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X-ray diffraction studies of multipole ordering in a strongly correlated electron system
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) 730–731 www.elsevier.com/locate/jmmm
X-ray diffraction studies of multipole ordering in a strongly correlated electron system Yoshikazu Tanaka RIKEN SPring-8 Center, Harima Institute, Sayo, Hyogo 679-5148, Japan Available online 30 October 2006
Abstract Non-resonant X-ray diffraction based on the intense synchrotron radiation source is demonstrated as a powerful tool for quantitative studies of the multipole ordering in strongly correlated system. The recent results of X-ray diffraction on CeB6 and Ce0:7 Pr0:3 B6 compounds are reported. A long-range multipole (quadrupole and hexadecapole) ordering is evidenced directly in the unit-cell structure factor of superlattice reflections as a function of the scattering vector q for CeB6 and Ce0:7 Pr0:3 B6 compounds. r 2006 Elsevier B.V. All rights reserved. PACS: 61.10.Nz; 71.20.Eh; 71.27.+a Keywords: Multipolar ordering; X-ray diffraction; CeB6 ; PrB6
In order to fully understand the physics in many correlated electron materials, such as the colossal magnetoresistance effect, the orbital degree of freedom must be considered on equal footing with the spin, charge and electron–lattice coupling. The occupancy of a particular valence orbital on an atomic site has an important role in determining the physical properties at low temperatures. The orbital degrees of freedom can be accurately studied by X-ray diffraction, and no other experimental technique in the science of materials offers quite the same scope. In rareearth (RE) ions, the orbital degrees of freedom are conventionally expressed by multipole moments, which are tensor operators that can be represented by products of total angular momentum J. The existence of a long-range multipole order has been discussed in RE and actinide compounds for a long time. In spite of the extensive studies of the multipole ordering, the lack of experimental methods for direct observation has prevented us from investigating the nature of multipoles in detail. A signature of the ordering of multipoles that breaks crystal translational symmetry is a class of superlattice Tel.: +81 791 58 2923; fax : +81 791 58 0808.
E-mail address: [email protected]. 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.112
reflections. The unit-cell structure factor, F ðqÞ, for these superlattice reflections is given by [1] X X K F ðqÞ ¼ ð4pÞ1=2 eiqd iK hj K ið1ÞQ Y K (1) Q ðqÞhT Q i, where d is the positions of atoms in a unit cell, hj K i is the Bessel function transform of the radial density in the valence shell and hT K Q i is an atomic tensor. The intensity of nonresonant X-ray diffraction is proportional to the square of F ðqÞ and the following important information is derived from the analysis of the data: the type of multipole moments that order and the amount of the moments. We demonstrate that an ordering of quadrupole and hexadecapole moments occurs in the dense Kondo material CeB6 [2,3]. We also present the first direct observation of an incommensurate multipole order in Ce0:7 Pr0:3 B6 [4]. Non-resonant X-ray diffraction experiments were performed at BL19LXU at SPring-8 [5]. A double-Si-crystal monochromator cooled by liquid N2 tuned the energy of incident X-rays to be 30 keV. The scattering plane was horizontal. An 8 T superconducting magnet and a 0.5 T electromagnet that could apply a magnetic field perpendicular and parallel to the scattering vector, respectively, were used to investigate the response of multipole ordering to the external magnetic field. Plate-like single crystals of CeB6 and
ARTICLE IN PRESS Y. Tanaka / Journal of Magnetism and Magnetic Materials 310 (2007) 730–731
