Studies of magneto-electric crystals using spherical neutron polarimetry

Solid State Sciences 7 (2005) 682–689 www.elsevier.com/locate/ssscie

Studies of magneto-electric crystals using spherical neutron polarimetry P. Jane Brown a,c,∗ , J. Bruce Forsyth b , Francis Tasset a a Institut Laue Langevin, BP 156, 38042 Grenoble, France b Rutherford Appleton Laboratory, Chilton Oxon, OX 11 0QX, UK c Department of Physics, Loughborough University, Loughborough LE11 3TU, UK

Received 4 November 2004; accepted 9 November 2004 Available online 21 April 2005

Abstract Spherical neutron polarimetry provides a new tool for the study of magneto-electric crystals since it allows the rotation of scattered polarisation, characteristic of this class of crystal to be measured. Measurement of this rotation not only allows the relative populations of the magneto-electric domains to be determined, but also permits precise determination of the antiferromagnetic structure factors and hence of the magnetisation distribution leading to magneto-electricity. The method was initially applied to Cr2 O3 ; in the present article we describe its use to study other magneto-electric crystals including LiCoPO4 and MnGeO3 .  2005 Elsevier SAS. All rights reserved. Keywords: Magneto-electricity; Antiferromagnetism; Neutron polarimetry

1. Introduction 1.1. The magneto-electric (ME) effect A linear magneto-electric (ME) effect in which magnetisation can be induced by the application of an electric field can only exist in crystals with ordered magnetic structures having particular magnetic symmetries [1]. A necessary feature in centrosymmetric crystals is a magnetic structure with zero propagation vector; if the centre of symmetry is combined with time inversion in the magnetic structure an ME effect can exist [2]. The magnetic moments in such structures can order in one or other of two 180◦ domains differing only in that the magnetic moments have opposite directions with respect to the arrangement of their ligand atoms. Such domains are related to one another by the time reversal operation rather than by any spatial rotation or inversion. E.F. Bertaut was one of the first to appreciate the relationship between the symmetry constraints on the ME effect and the symmetry properties of magnetic structures [3]. His * Corresponding author.

E-mail address: [email protected] (P.J. Brown). 1293-2558/$ – see front matter  2005 Elsevier SAS. All rights reserved. doi:10.1016/j.solidstatesciences.2004.11.014

group in Grenoble was active in identifying and characterising most of the ME materials which are the subject of this paper. Experiments have shown that although ME susceptibilities have a unique temperature dependence, their magnitudes and even their signs are specimen dependent. This variability is due to the presence of 180◦ antiferromagnetic domains which have opposite ME coefficients. The measured ME susceptibility αobs is related to the intrinsic susceptibility α0 by αobs = ηα0

with η = (v1 − v2 )/(v1 + v2 )

(1)

where v1 and v2 are the volumes of crystal belonging to each of the two domains. Magnetic annealing, in which the crystal is cooled through its Néel temperature in a magnetic field or in combined electric and magnetic fields, can strongly enhance the ME susceptibility by favouring the growth of one or other of the 180◦ domains. Until recently it was not possible to determine the degree to which the 180◦ domain populations were unequal, and hence the absolute values of the ME coefficients. However lately two techniques have become available which can distinguish these domains. In second harmonic generation spectroscopy, interference between

P.J. Brown et al. / Solid State Sciences 7 (2005) 682–689

the magnetic and the non-linear electric susceptibilities can lead to a different output of right and left circularly polarised light propagating along the optic axis. Since the non-linear electric susceptibility changes sign under time inversion, whilst the magnetic susceptibility does not, this difference is of opposite sign for different 180◦ domains; the resulting contrast enables images of the distribution and mobility of ME domains to be made [4]. The completely different technique of spherical neutron polarimetry does not have the spatial resolution of the latter method, but allows determination of the relative domain populations and a rather direct determination of the absolute spin configuration associated with the dominant domain.

with Ix = M 2 + N 2 + Px Jyz , Iy = M 2 + N 2 + Py Rny , Iz = M 2 + N 2 + Pz Rnz , I = M 2 + N 2 + Px Jyz + Py Rny + Pz Rnz N 2 = N N ∗,



P = PP + P



or in components P i = Pij Pj + P

i

∗ ) Jij = 2(M⊥i M⊥j



(N 2 − M 2 + Ryy )/Iy

Ryz /Iy

−Jny /Iz

Rzy /Iz

(N 2 − M 2 + Rzz )/Iz



−Jyz /I  P = Rny /I  Rnz /I 

∗ and Jni = 2(N M⊥i ).

