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Observation of correlations up to the micrometer scale in sliding charge-density waves
ARTICLE IN PRESS Physica B 404 (2009) 559–561
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Observation of correlations up to the micrometer scale in sliding chargedensity waves V.L.R. Jacques a,b, D. Le Bolloc’h a,, N. Kirova a, S. Ravy b, J. Dumas c a b c
ˆt. 510, 91405 Orsay Cedex, France Laboratoire de Physique des Solides (CNRS-UMR 8502), Universite´ Paris-sud, Ba Synchrotron SOLEIL, L’Orme des merisiers, Saint-Aubin BP 48, 91192 Gif-sur-Yvette Cedex, France ´el, CNRS/UJF, BP166 38042 Grenoble Cedex 9, France Institut Ne
abstract High-resolution coherent X-ray diffraction experiment has been performed on the Charge-Density Wave (CDW) system K0:3 MoO3 . The 2kF satellite reflection associated with the CDW has been measured with respect to external dc currents. In the sliding regime, the 2kF satellite reflection displays secondary satellites along the chain axis which corresponds to correlations up to the micrometer scale. This super long range order is 1500 times larger than the CDW period itself. & 2008 Elsevier B.V. All rights reserved.
1. Introduction Transport measurements are very efficient experiments to observe the sliding motion of a Charge-Density Wave (CDW) [1]. However, CDW systems are often heterogeneous materials and those macroscopic measurements give only few information about the microscopic origin of the sliding. X-ray diffraction seems to be the ideal technique to probe the consequences of sliding motion at the atomic scale. Unfortunately, performing diffraction experiments in the sliding regime of a CDW is always a difficult task because this technique has to face the issue of the phase in crystallography: any translational motion leaves the diffraction pattern invariant. However, few consequences of the sliding have been observed by using microscopic X-ray beams: in the sliding regime, the size of CDWs domains decreases along the direction perpendicular to the sliding [2]. The contraction of the CDW close to electrical contacts [3] or its rotation around a step [4] have also been observed. By using high-resolution and coherent X-ray diffraction, we show in this paper a new consequence of the sliding CDW: a new long range order, up to the micrometer scale, appears in the sliding regime of the CDW.
2. The blue bronze system In CDW systems, the nesting of the Fermi surface stabilizes the charge-density modulation and the band filling fixes the wave vector of the modulation at twice the Fermi wave vector kF , which may be incommensurate. The nesting properties of the K0:3 MoO3 Corresponding author.
E-mail address: [email protected] (D. Le Bolloc’h). 0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.11.046
blue bronze system is related to its quasi-1D structure: clusters of 10 MoO6 octahedra, form chains along the ½0 1 0 direction ˚ and layers along the ½1 0 2 direction [5]. The blue ðb ¼ 7:56 A), bronze stabilizes an incommensurate CDW modulation at 2kF ¼ 0:748 0:001 [6] along the chain axis. This incommensurate 2kF wave vector has a well-known consequence: the invariance by translation allows the electronic crystal to slide over the underlying lattice for currents greater than a threshold value. The signature of this sliding motion has been mainly observed by transport measurements [1,7]: a large broad band noise is observed, as well as periodic voltage oscillations [8]. At equilibrium, the CDW exhibits intrinsic defects like dislocations [9,10] and displays its own vibration modes [12,13].
3. Coherent diffraction setup The coherent X-ray diffraction experiment has been performed at the ID01 beamline at the ESRF. The beam quality and its transverse coherence length were tested by closing the entrance slit at 2 mm 2 mm: the expected cross-like diffraction pattern was observed in the Fraunhofer regime with strong contrast of fringes [14]. The degree of coherence has been estimated from the experimental setup around 20% at small angles. In this study, the main interest of using a coherent beam is the small beam size and 1 ˚ the excellent Q -resolution. A resolution of dq ¼ 0:7 104 A 4 along b (i.e. dq ¼ 0:8 10 in b units) was achieved at 7.5 keV ˚ by using 10 mm 10 mm entrance slits. We used the (l ¼ 1:65 A) same high quality single crystal as in our previous experiment [10]. Its electrical resistance has been carefully measured before and after the experiment. In both cases, a pronounced decrease of the differential resistance was observed at the threshold current,
ARTICLE IN PRESS 560
V.L.R. Jacques et al. / Physica B 404 (2009) 559–561
Fig. 1. 2D diffraction patterns (30 40 pixels) of the 2kF reflection in the (2a c ; b ) plane for I ¼ 0 mA (a), I ¼ 12Is (b) and I ¼ 16Is (c) (log scale). The secondary satellites are indicated by arrows. Corresponding profiles along 2a c in (d) and b in (e). The profiles have been rescaled.