Ce0:7 Pr0:3 B6 cut along the (1 1 1) plane were studied. The mosaic spread of each sample was about 0:02 . We show the unit-cell structure factor, F s ðqÞ, of superlattice reflections n2 n2 n2, where n ¼ 1; 3; 5; 7; 9, and 11, observed at temperature T ¼ 1:6 and 2:5 K and magnetic field H ¼ 0 T of CeB6 , together with the F ðqÞ of lattice reflections hhh measured at T ¼ 1:6 K and H ¼ 0 T, where h ¼ 1; 2; 3; 4, and 5, in Fig. 1. All intensities are corrected for Lorentz factor, polarization factor and the Debye–Waller factor. Superlattice intensities are about 106 of the lattice reflections, and have a maximum between 52 52 52 and 72 72 72. The solid and broken lines show the result of a fit to a formula ahj 2 i þ bhj 4 i, which is deduced from Eq. (1) for the CeB6 crystal [6]. Here, hj 2 i and hj 4 i are the Bessel function transform of the radial density for the quadrupole (QP) and hexadecapole (HDP) moments of Ce 4f valence shell, respectively, and the coefficients a and b are the magnitudes of the QP and HDP components, respectively, that contribute at reflections n2 n2 n2. A direct evidence of multipole ordering in CeB6 is found in the good fitting curves with a ¼ 0:137 0:002 and b ¼ 0:283 0:005 in phase III, and a ¼ 0:036 0:002 and b ¼ 0:132 0:005 in phase II. Note that diffraction at reflections n2 n2 n2 sets in at the T Q ¼ 3:2 K, as a consequence of a reduction in the symmetry of sites occupied by Ce atoms, and it is absent in the structure adopted by CeB6 at room temperature. One might argue that these superlattice reflections could be due to a lattice distortion. For RE compounds, the q value at which the structure factor for a lattice distortion is a maximum is much larger than that where F s ðqÞ for a QP and HDP ordering shows a peak. We also note that the intensity of the superlattice reflection 52 52 52 at phase II depends strongly on H. It decreases with increasing H and disappears at H ¼ 0:2 T when H ? q [2], while it increases with increasing H when Hkq [3]. This means that the octupole moment induced by the external field rotates the QP and HDP moments due to the spin–orbit interaction. The alloy, Cex Pr1x B6 is an excellent candidate material for a possible complex interplay of f-electron multipoles arising
0
n for reflection nnn 2 3 4
1
5
6
0.06
60
731
Fig. 2. F ðqÞ for incommensurate reflections in Ce0:7 Pr0:3 B6 measured at T ¼ 1:7 K and zero applied magnetic field. Triangles: 3 2d 2.5 L, squares: 3 þ 2d 3.5 L, inversed triangles: 4 2d 3.5 L, diamonds: 4 þ 2d 4.5 L and solid circles: 5 2d 4.5 L. Lines are drawn as a guide to eyes.
from Kramers (Ce3þ ) and non-Kramers (Pr3þ ) ions. PrB6 and CeB6 have the same CsCl-type crystal structure at room temperature. PrB6 exhibits a first-order transition at T N ¼ 6:9 K to an incommensurate magnetic state with k ¼ ½d; 1=4; 1=2, where d ¼ 0:20 [7]. On decreasing the temperature further, a lock-in transition occurs at T IC ¼ 3:9 K with k ¼ ½1=4; 1=4; 1=2. The magnetic field-temperature phase diagram of Ce0:7 Pr0:3 B6 , in which Oxy -type antiferro-quadrupole ordering is believed to be dominant as in CeB6 , has been constructed from the measurement of the electrical resistivity and magnetization [8]. We observed commensurate superlattice reflections n2 n2 n2, where n ¼ 1; 3; 5; 7; 9, and 11, together with incommensurate superlattice reflections h 2dk2L with odd k, and integer h and L at T ¼ 1:7 K and H ¼ 0 T. The F s ðqÞ for incommensurate reflections shown in Fig. 2 has a maximum at a non-zero q as is also the case in the commensurate reflection shown in Fig. 1. From these results we confirm that the incommensurate reflections come from multipole moments. In conclusion, we have demonstrated that non-resonant X-ray diffraction provides a powerful tool to study the multipole ordering in CeB6 and Ce0:7 Pr0:3 B6 compounds. The multipole ordering is evidenced directly in the unit-cell structure factor of superlattice reflections.
0.05
50
0.04
40
0.03
30
0.02
20
0.01
F(q)
Fs(q)
T=1.6 K phase III
10
T=2.5 K phase II
0.00
0 0
2
4
6
8
10
12
14
16
q[Å-1]
Fig. 1. F s ðqÞ of CeB6 for superlattice reflections measured at T ¼ 1:6 K and H ¼ 0 T (circles) and at T ¼ 2:5 K and H ¼ 0 T (squares) as a function of the scattering vector q scaled by the left axis. Also shown are F ðqÞ for lattice reflection measured at T ¼ 1:6 K and H ¼ 0 T (triangles) scaled by the right axis. The top axis is scaled by an index n for the nnn reflection. The solid and dashed lines show the results of a fit to a formula discussed in the text. The dot line shows the F ðqÞ calculated for the CeB6 crystal structure.