(5)

The polarisation in the plane containing M⊥ (k) and the scattering vector (x axis) is rotated. The direction of rotation depends on the relative phases of N and M⊥ and is opposite for the two 180◦ domains. The experimental quantities which are obtained in an SNP experiment are the components Pij of a polarisation matrix Pij =

(2)

−Jnz /Iy

∗ Rni = 2(N M⊥i ),

The off-diagonal components of P give components of scattered polarisation which are not parallel to the incident direction, and hence describe the rotation of the polarisation in the scattering process. For centro-symmetric ME crystals the origin can be chosen at a centre of symmetry so that the nuclear structure factors are real; the symmetry conditions governing the appearance of an ME effect then ensure that the magnetic structure factors will be pure imaginary quantities. In this case all the Rni and Jij are zero so that P = 0 and no polarisation is created by scattering. If z is chosen to be parallel to M⊥ , the polarisation tensor has the simple form:
 ξ ζ 0 1−γ2 2γ ; ξ = . (6) P = −ζ ξ 0 with ζ = 1+γ2 1+γ2 0 0 1

where Pij is a tensor describing the rotation of the polarisation in the scattering process and P is the polarisation created. A set of right handed orthogonal axes are defined with x parallel to the scattering vector k and y and z in the plane perpendicular to it (usually z is chosen to be vertical). With this definition there is no component of the magnetic interaction vector M⊥ parallel to x. Using the equations given above, and writing the ith component (i = y, z) of M⊥ as M⊥i , the components of P and P on these axes can be expressed as:
(N 2 − M 2 )/Ix  Jnz /Ix Jny /Ix P=

M 2 = M⊥ · M∗⊥ ,

∗ ), Rij = 2(M⊥i M⊥j

Spherical neutron polarimetry (SNP) is the name given to the class of polarisation analysis experiments in which the polarisation of the incident beam can be chosen in any arbitrary direction, as can the direction in which the polarisation of the scattered beam is analysed. The two directions need not be parallel, and it is this ability to determine rotation of the polarisation direction which distinguishes SNP from xyz polarisation analysis [5]. At the present time the cryogenic polarisation analysis device CRYOPAD [6], is the only instrument capable of making routine SNP experiments at finite scattering angles, although ways of effecting SNP using precessing polarisation have been suggested [7]. The fundamental equations which describe the way in which a polarised neutron beam is scattered were developed independently by Blume and Maleev in the early 60s [8,9]. For the purposes of polarisation analysis the relationship between the incident polarisation P and the scattered polarisation P implied by these equations can conveniently be described by a tensor equation [10]: 

(4)

where

1.2. Spherical neutron polarimetry



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σij++ − σij+− σij++ + σij+−

(7)

where σij++ and σij+− represent the partial cross-sections for neutrons incident with spins parallel to j , scattered with spins parallel and antiparallel to i respectively. The polarisation matrix is obtained from the polarisation tensor by averaging over all magnetic domains which contribute to the scattering. For a magneto-electric crystal, in which the only magnetic domains are of the 180◦ type with domain population η (Eq. (1)), the polarisation matrix has the same form as the polarisation tensor of Eq. (6) except that ζ becomes 2γ η/(1 + γ 2 ), ξ is unchanged. ,

(3)

2. Experiments on Cr2 O3 Cr2 O3 is one of the best known magneto-electric crystals and has a Néel transition (307 K) conveniently close to ambient temperature; it was therefore chosen for the first SNP

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Fig. 1. The double octahedral coordination polyhedra found in Cr2 O3 showing the moment directions of the Cr3+ ions after cooling in (a) parallel, (b) antiparallel electric and magnetic fields.