accompanied by a rapid increase of the broad band noise, which is a signature of the sliding state. The transition temperature (T c ¼ 180 K) and the threshold current ðIs ¼ 1:2 mA at 75 K) remained unchanged after the experiment: X-rays do not altered the macroscopic properties of the CDW. The 0:5 2 0:2 mm3 sample was mounted in a top-loading Cryostat cooled down to 75 K with He exchange gas. The sample was aligned with the reciprocal vector b , which runs along the chains, vertical, and the 2a c axis in the horizontal scattering plane. The patterns were recorded on a direct illuminated CCD camera (22 mm 22 mm pixel size) located 1.20 m from the sample position. The experiment consisted in recording the 2D diffraction patterns of the Q s ¼ ð5; 1; 3Þ þ qc satellite reflection, and the ð6; 0; 3Þ fundamental Bragg peak, far from any electric contact. Both reflections have been measured successively after each current variation. Several CCD acquisitions have been recorded for different incident y angles, with dy ¼ 103 degree steps. The 2D patterns of the Fig. 1 have been obtained from this 3D volume. It corresponds to a section of the 2kF satellite reflection along the 2a c and b directions.
4. Appearance of secondary satellites under current The host lattice seems not to be sensitive to the applied current: the ð6; 0; 3Þ fundamental Bragg peak remains unchanged under the applied current within the experimental resolution (Fig. 1b). On the other hand, the CDW modulation displays original features for large enough currents. Under current, a broadening of
the 2kF reflection is observed along the 2a c transverse direction which corresponds to decreasing CDW correlation lengths from xt ¼ 2p=dq ¼ 1:2 mm at I ¼ 0 mA, to xt ¼ 0:6 mm at I ¼ 12Is and to xt ¼ 0:4 mm at I ¼ 16 Is (Fig. 1d). As mentioned in the introduction, the loss of transverse order under external currents, along the softest direction of blue bronze, has already been observed in several studies [2]. The most striking feature appears along the b chain axis. Without current, the 2kF reflection displays a single peak, the width of which corresponds to the entrance slit aperture: the CDWs domain is bigger than 10 mm along b (Fig. 1a). Above approximately 12 times the threshold current, a broadening of the Q s satellite appears along b (Fig. 1b). This broadening increases continuously for larger currents. At a current 16 times larger than the threshold, several maxima located at regular positions along b are clearly distinguished from the 2kF reflection (see Fig. 1c). All of them display approximately the same width which corresponds to the entrance slit aperture. The secondary satellites are located at ð6; 0:252 ndqs ; 3:5Þ with dqs ¼ 4:9 104 b units. The reduced wave vector dqs leads to a period of L ¼ 2p=dqs ¼ 1:5 mm. We did not increase the current further for fear of destroying the electrical contacts. The appearance of secondary satellites is reversible: after heating up the sample above T c under zero field and cooling down to 75 K again, the same diffraction pattern has been observed. Note the strong intensity of higher-order satellites and the asymmetry of the profile. During the experiment, we also noticed the absence of any speckle either on the 2kF reflection or on secondary fringes, despite the coherence properties of our X-ray beam.
ARTICLE IN PRESS V.L.R. Jacques et al. / Physica B 404 (2009) 559–561
561
5. Interpretation
Acknowledgments
The appearance of secondary satellites cannot be due to the fragmentation of CDW in isolated CDWs domains or to phase shifts randomly distributed in the volume [11]. The secondary satellites is related but to the presence of long-range correlations along the b -axis. The model has to reproduce the different features observed experimentally: 2kF is constant, the secondary satellites are intense, there is a strong asymmetry between the two secondary satellites, and dq, corresponding to micrometer scale in real space, increases with current. Several explanations are possible to explain the electronic correlations up to micrometer scale. First, a static scenario based on the presence of a soliton lattice is interesting since the variation of only one parameter (the CDW constant force C) can reproduce the main features of the observed data. Strictly speaking, in this static model, the presence of the secondary satellites is not directly due to the sliding motion of the CDW, but to a strong variation of the CDW force constants by screening effect of free carriers. However, the large variation of C necessary to reproduce the secondary satellites remains to be explained. Moreover, this model implies a shift of the CDW reflection under current, which is not seen in our measurements. Still working on the phase of the CDW, if jðxÞ is a saw-toothed function, the data are well reproduced, particularly the asymmetry between the secondary satellites. However, the appearance in the sliding regime of such a phase profile remains to be explained. We cannot exclude either an amplitude modulation, as the formation of a CDW stripe lattice appearing along b , with a distance of 1:5 mm between each stripe. This scenario has some analogy with vortices lattices under magnetic field in type II superconductors. Besides, micrometric correlations take place in the two cases, which is not a very common feature. However, this approach based on an amplitude modulation cannot explain the assymmetry of secondary satellites. In any cases, additional measurements are necessary to understand the origin of those long-range electronic correlations induced by sliding.
