Neutron Scattering from Magnetic Materials part 8 pptx

In this configuration, if we takeapart the nuclear spin incoherent scattering, the non-spin-flip scattering is only nuclear andthe spin-flip scattering is only magnetic and due to the tr

206 J Schweizer

magnetic scattering, shows that a large fraction of the magnetic moments are not alignedalong the applied magnetic field but is canted away from the field direction The randomlydirected strong crystalline fields in the amorphous material has the effect that the appliedfield can orient only the component of the magnetic moment that is parallel to it Moreoversome correlations exist between the directions of crystalline fields in adjacent cells, enough

to produce a ferromagnetic short-range order of the transverse components of the magneticmoments

A more subtle arrangement of the transverse components is met in magnetically orderedamorphous materials which contain rare-earth atoms associated with a transition metal Inthese systems with two types of magnetic atoms, the huge anisotropy of the rare earth plays

a fundamental role For instance, in the Er–Co amorphous alloys, the cobalt moments areferromagnetically coupled and are parallel to one another, aligned by the external field,and the Er moments are on the average antiparallel to the Co moments, but strongly cor-related to the local crystal field axes, in spite of the fact that these axes are distributed inall directions Such a structure with rare-earth moments distributed in space, but still more

or less antiparallel to the cobalt moments, has been called “sperimagnetic” [52] The earth moments are also aligned when a magnetic field is applied, but only partly, because

rare-of their magnetic interaction with the cobalt moments; the magnetic anisotropy, very large

at low temperatures, prevents the alignment from being complete Concerning their verse components, are they randomly distributed or are they correlated? The question hasbeen answered by a uniaxial polarization analysis experiment [53] In this experiment, thepolarization was parallel to the scattering vector, with a horizontal field of 0.2 T, enough toalign the magnetization of the sample The non-spin-flip and the spin-flip scattering at lowtemperature are reported in Figures 30(a) and 30(b)–(c) In this configuration, if we takeapart the nuclear spin incoherent scattering, the non-spin-flip scattering is only nuclear andthe spin-flip scattering is only magnetic and due to the transverse components of the mo-ments (formulae (119) and (120)) There are two striking results concerning the magneticscattering First, a noticeable amount of these transverse components are correlated, about25% of the transverse magnetic scattering Second, there are “ferromagnetic” rings, at thesame position as those which exist for the nuclear scattering, but there are also, in betweenthe ferromagnetic rings, other rings that have been called “antiferromagnetic” These twotypes of rings are been explained by the following exchange scheme:

trans-• large and positive exchange JCo–Co is responsible for the long range order of the

Co moments,

• lower and negative exchange JCo–Erforces most of the Er moments to be opposed

to the Co ones However, because of their very strong anisotropy, the Er moments lievery close to their easy magnetization axes,

• the last exchange JEr–Er, being smaller and negative, when the angle θ between the

Co moments and the Er local axes is small, the Co–Er interactions dominate the Er–Erinteractions and the Er moments order ferromagnetically at short distances as shown

in Figure 31(a), but on the contrary, for large values of θ , the Co–Er interactions

are weakened as in Figure 31(b), and the negative Er–Er interactions cause the Ermoments to order antiferromagnetically

Polarized neutrons and polarization analysis 207

Fig 30 (a) Non-spin-flip; (b) spin-flip; (c) spin-flip corrected for the erbium form factor.

Fig 31 Ferromagnetic (a) and antiferromagnetic (b) short range distance arrangements of the Er moments.

208 J Schweizer

5.6 Investigation of antiferromagnetic structures

In their paper, Moon et al [3] illustrated the separation between the nuclear and the netic scattering of an antiferromagnet in showing the nuclear and the magnetic patterns of

mag-a powder of α-Fe2O3by applying the polarization along the scattering vector (Figure 32).Considering that equations (114)–(117) separate the components of M⊥which are par-allel or perpendicular to the uniaxial polarization P , we can expect specific informations

to analyze complex magnetic structures

A first example of information brought by the polarization analysis concerns thecollinearity of the antiferromagnetic structure in dilute semiconductors (Cd, Mn)Te Thecrystal structure of CdTe is of zincblende type with the Cd atoms occupying one FCClattice and the Te atoms the second FCC lattice Mn can replace the Cd atoms up to

Fig 32 α-Fe2O3nuclear and magnetic powder patterns separation obtained with the polarization along the

scattering vector [3].

