Neutron Scattering from Magnetic Materials part 7 ppsx

den-A first and natural model to represent a spin density is to consider it as the square of thewave function of the magnetic electrons.. where the sum runs over all the magnetic atoms.T

176 J Schweizer

Fig 8 Enaminoketon C 18 H 28 O 2 N 2 Cu(NO) 2spin density projection along the axis [24]: (a) T= 40 K, contour

step 0.008µB/Å2, (b) T = 4 K, contour step 0.090µB /Å2.

striking fact is the quasidisappearance of the spin density on N2O2and N6O6 These two

NO groups belong to the two different molecules: N2O2to molecule A and N6O6to cule B They are facing each other at a distance of 3.40 Å It is obvious that the negativecoupling between adjacent molecules, revealed by the drop of the magnetic susceptibility,corresponds actually to a negative coupling between these two spin carriers which provokes

mole-an almost complete dimerization of their spins

4.2.3 Modeling the magnetic wave function. A convenient way to retrieve the spin sity distribution, while avoiding the problems due to Fourier inversion, is to model the spindensity and to determine the parameters involved in the model by a refinement procedurefrom the experimental data

den-A first and natural model to represent a spin density is to consider it as the square of thewave function of the magnetic electrons The structure factor can be expressed by

where the sum runs over all the magnetic atoms.

Two types of terms enter the expression of F M:

• one center integrals



ϕ∗

jei Q r ϕ

which represent the main contribution;

• two center integrals

magnetic form factor and we shall consider successively the case of p and d electrons with

a magnetic moment which is mainly of spin origin, and the case of f electrons (rare earths

and actinides) where spin and orbit couple together to give a total angular momentum.The general treatment can be found in [25,26] or in [27]

(i) Form factor of p and d electrons in the pure spin case.

For the pure spin case, the magnetic form factor f (  Q) can be defined by

µf Q

=



where µ is the magnetic moment The one electron atomic wave function ϕ is expanded in

a radial part R(r) and in an angular part

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

 (θ, ϕ) are the usual spherical harmonics In this sum  = 0, 1, 2, for s, p, d, electrons In most of the cases only one value of  is concerned by magnetism.

One expands the exponential ei Q ·r,

where, besides θ , ϕ, the angular coordinates of r, θ Q and ϕ Qare the angular coordinates

of Q and where the j (z) are the spherical Bessel functions (see the Appendix).

an expression which depends on the first quantum number na and where the Slater

ex-ponent α, characteristic of the two quantum numbers na and  of the orbital, have been

tabulated, for instance, by Clementi and Roetti [29] In these conditions, the radial

in-tegrals can be expressed as analytical functions of Q and α (see the Appendix for the

expressions ofj0 and j2 for the 2p electrons and expressions of j0, j2 and j4 for

the 3d electrons) Such radial integrals are displayed in Figure 9 for the 2p electrons of

oxygen

Fig 9 The radial integrals and the form factor f (  Q) calculated for oxygen.

Coming back to (84), the coefficients C LM are given by

Because of the triangular relations which exist for the 3j symbols

From the knowledge of the magnetic wave function, the calculation of the form factor is

straightforward In particular, for a 2p orbital of oxygen, aligned along Oz (a 2p zorbital)the form factor is expressed as

f Q

= j0(Q) +1− 3 cos2θ Q

j2(Q)

It is a very anisotropic form factor, depending on θ Q, the angle between Q and the Oz

di-rection It is limited by the two functionsj0(Q)  − 2j2(Q)  and j0(Q)  + j2(Q) for

θ Q = 0 and θ Q = π/2, as represented by Figure 9.

In a general way, the wave function is a very convenient model to describe a spin density

distribution The adjustable parameters such as the wave function coefficients a m, or the

Slater exponents α which modify the expansion of the radial part R(r), can be refined In

particular, such a model is very suitable to detect the weak spin density which exists oncertain atoms

An example of such a treatment is given in Figure 10 It concerns the spin density tained by wave function modeling of the NitPy (C≡C−H) free radical [30], a compoundwhere the shortest contact (2.14 Å) between molecules correspond to a weak C≡C−H · · · Ohydrogen bond The main part of the spin density is carried by the ONCNO fragment, with

ob-a negob-ative sign on the centrob-al cob-arbon, but weob-aker contributions, with ob-alternob-ated signs, ob-alsoappear on the skeleton of the molecule A significant spin density has been found on thehydrogen atom of the hydrogen bond, an indication of the active role played by this bond

in the magnetic coupling of the molecules

180 J Schweizer

Fig 10 Projection onto the nitroxide mean plane of NitPy (C ≡C−H) of the spin density as analyzed by wave

function modeling: (a) high-level contours (step 0.04µB/Å2); (b) low-level contours (step 0.008µB/Å2).