The authors would like to acknowledge the ID01 beamline staff at the ESRF, especially C. Mocuta and T. Metzger, and P. Van den Linden for technical support. N. Kirova acknowledges the financial support of ANR grant LoMaCoCup. References ¨ ner (Eds.), Charge Density Waves in Solids, [1] For a review: L.P. Gor’kov, G. Gru North-Holland, Amsterdam, 1989. [2] T. Tamegai, et al., Solid State Commun. 51 (1984) 585; R.M. Fleming, R.G. Dunn, L.F. Schneemeyer, Phys. Rev. B 31 (1985) 4099. [3] S. Brazovskii, N. Kirova, H. Requardt, F.Ya. Nad, P. Monceau, R. Currat, J.E. Lorenzo, G. Grubel, Ch. Vettier, Phys. Rev. B 61 (2000) 10640; H. Requardt, F.Ya. Nad, P. Monceau, R. Currat, J.E. Lorenzo, S. Brazovskii, N. Kirova, G. Gruebel, Ch. Vettier, Phys. Rev. Lett. 80 (1998) 5631. [4] A.F. Isakovic, P.G. Evans, J. Kmetko, K. Cicak, Z. Cai, B. Lai, R.E. Thorne, Phys. Rev. Lett. 96 (2006) 046401. [5] Let uðrÞ ¼ u0 cosðqc r þ FðrÞÞ be the periodic lattice distortion in quadrature with the CDW, where qc ¼ ð1; 2kF ; 0:5Þ is defined as the wave vector normal to the CDW wave fronts. [6] J.-P. Pouget, C. Noguera, A.H. Moudden, R. Moret, J. Phys. France 46 (1985) 1731; J.-P. Pouget, in: C. Schlenker (Ed.), Low Dimensional Electronic Properties of Molybdenum Bronzes and Oxides, Klu¨wer Academic, Dordrecht, 1989, p. 87. [7] For a review on the blue bronze see: C. Schlenker (Ed.), Low Dimensional Electronic Properties of Molybdenum Bronzes and Oxides, Klu¨wer Academic, Dordrecht, 1989. [8] R.M. Fleming, C.C. Grimes, Phys. Rev. Lett. 42 (1979) 1423; Z.Z. Wang, M.C. Saint Lager, P. Monceau, M. Renard, P. Gressier, A. Meeschaut, L. Guemas, J. Rouxel, Solid State Commun. 46 (1983) 325; P. Monceau, M. Renard, Europhys. News 17 (1986) 99. [9] P.A. Lee, T.M. Rice, Phys. Rev. B 19 (1979) 3970. [10] D. Le Bolloc’h, S. Ravy, J. Dumas, J. Marcus, F. Livet, C. Detlefs, F. Yakhou, L. Paolasini, Phys. Rev. Lett. 95 (2005) 116401. [11] D. Le Bolloc’h, V.L.R. Jacques, N. Kirova, J. Dumas, S. Ravy, J. Marcus, F. Livet, Phys. Rev. Lett. 100 (2008) 96403. [12] B. Hennion, J.-P. Pouget, M. Sato, Phys. Rev. Lett. 68 (1992) 2374; J.-P. Pouget, B. Hennion, C. Escribe-Fillipini, M. Sato, Phys. Rev. B 43 (1991) 8421. [13] S. Ravy, H. Requardt, D. Le Bolloc’h, P. Foury-Leylekian, J.-P. Pouget, R. Currat, P. Monceau, M. Krisch, Phys. Rev. B 69 (2004) 115113. [14] D. Le Bolloc’h, F. Livet, F. Bley, T. Schulli, M. Veron, T.H. Metzger, J. Synchrotron Rad. 9 (2002) 258.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Observation of correlations up to the micrometer scale in sliding chargedensity waves V.L.R. Jacques a,b, D. Le Bolloc’h a,, N. Kirova a, S. Ravy b, J. Dumas c a b c
ˆt. 510, 91405 Orsay Cedex, France Laboratoire de Physique des Solides (CNRS-UMR 8502), Universite´ Paris-sud, Ba Synchrotron SOLEIL, L’Orme des merisiers, Saint-Aubin BP 48, 91192 Gif-sur-Yvette Cedex, France ´el, CNRS/UJF, BP166 38042 Grenoble Cedex 9, France Institut Ne
abstract High-resolution coherent X-ray diffraction experiment has been performed on the Charge-Density Wave (CDW) system K0:3 MoO3 . The 2kF satellite reflection associated with the CDW has been measured with respect to external dc currents. In the sliding regime, the 2kF satellite reflection displays secondary satellites along the chain axis which corresponds to correlations up to the micrometer scale. This super long range order is 1500 times larger than the CDW period itself. & 2008 Elsevier B.V. All rights reserved.