Polarized neutrons and polarization analysis 209about 70% In the concentration range between 17% and 60%, a spin-glass behavior isfound, attributed to the lattice frustration mechanism Between 60% and 70% Mn, addi-tional peaks are observed at low temperatures, indicating the onset of antiferromagneticordering, but a short-range ordering only, as the line width of the magnetic peaks aremuch larger than the experimental resolution width The propagation vector is of the type

(1/2, 0, 0), permitting the existence of 3 magnetic domains.

Among other experiments, Steigenberger and Lindley [54] investigated the polarization

of the magnetic reflections of a single crystal of composition Cd0.35Mn0.65Te with a polarized incident neutron beam They found that, for a number reflections with the scat-tering vector along the direction of the polarization analysis, the scattered intensities had

non-a nonzero polnon-ariznon-ation, but on the contrnon-ary, for those reflections with the scnon-attering vectorperpendicular to this direction, the scattered reflections were not polarized As explained

before, the only term able to produce a polarization is the cross-term i(  M⊥∧ M∗

⊥), a larization that is parallel to the scattering vector If a polarization is detected, such a termexists, and if such a term exists, the magnetic structure is chiral, not necessarily helicoidal,but at least not collinear This is the case for Cd0.35Mn0.65Te

po-A second example concerns MnP for which Moon [55] took advantage of uniaxial ization analysis to answer the questions which remained open about its helicoidal magneticstructure In this orthorhombic crystal, an incommensurate helix propagates at low temper-atures alonga, the hard axis, and the moments rotate in the (b, c) plane The questions that

polar-arose were:

(i) does the anisotropy between the b and the c axes of the crystal modify the helix?

(ii) if yes, would it result in an elliptical helix with different b and c components, or

in a circular but distorted helix with equal components of the moments but with abunching of the moments along the easy directionc as proposed by Hiyamizu and

Nagamiya [56]? In this last case, third-order satellites should exist, but they had notbeen seen

Before Moon’s experiment, the best neutron data collection [57] had given, in the frame

of the elliptical model, 1.29 ± 0.10µB along c and 1.20 ± 0.05µB along b, giving no

clear answer to the open questions To remove this uncertainty, Moon compared the

spin-flip and the non-spin-spin-flip intensities of the satellites (2 ± δ, 0, 0) of a single crystal For

this comparison, the b axis of the crystal was put vertical and the uniaxial polarization as

well (Figure 33) With this arrangement (formulae (104)–(107)), and still in the frame ofthe elliptical model, the spin-flip cross-section measures the component of the momentsalong c while the non-spin-flip cross-section measures the component along b For the

two satellites, the spin-flip intensity associated withc was found higher than the

non-spin-flip intensity associated with b, and their ratio was measured with an excellent accuracy (R = 1.091 ± 007).

But the observed difference could also result of the bunching of the moments If thiswere the case, the bunching parameter, which is related to the exchange and anisotropyconstants, can be determined from the former measurement Its value would imply that thethird-order satellites are 10−3or 10−4the first-order satellites With unpolarized neutrons,Moon looked for these very weak reflections and he then concluded that the bunchingmodel of Hiyamizu and Nagamiya was correct

210 J Schweizer

Fig 33 Neutron polarization analysis of the satellites (2 ± δ, 0, 0) of MnP [55].

5.7 Conclusions on the uniaxial polarization analysis

Uniaxial polarization analysis opened huge possibilities of investigations in magnetic tering The main fields of application are the separation between nuclear and magneticscattering and the study of all the magnetic contributions for which the separation from thenuclear scattering is not straightforward: paramagnetic scattering, magnetic short rangeorder, transverse components It is also very useful to investigate complex magnetic struc-tures, in spite of the fact that today spherical polarization analysis offers more possibilities

scat-as will be explained in the next chapter [10]

Uniaxial polarization is rather simple to adapt on a spectrometer Considering that larizing monochromators and analyzers are less efficient in term of luminosity than thenonpolarizing ones, the cost in neutron intensity is rather high, particularly at short wave-lengths However, with the development of polarizing filters, particularly the3He polariz-ing filters, this inconvenience is not as strong as it was and uniaxial polarization analysis is

po-a very powerful tool, po-able to solve mpo-any problems in mpo-agnetism

Appendix: Atomic Slater functions and radial integrals

As seen in the text, the one electron atomic wave function can be expanded in a radial partand an angular part:

Polarized neutrons and polarization analysis 211

where θ , ϕ are the angular coordinates of r and where the Y m

 are the usual sphericalharmonics

Very commonly, Ra (r) is expressed as an atomic Slater function,

a(r)=23/2 α 7/2

3√

5 r2e−αr . The atomic Slater exponent α is characteristic of the two quantum numbers naand .