experiments. Cr2 O3 has the corundum structure in which the Cr is octahedrally coordinated by oxygen but the Cr sites are not centres of symmetry. Pairs of octahedra related by a centre of symmetry share a common face as illustrated in Fig. 1; they are linked to other pairs in the three dimensional structure by sharing their free vertices. In the antiferromagnetic state the Cr spins are parallel to the triad axis, atoms with opposite spins being related by the centres of symmetry. In structural terms the 180◦ domains are distinguished by the direction in which the spins point relative to the shared face of the octahedral pairs. We have investigated the longitudinal ME effect in Cr2 O3 and determined that the magnetic configuration stabilised by cooling a sample through its Néel point with parallel magnetic and electric fields applied along the triad axis is that in which the spins point towards the shared face [11]. Eq. (6) shows that the polarisation matrices for Bragg reflections from ME crystals depend sensitively on the ratio γ between the magnetic and nuclear structure factors. This property was exploited to determine the Cr2+ form-factor in Cr2 O3 by measurement of the polarisation rotation in magnetically annealed crystals [12]. The points on the Cr3+ form factor obtained from the measured structure factors are plotted in Fig. 2 where they are compared with the Cr3+ free ion form factor. It can be seen that for most reflections an extremely good precision was obtained. Exceptions occur when either the nuclear structure factor is small so that γ  1 or when the geometric structure factor of the magnetic atoms is small so that the reflection is insensitive to the form factor.

3. Transition metal lithium orthophosphates The lithium orthophosphates LiT PO4 (T = Mn, Fe, Co, Ni) form an isomorphous series with the olivine structure, space group Pnma. All four compounds order antiferromagnetically with Néel temperatures ranging from 50 K for LiFePO4 to 20 K for LiNiPO4 . Their magnetic structures were determined from powder neutron diffraction [13–15], and provide one of the early examples of structures to which Bertaut’s group theoretical method was applied [3].

Fig. 2. The experimental values of the magnetic form factor measured at the h0. Bragg reflections of Cr2 O3 . The smooth curve is the spin-only free-ion form factor for Cr2+ normalised to the experimental value at the ¯ lowest angle reflection (1012).

Fig. 3. Schematic representation of the magnetic structures of the transition metal lithium orthophosphates projected down [010], showing the two different 180◦ domains in each case; (a) LiMnPO4 and LiFePO4 , (b) LiCoPO4 , (c) LiNiPO4 .

The magnetic atoms occupy the 4(c) positions of the space group 1: (x, 1/4, z), 2: (x, ¯ 3/4, z¯ ), 3: (1/2 − x, 3/4, 1/2 + z), 4: (1/2 + x, 1/4, 1/2 − z); with x ≈ 0.28 and z ≈ 0.97. If the spins on the four sites are represented by the vectors Si , (i = 1, 4) the magnetic structure of all four compounds corresponds to the combination S1 − S2 − S3 + S4 . In LiMnPO4 and LiFePO4 the moments are parallel to a and in LiCoPO4 and LiNiPO4 to b and c respectively. The three different structures are illustrated in Fig. 3. The corresponding magnetic space groups are Pn m a , Pn m a , Pnma , Pnm a for LiMnPO4 , LiFePO4 , LiCoPO4 and LiNiPO4 respectively. Since in all four compounds the inversion operation which relates S1 to S2 reverses the spin, the ME effect is allowed. It has been demonstrated in LiMnPO4 , LiCoPO4 and LiNiPO4 [16,17]. The group theoretical constraints predict a parallel ME effect in LiMnPO4 with three independent coefficients