1. Introduction Transport measurements are very efficient experiments to observe the sliding motion of a Charge-Density Wave (CDW) [1]. However, CDW systems are often heterogeneous materials and those macroscopic measurements give only few information about the microscopic origin of the sliding. X-ray diffraction seems to be the ideal technique to probe the consequences of sliding motion at the atomic scale. Unfortunately, performing diffraction experiments in the sliding regime of a CDW is always a difficult task because this technique has to face the issue of the phase in crystallography: any translational motion leaves the diffraction pattern invariant. However, few consequences of the sliding have been observed by using microscopic X-ray beams: in the sliding regime, the size of CDWs domains decreases along the direction perpendicular to the sliding [2]. The contraction of the CDW close to electrical contacts [3] or its rotation around a step [4] have also been observed. By using high-resolution and coherent X-ray diffraction, we show in this paper a new consequence of the sliding CDW: a new long range order, up to the micrometer scale, appears in the sliding regime of the CDW.
2. The blue bronze system In CDW systems, the nesting of the Fermi surface stabilizes the charge-density modulation and the band filling fixes the wave vector of the modulation at twice the Fermi wave vector kF , which may be incommensurate. The nesting properties of the K0:3 MoO3 Corresponding author.
E-mail address: [email protected] (D. Le Bolloc’h). 0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.11.046
blue bronze system is related to its quasi-1D structure: clusters of 10 MoO6 octahedra, form chains along the ½0 1 0 direction ˚ and layers along the ½1 0 2 direction [5]. The blue ðb ¼ 7:56 A), bronze stabilizes an incommensurate CDW modulation at 2kF ¼ 0:748 0:001 [6] along the chain axis. This incommensurate 2kF wave vector has a well-known consequence: the invariance by translation allows the electronic crystal to slide over the underlying lattice for currents greater than a threshold value. The signature of this sliding motion has been mainly observed by transport measurements [1,7]: a large broad band noise is observed, as well as periodic voltage oscillations [8]. At equilibrium, the CDW exhibits intrinsic defects like dislocations [9,10] and displays its own vibration modes [12,13].
3. Coherent diffraction setup The coherent X-ray diffraction experiment has been performed at the ID01 beamline at the ESRF. The beam quality and its transverse coherence length were tested by closing the entrance slit at 2 mm 2 mm: the expected cross-like diffraction pattern was observed in the Fraunhofer regime with strong contrast of fringes [14]. The degree of coherence has been estimated from the experimental setup around 20% at small angles. In this study, the main interest of using a coherent beam is the small beam size and 1 ˚ the excellent Q -resolution. A resolution of dq ¼ 0:7 104 A 4 along b (i.e. dq ¼ 0:8 10 in b units) was achieved at 7.5 keV ˚ by using 10 mm 10 mm entrance slits. We used the (l ¼ 1:65 A) same high quality single crystal as in our previous experiment [10]. Its electrical resistance has been carefully measured before and after the experiment. In both cases, a pronounced decrease of the differential resistance was observed at the threshold current,
ARTICLE IN PRESS 560
V.L.R. Jacques et al. / Physica B 404 (2009) 559–561
Fig. 1. 2D diffraction patterns (30 40 pixels) of the 2kF reflection in the (2a c ; b ) plane for I ¼ 0 mA (a), I ¼ 12Is (b) and I ¼ 16Is (c) (log scale). The secondary satellites are indicated by arrows. Corresponding profiles along 2a c in (d) and b in (e). The profiles have been rescaled.