They have been tabulated for all the electronic shells of all the elements by Clementi and

Roetti [29] (attention: values given in atomic units and not in Å−1).

Radial integrals are the Bessel–Fourier transform of these radial functions,

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[14] J.X Boucherle, F Givord, P Lejay, J Schweizer and A Stunault, Physica B 156–157 809 (1989) [15] A Zheludev, M Bonnet, E Ressouche, J Schweizer, M Wan and H Wang, J Magn Magn Mater 135

147 (1994).

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[18] C Shannon, Bell System Tech J 27 379, 623 (1948).

[19] R Papoular and B Gillon, Europhys Lett 13 429 (1990).

[20] R Papoular and B Gillon, in: Neutron Scattering Data Analysis, ed M.W Johnson, p 101, Hilger, Bristol (1990).

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[23] P Schleger, A Puig-Molina, E Ressouche, O Rutty and J Schweizer, Acta Crystallogr A 53 426 (1997) [24] Y Pontillon, V.I Ovcharenko, E Ressouche, P Rey, P Schleger and J Schweizer, Physica B 234–236 785

(1997).

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[25] W Marshall and S.W Lovesey, in: Theory of Thermal Neutron Scattering, Clarendon, Oxford (1971) [26] S.W Lovesey, in: Theory of Neutron Scattering from Condensed Matter, Clarendon, Oxford (1983) [27] E Balcar and S.W Lovesey, in: Theory of Magnetic Neutron and Photon Scattering, Clarendon, Oxford (1989).

[28] P.J Brown, in: International Tables for Crystallography, Volume C: Mathematical, Physical and Chemical Tables, ed A.J.C Wilson, p 391, Kluwer Academic (1992).

[29] E Clementi and C Roetti, At Data Nucl Data Tables 14 178 (1974).

[30] Y Pontillon, E Ressouche, F Romero, J Schweizer and R Ziessel, Physica B 234–236 788 (1997) [31] G.H Lander and T.O Brun, J Chem Phys 53 1387 (1970).

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[35] P.J Brown, A Capiomont, B Gillon and J Schweizer, J Magn Magn Mater 14 289 (1979).

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J Am Chem Soc 116 7243 (1994).

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[43] O Schaerpf and H Capellmann, Phys Status Solidi 135 35 (1993).

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CHAPTER 5

Spherical Neutron Polarimetry

P.J Brown

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

and Physics Department, Loughborough University, Loughborough, UK

E-mail: brown@ill.fr

Contents

Notation 217

1 Neutron polarimetry 217

1.1 Neutron precession in an external field 218

1.2 Classical polarisation analysis 219

1.3 Multidirectional polarisation analysis 220

1.4 Spherical neutron polarimetry 220

2 Polarised neutron scattering 222

2.1 Magnetic scattering 222

2.2 Nuclear scattering 223

2.3 Nuclear–magnetic interference 224

2.4 The Blume–Maleev equations 225

2.5 Tensor representation of the scattered polarisation 225

3 Magnetic domains 226

3.1 Configuration domains 227

3.2 180 ◦domains 227

3.3 Orientation domains 229

3.4 Chirality domains 230

4 Magnetic structure determination using SNP 232

4.1 Experimental strategy 232

4.2 Commensurate structures with nonzero propagation vectors 233

4.3 Incommensurate structures 234

4.4 Magnetic structures with zero propagation vector 237

5 Determination of antiferromagnetic form factors 241

References 244

NEUTRON SCATTERING FROM MAGNETIC MATERIALS

Edited by Tapan Chatterji

© 2006 Elsevier B.V All rights reserved

215

Spherical neutron polarimetry 217

Notation

a, b, c unit cell vectors

θB Bragg angle

g a reciprocal lattice vector

γN gyromagnetic ratio of the neutron

κ crystallographic scattering vector Q= −Q

l a real space lattice vector

mN neutron mass

M magnetic moment vector

M(r) magnetisation at r

M(Q) magnetic structure factor at Q

M⊥(Q) magnetic interaction vector at Q

M⊥ magnetic interaction vector (shorthand form)