In the former example, the magnetic wave function was refined from the experimental

magnetic structure factors F M (h, k, l) It is also possible to use programs which are adapted for acentric structures as they start directly from the flipping ratios R.

(ii) Form factors of p and d electrons, with an orbital moment contribution.

When the orbital contribution to the magnetic moment is not completely quenched, themagnetic moments includes not only a spin part but also an orbital part,

Let us note that as the orbital magnetization is being produced by orbital currents, it

is more localized than the spin magnetization due to the same magnetic electrons andtherefore its form factor falls down less rapidly in the reciprocal space

(iii) Form factors of f electrons: Rare earths and actinides.

For rare earths and actinides the spin-orbit coupling is large Spin and orbit couple gether to give a total angular momentum,

to-

J = L + S·

J is a good quantum number (J = L − S for the first half and L + S for the second half ),

and the magnetic moment can be expressed as

A simplified expression for the form factor is given by the spherical approximation

Table 1 displays the values of c2for the different fillings of the f shell One can note

the particular case of five electrons where the spin part and the orbital part almost cancel,giving unusual shapes for the form factor This is illustrated in Figure 12 for SmCo5, where

the maximum at Q = 0 results from the different spatial extensions for the spin part andfor the orbital part, the sign of both contributions being opposed [33]

Fig 11 The form factor and the magnetization density of Ce 3+ calculated for the two cubic states Γ7

and Γ [32].

Fig 12 The form factor of Sm measured in SmCo5at 300 K [33].

4.2.4 Modeling the spin density: The multipolar expansion. It is possible to obtain amore versatile model of the magnetization (spin) distribution by parametrizing the magne-tization density directly rather than parametrizing the wave function A well-adapted modelresults from a multipolar expansion of the density around the nuclei at rest [34] It consists

of a superposition of aspherical atomic densities, each one described by a series expansion

in real spherical harmonic functions y  m ( ˆr),

where the P m are the population coefficients and R d(r) are radial functions of the spin

density of the Slater type As the density is the square of the wave function, we can take

the radial functions Rd (r) of the density as the square of the radial functions R  (r) of the

amplitude and write them in the following way:

R d(r)= ζ (nd+3)

(nd+ 2)! r

with ζ = 2α and nd= 2(na− 1) This would give for 1s orbitals (na= 1) nd= 0, for

2s and 2p orbitals (na= 2) nd= 2, for 3s, 3p and 3d orbitals (na= 3) nd= 4,

The magnetic structure factors become

with the spherical Bessel functions j  (z) The thermal motion of the atoms is taken into

account through the term e−W The real spherical harmonics y m

 ( ˆr) are linear combinations

of the usual spherical harmonics Y  m ( ˆr),

An example of application of this method is given for the spin distribution of the tanolsuberate (C13H23O2NO)2 This molecule is a binitroxide free radical where the unpairedelectrons are localized on the NO groups located at the two ends of the chain molecules

The flipping ratios of reflections (0kl), up to sin θ/λ = 0.45 Å−1, were investigated [35].Actually, only reflections with a large nuclear amplitude were measured, giving a partial

set of 69 magnetic structure factors F Mobs Figure 13 compares two spin density structions obtained by Fourier inversion and by multiple expansion The last map clearlyshows less noise, an enhanced resolution and also values of the density closer to realitythan the partial Fourier summation

recon-Another example of interest is the reconstruction of the spin density of the radical-based

cyano-acceptor tetracyanoethylene (TCNE)•+[36] The main part of the density is

local-ized on the central sp2carbon atoms but, due to both the spin delocalization and the spin

184 J Schweizer

Fig 13 Tanol suberate: the comparison of the spin distribution projected along thea direction (a) by Fourier

inversion, (b) by multipole expansion [35].