accompanied by a rapid increase of the broad band noise, which is a signature of the sliding state. The transition temperature (T c ¼ 180 K) and the threshold current ðIs ¼ 1:2 mA at 75 K) remained unchanged after the experiment: X-rays do not altered the macroscopic properties of the CDW. The 0:5 2 0:2 mm3 sample was mounted in a top-loading Cryostat cooled down to 75 K with He exchange gas. The sample was aligned with the reciprocal vector b , which runs along the chains, vertical, and the 2a c axis in the horizontal scattering plane. The patterns were recorded on a direct illuminated CCD camera (22 mm 22 mm pixel size) located 1.20 m from the sample position. The experiment consisted in recording the 2D diffraction patterns of the Q s ¼ ð5; 1; 3Þ þ qc satellite reflection, and the ð6; 0; 3Þ fundamental Bragg peak, far from any electric contact. Both reflections have been measured successively after each current variation. Several CCD acquisitions have been recorded for different incident y angles, with dy ¼ 103 degree steps. The 2D patterns of the Fig. 1 have been obtained from this 3D volume. It corresponds to a section of the 2kF satellite reflection along the 2a c and b directions.
4. Appearance of secondary satellites under current The host lattice seems not to be sensitive to the applied current: the ð6; 0; 3Þ fundamental Bragg peak remains unchanged under the applied current within the experimental resolution (Fig. 1b). On the other hand, the CDW modulation displays original features for large enough currents. Under current, a broadening of
the 2kF reflection is observed along the 2a c transverse direction which corresponds to decreasing CDW correlation lengths from xt ¼ 2p=dq ¼ 1:2 mm at I ¼ 0 mA, to xt ¼ 0:6 mm at I ¼ 12Is and to xt ¼ 0:4 mm at I ¼ 16 Is (Fig. 1d). As mentioned in the introduction, the loss of transverse order under external currents, along the softest direction of blue bronze, has already been observed in several studies [2]. The most striking feature appears along the b chain axis. Without current, the 2kF reflection displays a single peak, the width of which corresponds to the entrance slit aperture: the CDWs domain is bigger than 10 mm along b (Fig. 1a). Above approximately 12 times the threshold current, a broadening of the Q s satellite appears along b (Fig. 1b). This broadening increases continuously for larger currents. At a current 16 times larger than the threshold, several maxima located at regular positions along b are clearly distinguished from the 2kF reflection (see Fig. 1c). All of them display approximately the same width which corresponds to the entrance slit aperture. The secondary satellites are located at ð6; 0:252 ndqs ; 3:5Þ with dqs ¼ 4:9 104 b units. The reduced wave vector dqs leads to a period of L ¼ 2p=dqs ¼ 1:5 mm. We did not increase the current further for fear of destroying the electrical contacts. The appearance of secondary satellites is reversible: after heating up the sample above T c under zero field and cooling down to 75 K again, the same diffraction pattern has been observed. Note the strong intensity of higher-order satellites and the asymmetry of the profile. During the experiment, we also noticed the absence of any speckle either on the 2kF reflection or on secondary fringes, despite the coherence properties of our X-ray beam.
ARTICLE IN PRESS V.L.R. Jacques et al. / Physica B 404 (2009) 559–561
561
5. Interpretation
Acknowledgments
The appearance of secondary satellites cannot be due to the fragmentation of CDW in isolated CDWs domains or to phase shifts randomly distributed in the volume [11]. The secondary satellites is related but to the presence of long-range correlations along the b -axis. The model has to reproduce the different features observed experimentally: 2kF is constant, the secondary satellites are intense, there is a strong asymmetry between the two secondary satellites, and dq, corresponding to micrometer scale in real space, increases with current. Several explanations are possible to explain the electronic correlations up to micrometer scale. First, a static scenario based on the presence of a soliton lattice is interesting since the variation of only one parameter (the CDW constant force C) can reproduce the main features of the observed data. Strictly speaking, in this static model, the presence of the secondary satellites is not directly due to the sliding motion of the CDW, but to a strong variation of the CDW force constants by screening effect of free carriers. However, the large variation of C necessary to reproduce the secondary satellites remains to be explained. Moreover, this model implies a shift of the CDW reflection under current, which is not seen in our measurements. Still working on the phase of the CDW, if jðxÞ is a saw-toothed function, the data are well reproduced, particularly the asymmetry between the secondary satellites. However, the appearance in the sliding regime of such a phase profile remains to be explained. We cannot exclude either an amplitude modulation, as the formation of a CDW stripe lattice appearing along b , with a distance of 1:5 mm between each stripe. This scenario has some analogy with vortices lattices under magnetic field in type II superconductors. Besides, micrometric correlations take place in the two cases, which is not a very common feature. However, this approach based on an amplitude modulation cannot explain the assymmetry of secondary satellites. In any cases, additional measurements are necessary to understand the origin of those long-range electronic correlations induced by sliding.