M∗

⊥(Q) complex conjugate of magnetic interaction vector at Q

M∗

⊥ complex conjugate of magnetic interaction vector (shorthand form)

M ⊥i ith component of magnetic interaction vector

N (Q) nuclear structure factor at Q

N nuclear structure factor (shorthand form)

N∗ complex conjugate of nuclear structure factor

ΩL Larmor precession frequency

P incident polarisation

P i ith component of incident polarisation

P scattered polarisation

P

i ith component of scattered polarisation

P polarisation created by scattering

P

i ith component of polarisation created by scattering

P polarisation tensor

Pij ij th element of the polarisation tensor

P polarisation matrix (the experimental result)

P ij ij th element of the polarisation tensor

Q momentum transfer vector Q = ki − kf

r a vector in real space

τ magnetic propagation vector

ˆu a unit vector parallel to u

218 P.J Brown

it is again with respect to a direction fixed by an applied field Between polariser andanalyser the evolution of the polarisation is subject to its interaction with magnetic fieldsnot only in the scattering sample, but also in the incident and scattered beam paths Theselatter interactions must be carefully controlled in order to obtain meaningful results inpolarisation analysis experiments

The results of a polarisation analysis experiment may be expressed in terms of

gener-alised cross-sections I ij The indices i and j each refer to one of the three orthogonal

directions defined by the experiment: the second superscript gives the direction of isation and the first the direction of analysis They are related to the general polarisationmatrix by

The polarisationsP ij can be measured with much higher precision that can the ual cross-sections because they are obtained from ratios of intensities measured withouthaving to move the sample

individ-1.1 Neutron precession in an external field

The behaviour of a neutron with spin S and gyromagnetic ratio γNinclined at an angle θ to

a uniform magnetic field B can be represented classically as the precession of a magnetic

dipole (γNS) about the field direction The precession frequency ΩLis given by the Larmor

equation and the angle of precession φ of a neutron of wavelength λ depends upon the path

Exactly the same result can be obtained quantum mechanically by using simple

per-turbation theory to determine the evolution with time of the x and y components of the

spin of a neutron when a magnetic field B is switched on at t= 0 Taking the quantisation

axis z of angular momentum in the direction of B; the wave function of a neutron with its

spin inclined at an angle θ= 2 tan−1(a/b) to z can be written ψ(0) = aψ++ bψ− The

x and y components spin are &(ab∗) and '(ab∗), respectively, so that if a and b are real

the moment lies in the x–z plane The matrix elements of the perturbing Hamiltonian are

all are zero for t < 0.

If the unperturbed energy is E0, the perturbed wave function after time t becomes

Spherical neutron polarimetry 219

and the x and y components of the spin are

S x= &abe iγNBt

When considering the effect of a nonuniform magnetic field, two special situations can

be recognised Firstly, if the change in field direction during one cycle of Larmor sion (2π/ΩL) is negligibly small compared with the field itself, then transitions between

preces-the two spin states do not occur and preces-the wave function changes slowly remaining an

eigen-state of the new Hamiltonian This is the adiabatic approximation in which the neutron polarisation follows the field direction The second simple situation, the abrupt transi-

tion, occurs when the field changes rapidly from one uniform value to another in a timemuch less than that needed for a complete Larmor precession In such a case the neutronpolarisation does not change at the boundary between the two field regions, but passes in-

stantaneously from one precession regime to the other Adiabatic and abrupt changes in

the directions of magnetic fields, together with controlled precession, are the means used

to manipulate the neutron polarisation in polarisation analysis experiments

1.2 Classical polarisation analysis

The earliest and the simplest experimental arrangement for polarisation analysis is thatdescribed by Moon, Riste and Koehler [1] in the late sixties In this arrangement, sketched

in Figure 1 all the magnetic fields are parallel to a single direction usually vertical, and theneutrons are polarised and analysed with respect to this direction In the figure, a neutron

Fig 1 Triple-axis spectrometer for polarisation analysis in a vertical magnetic field.