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αxx , αyy and αzz , whereas in LiCoPO4 and LiNiPO4 the effect is transverse with two non-zero coefficients in each case: αxy , αyx for LiCoPO4 and αxz , αzx for LiNiPO4 . More recently the ME polarisabilities of LiCoPO4 and LiNiPO4 have been studied in high magnetic fields [18,19] and it has been shown that, contrary to what is expected for this class of antiferromagnetic structure, in LiCoPO4 a single domain state can be induced by application of magnetic field alone. We have used SNP to study the spin configurations favoured by applying fields in different directions whilst cooling LiCoPO4 and LiNiPO4 through their Néel transitions. For LiCoPO4 with magnetic space group Pnma the h0l reflections with h even contain both nuclear and magnetic contributions; they are absent for h odd. A LiCoPO4 crystal of about 2 × 2 × 5 mm with well formed faces was fixed with [001] vertical on an insulating glass rod between two parallel plane aluminium electrodes ≈ 10 × 5 mm in area placed 5 mm apart and connected to a radio-frequency generated H.T. supply. The whole assembly was mounted inside a thin-tailed He flow cryostat. Once the sample space was cooled and evacuated to a pressure of a few mb it was possible to apply a voltage of 3500 V to the condenser plates before breakdown occurred. The cryostat tail was placed between the poles of an electromagnet giving a horizontal field of 1.6 T. Initially the electric field was parallel to ¯ ¯ The crystal was cooled [100] and the magnetic field to [010]. through its Néel temperature (23 K) with this arrangement of crossed electric and magnetic fields. The cryostat with the sample was then removed from the magnet and placed in the cylindrical zero-field region of Cryopad II mounted on the polarised triple axis spectrometer IN20 of the Institut Laue Langevin. Measurements were made of the polarisation scattered by a set of mixed magnetic and nuclear reflections. Significant rotation of the polarisation and very little depolarisation was observed indicating that the sample was nearly a single 180◦ domain. The sample was then heated to some 10◦ above its Néel point of 21 K and recooled in zero field. The polarisation scattered by mixed reflections was no longer rotated and strong depolarisation occurred. This behaviour is typical of reflections from an equi-domain crystal when |γ | ≈ 1 giving ξ ≈ 0. The cryostat was removed from the Cryopad and the field cooling cycle repeated with the electric field reversed. Again a mono-domain was formed but, as expected, it was of the opposite type from that stabilised previously and rotated the polarisation in the opposite sense. Further combinations of crossed electric and magnetic fields were tried with the results given in Table 1. Finally the Pxz component of the polarisation matrix was measured as a function of temperature to obtain a precise value for the Néel temperature. The results, shown in Fig. 4, give TN = 21.7(1) K. For LiNiPO4 ME coefficients αzx , αxz are expected. The LiNiPO4 crystal was therefore mounted with [010] vertical and the fields applied along [100] and [001]. The same values of electric and magnetic fields were used, but for LiNiPO4 the treatment did not yield a single 180◦ domain. In

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Table 1 Domain stabilised in LiCoPO4 by different combinations of electric and magnetic fields applied during cooling. Domain A is that for which the spin on the Co2+ ion at (0.28, 1/4, 0.97) points in the positive b direction E ¯ [100] [100] [010] ¯ [010]

H

α

Domain

¯ [010] ¯ [010] ¯ [100] ¯ [100]

αxy −αxy −αyx αyx

A B A B

Fig. 4. The temperature dependence, near to the Néel transition, of the po¯ reflection larisation scattered parallel to the scattering vector by the 410 from LiCoPO4 for incident polarisation parallel to [001].

¯ ¯ fact only one set of cooling conditions: E [100]H [001], produced even a small imbalance in the domain populations. The magnetic scattering observed in this one case, allowed the magnetic structure suggested by [5] to be confirmed and the Néel temperature 20.9(1) K to be determined.

4. The magnetisation distribution in LiCoPO4 SNP measurements made on the LiCoPO4 crystal in magnetically annealed states have been used to study the Co2+ form factor in LiCoPO4 . The experiment was carried out using Cryopad mounted on the polarised neutron diffractometer D3 at ILL. The polarisation of the scattered beam was measured by a 3 He neutron spin filter placed inside a cryogenic constant field enclosure [20]. The transmission of the filter was ≈ 22% and the analysing efficiency varied from ≈ 60% for new cells to a worst case of 25% after 48 h. The 3 He cell was changed at roughly 24 hour intervals throughout the experiment. The characteristics of each cell were determined from measurements, made periodically, of the unrotated polarisation scattered by a pure nuclear reflection