polarization, the spin density extends also on the other atoms: 33%,−5% and 13% of the

total spin on the sp2carbons, sp carbons and the nitrogen atoms, respectively However,

when reconstructing the spin density from a refinement of the magnetic molecular wavefunction, the results were not satisfactory as the agreement between observed and calcu-lated structure factors was by far too poor To know the reason for this mismatch, moreflexibility was given to the model, and a multipolar expansion of the spin density was per-formed and refined up to the octupoles The result is presented in Figure 14, which shows

clearly what happens The π∗molecular orbital is antibonding and, on the sp2carbons, theprojected spin density is pushed away from the center of the C–C bond Such an effect wasimpossible to see in the molecular orbital refinement, the model being not flexible enough.This method of modeling the spin density can be extended to acentric structures(see [37]) for the DPPH (diphenyl picryl hydrazil) spin density In such cases, the least-square refinement compares directly the experimental and the calculated flipping ratios

4.3 Investigation of noncollinear magnetic structures

The flipping ratio method can also be used to investigate complex distributions, either whenthe magnetic structures are naturally canted or when the noncollinearity results from theapplied magnetic field In such situations, expressions (61) and (62) of the flipping ratioare no more right for two reasons:

(i) the scheme of Figure 3 being not legitimate, relations (59) and (60) are not fulfilledand the sin2α simplification cannot be applied;

(ii) for noncollinear structures, F M⊥ is not supposed to be parallel to F∗

M⊥, and the

Fig 14 Projection of the spin density of the tetracyanoethylene plane, projection reconstructed by multipolar

expansion [36].

chiral term (  F M⊥∧ F∗

M⊥) zcannot be ignored This term, being of the order two in magnetic

amplitude, could be neglected only if F M is small compared to F N

Therefore, the exact expression (57) for the flipping ratio has to be applied, and the flippingratio method is still very accurate and may be very useful

This has been done to understand the action of a magnetic field on the very anisotropiccompound Ce3Sn7 This compound is unusual As already seen (Figure 5), it is an interme-diate valence compound: there are two sites for the cerium atoms, but only atoms of the site

CeIIcarry a moment; atoms CeIdo not carry any moment Below 5.1 K, the moments of

CeIIorder antiferromagnetically, splitting the four positions of this site in two sublattices:

m1and m3in one direction and m2and m4in the opposite direction The unusual feature

is that axisc is the spontaneous axis, the axis along which the antiferromagnetic moments align themselves, but with a very small moment of 0.36µBonly When a field is appliedalong the a axis, the antiferromagnetic structure is broken and an average magnetization

of 3.5µB/Ce3Sn7unit is already reached at 1.5 T, while when the same field is appliedalong the spontaneous axisc, the induced magnetization is lower by almost two orders of

magnitude (see Figure 15(a)) Furthermore, at higher fields, two transitions occur Flippingratios were measured below the first transition under a field of 4.6 T applied along thespontaneous axisc [38] The result is surprising and displayed in Figure 15(b): though the

field is applied alongc, both sublattices lean over axis a, which is not the spontaneous axis

but seems to be “easy” in the sense that moments, when they are aligned along this axis,are larger than when they are aligned, even spontaneously, along any other axis

Paramagnetic Sm3Te4is another case of very anisotropic compound The properties ofthis mixed valence compound were described by the superposition of magnetic Sm3+andnonmagnetic Sm2 + In the acentric cubic symmetry I ¯43d, the Sm atoms occupy a single

site (12a) with a ¯4-local symmetry and with an easy axis which is along x, y or z A valence

order between Sm3+and Sm2 +had been proposed, with an atom ratio Sm3 +: Sm2 +equal

to 2: 1 [39] For a field applied along [110], it has been observed that the Sm atoms are

186 J Schweizer

Fig 15 Ce 3 Sn 7 : (a) magnetization measured along the different axes of the crystal; (b) magnetic moments of

the 2 sublattices without field and with a 4.6 T field applied along the spontaneous axisc.

distributed on two subsites: SmI with a magnetic moment and SmII without a magneticmoment (Figure 16) The atomic ratio SmI: SmIIis 2: 1, supporting the idea of a valenceordering However, it has to be noted that atoms SmIhave an easy axis at an angle of 45◦with the applied field, while atoms SmIIhave their easy axis at 90◦ from the field In asecond experiment [40], for a field applied along[001], it has been observed that the Smatoms split into two subsites: a subsite Sm with a large induced moment (field parallel tothe easy axis) and a subsite Sm⊥with a small induced moment (orthogonality between thefield and the easy axis) In this case, the atomic ratio Sm : Sm⊥was 1: 2 instead of 2 : 1(Figure 17) This ruled out completely the valence order hypothesis