The authors would like to acknowledge the ID01 beamline staff at the ESRF, especially C. Mocuta and T. Metzger, and P. Van den Linden for technical support. N. Kirova acknowledges the financial support of ANR grant LoMaCoCup. References ¨ ner (Eds.), Charge Density Waves in Solids, [1] For a review: L.P. Gor’kov, G. Gru North-Holland, Amsterdam, 1989. [2] T. Tamegai, et al., Solid State Commun. 51 (1984) 585; R.M. Fleming, R.G. Dunn, L.F. Schneemeyer, Phys. Rev. B 31 (1985) 4099. [3] S. Brazovskii, N. Kirova, H. Requardt, F.Ya. Nad, P. Monceau, R. Currat, J.E. Lorenzo, G. Grubel, Ch. Vettier, Phys. Rev. B 61 (2000) 10640; H. Requardt, F.Ya. Nad, P. Monceau, R. Currat, J.E. Lorenzo, S. Brazovskii, N. Kirova, G. Gruebel, Ch. Vettier, Phys. Rev. Lett. 80 (1998) 5631. [4] A.F. Isakovic, P.G. Evans, J. Kmetko, K. Cicak, Z. Cai, B. Lai, R.E. Thorne, Phys. Rev. Lett. 96 (2006) 046401. [5] Let uðrÞ ¼ u0 cosðqc r þ FðrÞÞ be the periodic lattice distortion in quadrature with the CDW, where qc ¼ ð1; 2kF ; 0:5Þ is defined as the wave vector normal to the CDW wave fronts. [6] J.-P. Pouget, C. Noguera, A.H. Moudden, R. Moret, J. Phys. France 46 (1985) 1731; J.-P. Pouget, in: C. Schlenker (Ed.), Low Dimensional Electronic Properties of Molybdenum Bronzes and Oxides, Klu¨wer Academic, Dordrecht, 1989, p. 87. [7] For a review on the blue bronze see: C. Schlenker (Ed.), Low Dimensional Electronic Properties of Molybdenum Bronzes and Oxides, Klu¨wer Academic, Dordrecht, 1989. [8] R.M. Fleming, C.C. Grimes, Phys. Rev. Lett. 42 (1979) 1423; Z.Z. Wang, M.C. Saint Lager, P. Monceau, M. Renard, P. Gressier, A. Meeschaut, L. Guemas, J. Rouxel, Solid State Commun. 46 (1983) 325; P. Monceau, M. Renard, Europhys. News 17 (1986) 99. [9] P.A. Lee, T.M. Rice, Phys. Rev. B 19 (1979) 3970. [10] D. Le Bolloc’h, S. Ravy, J. Dumas, J. Marcus, F. Livet, C. Detlefs, F. Yakhou, L. Paolasini, Phys. Rev. Lett. 95 (2005) 116401. [11] D. Le Bolloc’h, V.L.R. Jacques, N. Kirova, J. Dumas, S. Ravy, J. Marcus, F. Livet, Phys. Rev. Lett. 100 (2008) 96403. [12] B. Hennion, J.-P. Pouget, M. Sato, Phys. Rev. Lett. 68 (1992) 2374; J.-P. Pouget, B. Hennion, C. Escribe-Fillipini, M. Sato, Phys. Rev. B 43 (1991) 8421. [13] S. Ravy, H. Requardt, D. Le Bolloc’h, P. Foury-Leylekian, J.-P. Pouget, R. Currat, P. Monceau, M. Krisch, Phys. Rev. B 69 (2004) 115113. [14] D. Le Bolloc’h, F. Livet, F. Bley, T. Schulli, M. Veron, T.H. Metzger, J. Synchrotron Rad. 9 (2002) 258.