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with incident polarisation parallel to the z (vertical) direction. The LiCoPO4 crystal was mounted with the moment direction [010] vertical. Initially it was cooled with the electric field E [100] and the magnetic field H [001]. This combination did not lead to much domain imbalance, but values of γ for five reflections with large magnetic contributions were obtained from measurements of the diagonal components of the polarisation matrices. Subsequently the crystal was recooled with E in the same direction, but with H vertical and parallel to [010]. As expected, this combination of fields yielded a nearly single domain crystal (η = 0.84) and a −1 set of 24 h0l reflections (sin θ/λ < 0.5 Å ) were measured with the crystal in this state. In order to obtain a reasonable accuracy in determination of the scattered polarisation it was found necessary to spend approximately 2 hours measuring each of the stronger reflections and twice this time for the weaker ones. The consistency between measurements of equivalent reflections and between measurements of the same reflections made with different 3 He analyser efficiencies was good. The ratio γ of magnetic to nuclear scattering was determined using the method outlined in [11] and also taking into account the varying efficiency of the 3 He analyser. In this method initial values of γ for each reflection are calculated from the diagonal components of scattered polarisation Pxx and Pzz , and used with the corresponding rotated components Pxz and Pzx to obtain values for the domain ratio η. The weighted mean of η is then used to recalculate all the γ s. Standard deviations are obtained from the deviations of individual values from the mean. An effective form-factor for Co2+ shown by the filled triangles in Fig. 5 was derived from these results by multiplying γ by the nuclear structure factor and dividing by the unitary magnetic structure factor MU = (2m/eh¯ ] n sin 2π(hxn + lzn ) where n runs over the four Co2+ ions in the unit cell of LiCoPO4 . The points on the form factor obtained in this way by no means follow a smooth curve. In particular several reflections measured with good precision give points which lie well below the mean curve. In a further experiment on the same crystal, the measurements were extended to include the stronger h0l reflections −1 with 0.5 > sin θ/λ > 0.6 Å . Further checks were also made on the reliability of the results by measuring a number of reflections with the crystal in different ME domain states: η in the range 0.3–0.6. Initial measurements seemed to confirm the previous anomalous results, however it was observed that the spread in the values of the domain ratio obtained using different reflections was larger than would be expected from statistical errors alone. The effect was most apparent in reflections in which the ratio γ between the magnetic and nuclear scattering amplitudes was greater than unity (|γ | > 1). These are in general those for which the nuclear structure factors are rather small; they are therefore those in which the effects of multiple scattering may be significant. To take account of multiple scattering it has been

Fig. 5. Magnetic scattering per Co2+ ion in LiCoPO4 . The triangles show the values obtained in the first experiment. The filled circles show the mean values for the same reflections obtained from all the measured data and including the multiple scattering correction. The solid curve is the Co2+ free ion form factor scaled to 3.6 µB .

assumed that it arises predominantly from multiple nuclear scattering processes and has therefore no effect on the neutron polarisation. If the intensity of multiple scattering is S then the Pxx and Pzz components of the polarisation matrix become +− σ ++ − σxx N 2 + S − M⊥z 2 Pxx = −Pzz = xx ++ +− = N 2 + S + M 2 σxx + σxx ⊥z

=

1−γ2 +ν 1+γ2 +ν

with ν = S/N 2 .

(8)

In the Pxy and Pyx components on the other hand half of the multiple scattering is recorded in each channel and so cancels out in the numerator and only enters into the denominator. Pxz =

++ − σ +− σxz xz ++ σxz

+− + σxz

=

2ηγ 2ηFN FM = . 2 + FM + S 1 + γ 2 + ν

FN2

(9)

The value of γ including multiple scattering can be written in terms of the value obtained without a correction (γ  ) as γ = γ  (1 + ν)1/2 from (8),    1/2 1/2  1 ± 1 − ξ2 γ = γ  1 ± 1 − ξ 2 (1 + ν) from (9) with ξ = Pxy /η.

(10)

When |γ | > 1, the positive root in 10 is appropriate and the ratio γ /γ  is greater than 1 for Eq. (8) and less than 1 for Eq. (9), a result which can explain the large spread in the values of γ obtained. A correction for multiple scattering in the reflections with |γ | > 1 was obtained by determining the value of η using

P.J. Brown et al. / Solid State Sciences 7 (2005) 682–689

¯ of Fig. 6. Maximum entropy reconstruction of the projection down [010] the antiferromagnetic magnetisation distribution in LiCoPO4 . Contours are drawn at intervals of 0.7 µB Å−2 , negative contours are shown as dashed lines. The projected positions of the Co2+ ions and their coordinating O2− ions are indicated.

only strong reflections with 0.2 < |γ | < 1. This value was then used to determine the values of ν and γ for which Eqs. (8) and (9) were consistent. This technique yielded plausible values for the multiple scattering: ν ≈ 2 in the weakest reflection (501), which is the same order of magnitude as that observed in integrated intensity measurements made with the same crystal and wavelength. The points on the form factor curve obtained after correction for multiple scattering are shown as the circles in Fig. 5. The results of all the measurements have been concatenated to obtain a set of magnetic structure factors for 33 independent h0l reflections. These have been used to make a maximum entropy reconstruction of the projection down ¯ of the magnetisation distribution in LiCoPO4 . The [010] map obtained is shown in Fig. 6. The magnetisation is essentially confined to the Co2+ ions, which show an anisotropy which can be related to the positions of the coordinating oxygens. The acentricity of the distribution, which is fundamental to the ME effect, is very clear.