A very important achievement has been made by Gukasov and Brown to treat the ping ratio data in the case of anisotropic paramagnetic compounds [41] Noting that, insuch a case, when a magnetic field is applied, the induced magnetic moment does not,

flip-in the general case, align along the magnetic field, they propose to use the susceptibilitytensor ¯¯χ and write

m i=

j

The number of independent components of ¯¯χ depends on the symmetry of the magnetic

site: it ranges from one for a cubic site symmetry to six in the most general case netic atoms on the different positions of a same crystal site have ¯¯χ tensors related by the

Mag-symmetry operators which connect these positions in the same way as the vibration sor parameters are connected Therefore, instead of refining the magnetic moments on thedifferent positions of all the magnetic atoms, it is enough to refine only the independentcomponents of the susceptibility tensors of the different magnetic sites The number ofparameters to refine may be very much reduced Furthermore, these parameters depend

ten-Fig 16 Magnetization in Sm3Te4: the magnetic field is along [110] (perpendicular to the page).

Fig 17 Magnetization in Sm3Te4: the magnetic field is along [001] (also perpendicular to the page).

neither on the direction nor on the strength of the field, which allows to process togetherdifferent sets of measurements

The application of this approach to the Nd atoms in Nd3−xS4, isomorphous of Sm3Te4

is convincing [41] As for Sm, the local symmetry of Nd is ¯4, and therefore the

susceptibil-ity tensor contains only two independent parameters: χ11= χ22and χ33 The refinement

of a first experiment with 116 flipping ratios measured with the field applied along[001]

gave, for a field of 7 T, χ11H = χ22H = 1.45(5)µBand χ33H = 0.55(5)µB The ment of a second set of 122 flipping ratios measured for a field applied along[421] yielded

refine-for the same field magnitude χ11H = χ22H = 1.43(8)µBand χ33H = 0.76(16)µB Theparameters, for the two sets of data and for the two directions of the field, are the samewithin the error bars The atomic ratio between those atoms with a large induced moment

188 J Schweizer

Fig 18 Induced magnetic moments in Nd 3−x S4: the magnetic field is along [001] (along the page).

to those with a small induced moment (Figure 18) is here 2: 1 (ratio Nd2 : Nd1) If wecompare with Sm3Te4it was 1: 2 (for the atoms Sm : Sm⊥) This is due to the differencebetween the Sm3+and Nd3 +ions: Sm3 +has an easy magnetization axis and its suscepti-bility ellipsoid is elongated (prolate) while Nd3+has an easy magnetization plane and itssusceptibility ellipsoid is flattened (oblate) To conclude with this comparison, it is clearthat the refinement of the atomic susceptibility parameters would have been completelyadapted to process the data of Sm3Te4

5 Uniaxial (longitudinal) polarization analysis

In 1969, Moon, Riste and Koehler opened a new domain of investigation in neutrons tering In their paper [3], they showed that it is possible to increase the power of a polarizedneutron diffractometer by adding a polarization analyzer after the sample, an analyzer that

scat-is identical to the polarizing monochromator, adding also a second flipping device betweenthe sample and the analyzer, as shown in Figure 19 This way, it becomes possible to mea-sure four cross-sections, depending on the polarization of the incoming neutrons and onthe polarization of the outgoing neutrons

5.1 The four partial cross-sections and the polarization of the scattered beam

The characteristics of a polarization analysis experiment made on such an instrument are

that the neutrons are polarized in a given direction of space (labeled z in the paper of Moon

et al.) and are analyzed in the same direction of space This polarization analysis method

is therefore called uniaxial polarization analysis method Sometimes the appellation

“lon-gitudinal” polarization analysis is used instead of “uniaxial” polarization analysis

Fig 19 The main features of the uniaxial polarization analysis [3].

The partial cross-sections. The instrument described above allows us to select the spinstate of the incoming beam|+ or |− and the spin state |+ or |− of the scattered beam.Therefore, the four scattering amplitudes given by formulae (33)–(36) are relevant andpermit us to express the four partial cross-sections that are directly measured in such auniaxial polarization analysis method

Restricting first to states λ and λ of the target before and after the scattering, thesepartial cross-sections can be written as

cross-a sccross-attering which is only due to the components of M⊥perpendicular to P

The uniaxial polarization of the scattered beam. If we consider now the uniaxial

Fig... Γ7< /sub>

and Γ [32].