator crystals used areBe(002), Cu(111), Cu(200), Cu(220), Ge(111), Si(111) and Zn(002) For a more completelist the readers can consult Table 3.1 of Shirane et al [2] High intensity at the detectorcan be achieved by focusing the beam by curved monochromator and analyzer crystals

The vertical focusing of the monochromator crystal is commonly used since good Q

res-olution within the scattering plane is desired, while poor resres-olution in the vertical plane istolerated However, the analyzer crystal is often horizontally curved The focusing of the

Magnetic excitations 251monochromatic beam is more frequently achieved by an assembly of small single crystalpieces oriented in such a way as to focus the monochromatic beam on to the sample Thenature and characteristics of the crystal used as analyzer are similar to those of the mono-

chromator The analyzer d-spacing must be adapted to the scattered neutron energies to be

analyzed and the energy resolution required

The detector is generally a simple3He-gas proportional counter A counting efficiency ofabout 80–95% in the relevant neutron energy range is achieved by choosing the thickness

of the counter and the gas pressure Unlike the highly collimated X-ray beam from the chrotron sources the neutron beam from the reactor emerge in all directions Although theBragg diffraction from the monochromator and the analyzer crystals puts some constraint

syn-on the angular divergence of the beam, it is necessary to have additisyn-onal adjustable csyn-ontrol

of the beam divergence The horizontal beam divergence in the scattering is typically trolled by the Soller collimators The horizontal collimators are normally placed before themonochromator, before the sample, before the analyzer and also before the detector How-ever care should be taken regarding the compatibility of the collimation with the focusing

con-A variety of other devices can be inserted in the neutron beam, viz adjustable diaphragms,low efficiency counters to monitor the intensity of the beam incident on the sample (M1) orthe analyzer (M2), filters (oriented PG, polycrystalline Be or BeO, resonance filters, etc.)

to eliminate higher-order wavelength contaminations Cooled polycrystalline Be or BeOare generally used for the cold triple-axis spectrometers to eliminate all neutrons above theenergy corresponding to the Bragg cut-off The elimination of neutrons with unwanted en-ergies by the filters are achieved by the Bragg diffraction process This implies that, unlesscarefully shielded, the filters may cause increase in the background signal

2.2 Intensity and resolution function of TAS

The determination of the resolution function of a TAS spectrometer is quite complicated.Here we give the definition of the resolution function and some simple relations betweenthe intensity measured during a scan and the norm of the resolution function followingCurrat [3] and Dorner [4] closely The intensity or the neutron counts recorded by the de-

tector for a given spectrometer configuration corresponding to the nominal values (Q0 , ω0)

where A(ki ) gives the spectrum of the source, pi(ki) and pf(kf) refer to the transmission

of the monochromator (analyzer) crystal for each incident (scattered) neutron wave vector

and N is the number of scattering particles in the irradiated sample volume The variables

(k , k ) and (Q, ω) are related through the energy and momentum conservation relations

[8] D Yablonski, Physica C 171 454 ( 199 0).

[9] Yu.G Raydugin, V.E Naish and E.A Turov, J Magn Magn Mater 102 331 ( 199 1).... the neutron intensity incident on the sample The monochromator crys-tal selects a specific neutron wavelength from the incident polychromatic neutron beamfrom the reactor by Bragg diffraction from. ..

( 199 1).

[11] A Dommann and F Hullinger, J Less-Common Met 141 261 ( 198 8).

[12] A Dommann et al., J Less-Common Met 160 171 ( 199 0).