5. The magnetic structure of the ME compound Mn1−x Mgx GeO3 MnGeO3 exists in two polymorphs. The high temperature phase has the orthorhombic pyroxene structure; on cooling this phase transforms to one having the monoclinic clinopyroxene structure at a temperature around 200 ◦ C. This transition precludes the preparation of large single crystals of orthorhombic MnGeO3 , but it has been found that the low temperature phase transition can be suppressed by the addition

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of magnesium to form Mn1−x Mgx GeO3 [21]. The present experiments show that these substituted crystals are not really stable, but break up whilst partially transforming to the clinopyroxene phase. Orthorhombic MnGeO3 , has been shown to order magnetically at 16 K to a collinear structure with zero propagation vector, magnetic space group P b ca and moments parallel to b [22]. This magnetic group is one which can exhibit magneto-electricity and an ME susceptibility was subsequently observed below 11.1(3) K [23,24]. This temperature agrees well with a specific heat anomaly characteristic of magnetic order at 10.8(2) K. Single crystals with x = 0.10 and 0.15 were kindly provided by Professor Y. Endoh (Tohoku University, JAPAN). At the time of the experiment they had deteriorated, owing to the onset of the clinopyroxene transition, and were of rather poor quality. They were studied using unpolarised neutron diffraction on the 4-circle diffractometer D9 at ILL. The structural parameters and site occupancies given in Table 2 were obtained from integrated intensity measurements made at 20 K of 350 independent reflections from the x = 0.10 crystal. Measurements made below 10 K gave a strong indication that at low temperatures the Mg substituted compound orders with an incommensurate structure, based on the magnetic configuration given by Herpin et al. [22], with a propagation vector τ ∼ 0, 0, 0.2. A commensurate phase with zero propagation vector was found in the temperature range 8–5.5 K for the x = 0.15 crystal. Neutron Laue photography of a freshly zone melted crystal rod with x = 0.06 showed that its quality was far superior to either of the two crystals studied previously and so it was chosen for SNP. A rectangular parallelepiped was cut from the crystal rod ∼ 10 × 8 × 5 mm3 with axes parallel to a, b and c respectively. It was mounted with [100] vertical between two flat electrodes parallel to the (001) faces. The SNP experiment was carried out on D3 using the 3 He neutron spin filter, as in the experiment on LiCoPO4 . Measurement of the intensity of the 023 reflection while cooling showed Table 2 Structural parameters of the orthopyroxene structure obtained for Mn1−x Mgx GeO3 with x = 0.10. The isotropic temperature factors of the two Mn atoms and those of all oxygen atoms were constrained to be equal Atom x

y

z

B (Å2 )

Site b (fm)a

Ge1 Ge2 Mn1 Mn2 O1 O2 O3 O4 O5 O6

0.301(3) 0.033(3) 0.363(22) 0.353(12) 0.123(6) 0.170(9) 0.201(8) 0.030(6) 0.041(5) 0.310(5)

0.0263(2) 0.2302(2) 0.1233(11) 0.1216(7) 0.0559(4) 0.0628(3) 0.0676(3) 0.1800(4) 0.1907(3) 0.1927(3)

0.6632(8) 0.8424(8) 0.333(4) 0.989(2) 0.8132(11) 0.1591(10) 0.5140(9) 0.3422(10) 0.0096(10) 0.7784(9)

% Mgb

0.22(10) 0.33(11) 0.3(4) −1.72(20) 22(2) ” −2.94(21) 9(6) 0.59(8) ” ” ” ” ”

a The scattering length refined for the Mn sites. b Values calculated from the previous column using b Mn = −3.73 fm, bMg = 5.36 fm.

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a transition to the commensurate antiferromagnetic phase at TN ∼ 8 K followed by a further transition to the incommensurate phase τ ∼ 0, 0, 0.2 at TL ∼ 4.5 K. (Note: whilst the quality of this crystal remained good during the experiment it subsequently deteriorated and completely broke up after several months storage.) The ME susceptibilities allowed by the magnetic point group m mm are αyz and αzy . Cooling through TN in crossed magnetic and electric fields: 1.4 T [010] and 4000 V cm−1 [001] gave an almost single domain state, demonstrated by a strong Pxz component in the polarisation matrix of the 020 reflection. On cooling below TL and subsequent warming into the commensurate region this polarisation was lost, showing that memory of the domain structure is not retained in the incommensurate structure. It was however found that warming the crystal through TL in the same crossed electric and magnetic fields could induce a domain imbalance. There was some evidence that with the same field configuration, the opposite 180◦ domain was favoured when warming through TL to that favoured when cooling through TN although the results obtained in several heating and cooling cycles were not completely consistent. The temperature dependence of the components Pxz and Pzx of the polarisation matrix provides a particularly sensitive measurement of the transition temperatures, since the rotation of polarisation depends on coherent interference between the nuclear and magnetic scattering. The temperature dependence of Pxz for the 023 reflection is shown in Fig. 7, both warming through TN (triangles) and cooling through TL (circles). The Pxz , Pzx , Pxx and Pzz components of the polarisation matrix were measured for 24 independent re-