220 P.J Brown

beam from a reactor enters the instrument from the left Neutrons of the chosen wavelength,polarised parallel to the polarising field, are selected by the monochromator They areguided onto the spin flipper F1, which when activated causes their spins to rotate by 180◦.The beam scattered by the sample passes through a second flipper F2 and is guided ontothe analyser which allows only neutrons of the chosen wavelength, polarised parallel tothe analyser field, to pass to the detector The results of experiments using this technique

are usually analysed in terms of four cross-sections: I++, I+−, I−+ and I−− The plusand minus signs indicate polarisation parallel and antiparallel to the field direction, the firstsuperscript indicates in direction of incident polarisation, it is “+” when F1 is deactivatedand “−” when it is activated The second subscript indicates the direction of analysis,switched using F2 These cross-sections are related to the general cross-sections defined

above as I++= I zz , I+−= I −zz , , etc.

1.3 Multidirectional polarisation analysis

The simple technique described above may be extended to measure some of the other larised cross-sections This was achieved in the Moon, Riste and Koehler experiment byrotating the magnet providing the sample field Alternatively the sample field may be pro-vided by a set of three perpendicular Helmholtz coils centred at the sample position [2]

po-By adjusting the currents in the three coils the field at the sample may be made to be allel to any arbitrary direction To ensure that the polarisation of the beam is maintained,the guide fields between the polariser and the sample, and between the sample and theanalyser must be arranged so that the changes in the field direction take place sufficientlyslowly to satisfy the adiabacity condition With the three coil set-up, often referred to as

par-XYZ-polarisation analysis, all the cross-sections I ±i±i , i = x, y, z, can be measured

with-out moving the sample When used in conjunction with a multidetector it allows data fordifferent scattering vectors to be measured simultaneously

1.4 Spherical neutron polarimetry

Neither of the two techniques just described allow components of polarisation scatteredperpendicular to the incident polarisation direction to be measured If such scattering

occurs (I ij = 0, i = j) these components will precess around the sample field tion j and only their projection onto j will be measured by the analyser The polarisa-

direc-tionP jj = (I jj − I j −j )/(I jj + I j −j ) will be less than unity, but the experiment cannot distinguish whether the beam is still polarised, but not parallel to j , or whether is truly

depolarised because it has been rotated in different directions in different parts of the ple The off diagonal terms (P ij , i = j) in the polarisation matrix can be measured using a technique known as spherical neutron polarimetry (SNP) which has been implemented in

sam-the CRYOPAD device [3] In this device superconducting Meissner screens are exploited

to provide the conditions for an abrupt transition between different magnetic field regimes.

The principles of operation and the essential elements of the CRYOPAD device are trated in Figure 2 Figure 2(a) shows a horizontal section through the device; the cryostat

illus-Spherical neutron polarimetry 221

Fig 2 The cryogenic polarisation analysis device CRYOPAD II (a) Horizontal section through the beam path;

(b) schematic representation of the rotation of the neutron spins in the incident beam path.

containing two cylindrical Meissner shields has itself the form of a hollow cylinder at thecentre of which the sample and its independent sample environment is placed The twoprecession coils, wound from superconducting wire, lie between the two Meissner shields.The primary coil is a complete toroid and the secondary coil is part of a second toroid,wound over the primary coil, in the region through which the incident beam passes Fig-ure 2(b) illustrates the rotation of the neutron spin directions in the incident beam path.The incident beam is polarised along its direction of motion, represented by the arrow A,

it passes into the nutation region in which the field direction changes gradually from being parallel to A, to a direction perpendicular to A, making an angle θinwith the vertical Theneutron spins follow the field adiabatically and on arriving at the first Meissner screen areparallel to the arrow B Between the two Meissner screens the neutron spins precess about

a horizontal axis under the influence of the combined fields of the two precession coils and

at the inner Meissner screen, have precessed through an angle χinso that they are parallel

to the arrow C The inner Meissner screen isolates the zero field region from the sion fields; within it the spins remain parallel to C except in so far as they are changed byinteraction with the sample In the outgoing path, a similar process guides the spins in the

preces-scattered beam so that those oriented with the chosen χout, θoutare parallel to the field axis

at the analyser The angles θin, θoutare fixed by the angles of rotation of the magnetic fields

in the nutators, and the angles χin , χoutby suitable adjustment of the currents in the primary

and secondary precession coils For a reflection with Bragg angle θBthe directions of

inci-dent polarisation and of analysis can be made parallel to the orthogonal directions x, y, z with x