Fig. 7. Temperature dependence of the Pxz component of the polarisation scattered by the 023 reflection of Mn0.94 Mg0.06 GeO3 . The circles represent measurements made while cooling and the triangles those made while heating. The crystal had a different initial domain ratio for the two sets.

flections at 5 K with the crystal in a state with a strong domain imbalance. These data were used to determine the magnetic moments associated with the two independent Mn sites in the structure. In calculating the nuclear structure factors the positional parameters,and site preferences obtained for the x = 0.10 sample given in Table 2 were used. A least squares refinement gave 2.905(25) µB and 3.035(41) µB for the Mn1 and Mn2 sites respectively with a domain ratio 0.877(8). Allowing for the different occupancies, the mean moments of Mn atoms on the Mn1 sites are 3.23(3) µB whilst those of Mn on the Mn2 sites are 3.11(4) µB . To obtain the structure of the incommensurate phase the polarisation matrices for several 0kl ± τ reflections were determined. They were found in all cases to be diagonal with Pyy ≈ −Pxx ≈ −Pzz ≈ 1 showing that all moments are perpendicular to [100]. Finally the crystal was remounted with the b axis vertical and the polarisation matrices of a few h0l ± τ reflections measured. These, although still diagonal, showed strong depolarisation in the Pyy and Pzz components which was strongest for the h00 ± τ reflections. This behaviour is typical of a magnetic structure with moments rotating in the (100) plane in which the two different chirality domains are equally populated. It shows that the magnetic structure of the incommensurate phase of Mn0.94 Mg0.06 GeO3 is a cycloid with a nearly circular envelope. There were insufficient data to determine the ratios of the components of moment and the phase shift between the cycloids on the two Mn sub-lattices.

6. Conclusions The unique capability of SNP to measure components of scattered polarisation in directions orthogonal to the incident polarisation has been shown to be particularly apposite to the study of ME crystals, it provides information which is complementary to that obtained from domain imaging with second harmonic generation optical spectroscopy. There is a rather straightforward relationship between the direction of rotation of the scattered polarisation and the magnetic configuration of the more populous domain. Although the SNP data have not yet been combined with measurements of the ME coefficients, this relationship should allow the magnetic configuration to be correlated with the sense of ME polarisability to which it gives rise. The sensitivity of the rotated components of polarisation to the ratio γ between the magnetic and nuclear structure factors provides a method to measure the magnetisation distribution in ME crystals. In comparison with conventional polarised neutron diffraction, in which a polarisation dependent cross-section is measured, SNP has the advantage that the polarisation components are measured with a constant scattering cross-section. The effects of extinction are therefore largely eliminated. Exceptions can occur if the effective magnetic mosaic is very different from the nuclear one as has been demonstrated in TbAlO3 [25]. On the other hand,

P.J. Brown et al. / Solid State Sciences 7 (2005) 682–689

multiple scattering can be the source of significant error; its presence can be detected and appropriate corrections made if measurements are made with the crystal in different domain states. The absolute magnetic configuration of ME crystals, and the details of their magnetisation distributions are fundamental to understanding the ME effect. SNP has provided new insights in both these directions.

Acknowledgements We would like to thank E. Lelièvre-Berna, E. BourgeatLami, S. Pujol, J. Alibon and H. Humblot for their help with the technical aspects of these experiments. We are indebted to Dr. B. Wanklyn (Clarendon Laboratory, Oxford, UK) for supplying the crystals of the ortho-phosphates and to Professor Y. Endoh (Sendai, Japan) and Dr. D. Prabhakeran (Clarendon Laboratory, Oxford, UK) for their efforts to prepare stable crystals of MnGeO3 .

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