These studies have been done in collaboration with F. Iga, T. Ishikawa, K. Katsumata, A. Kikkawa, S. Kishimoto, H. Kitamura, Y. Kuramoto, J.E. Lorenzo, S.W. Lovesey, T. Nakamura, Y. Narumi, Y. Onuki, V. Scagnoli, M. Sera, S. Shimomura, U. Staub, and Y. Tabata. References [1] [2] [3] [4] [5] [6] [7] [8]
S.W. Lovesey, et al., Phys. Rep. 411 (2005) 233. Y. Tanaka, et al., Europhys. Lett. 68 (2004) 671. Y. Tanaka, et al., J. Phys. Soc. Japan 74 (2005) 2201. Y. Tanaka, et al., J. Phys. Soc. Japan 75 (2006) 073702. M. Yabashi, et al., Nucl. Instrum. Methods A 467–468 (2001) 678. S.W. Lovesey, J. Phys. Condens. Matter 14 (2002) 4415. P. Burlet, et al., J. Phys. (Paris) 49 (1988) C8-459. S. Kishimoto, et al., J. Phys. Soc. Japan 74 (2005) 2913.
Journal of Magnetism and Magnetic Materials 310 (2007) 730–731 www.elsevier.com/locate/jmmm
X-ray diffraction studies of multipole ordering in a strongly correlated electron system Yoshikazu Tanaka RIKEN SPring-8 Center, Harima Institute, Sayo, Hyogo 679-5148, Japan Available online 30 October 2006
Abstract Non-resonant X-ray diffraction based on the intense synchrotron radiation source is demonstrated as a powerful tool for quantitative studies of the multipole ordering in strongly correlated system. The recent results of X-ray diffraction on CeB6 and Ce0:7 Pr0:3 B6 compounds are reported. A long-range multipole (quadrupole and hexadecapole) ordering is evidenced directly in the unit-cell structure factor of superlattice reflections as a function of the scattering vector q for CeB6 and Ce0:7 Pr0:3 B6 compounds. r 2006 Elsevier B.V. All rights reserved. PACS: 61.10.Nz; 71.20.Eh; 71.27.+a Keywords: Multipolar ordering; X-ray diffraction; CeB6 ; PrB6
In order to fully understand the physics in many correlated electron materials, such as the colossal magnetoresistance effect, the orbital degree of freedom must be considered on equal footing with the spin, charge and electron–lattice coupling. The occupancy of a particular valence orbital on an atomic site has an important role in determining the physical properties at low temperatures. The orbital degrees of freedom can be accurately studied by X-ray diffraction, and no other experimental technique in the science of materials offers quite the same scope. In rareearth (RE) ions, the orbital degrees of freedom are conventionally expressed by multipole moments, which are tensor operators that can be represented by products of total angular momentum J. The existence of a long-range multipole order has been discussed in RE and actinide compounds for a long time. In spite of the extensive studies of the multipole ordering, the lack of experimental methods for direct observation has prevented us from investigating the nature of multipoles in detail. A signature of the ordering of multipoles that breaks crystal translational symmetry is a class of superlattice Tel.: +81 791 58 2923; fax : +81 791 58 0808.
E-mail address: [email protected]. 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.112
reflections. The unit-cell structure factor, F ðqÞ, for these superlattice reflections is given by [1] X X K F ðqÞ ¼ ð4pÞ1=2 eiqd iK hj K ið1ÞQ Y K (1) Q ðqÞhT Q i, where d is the positions of atoms in a unit cell, hj K i is the Bessel function transform of the radial density in the valence shell and hT K Q i is an atomic tensor. The intensity of nonresonant X-ray diffraction is proportional to the square of F ðqÞ and the following important information is derived from the analysis of the data: the type of multipole moments that order and the amount of the moments. We demonstrate that an ordering of quadrupole and hexadecapole moments occurs in the dense Kondo material CeB6 [2,3]. We also present the first direct observation of an incommensurate multipole order in Ce0:7 Pr0:3 B6 [4]. Non-resonant X-ray diffraction experiments were performed at BL19LXU at SPring-8 [5]. A double-Si-crystal monochromator cooled by liquid N2 tuned the energy of incident X-rays to be 30 keV. The scattering plane was horizontal. An 8 T superconducting magnet and a 0.5 T electromagnet that could apply a magnetic field perpendicular and parallel to the scattering vector, respectively, were used to investigate the response of multipole ordering to the external magnetic field. Plate-like single crystals of CeB6 and
ARTICLE IN PRESS Y. Tanaka / Journal of Magnetism and Magnetic Materials 310 (2007) 730–731
Ce0:7 Pr0:3 B6 cut along the (1 1 1) plane were studied. The mosaic spread of each sample was about 0:02 . We show the unit-cell structure factor, F s ðqÞ, of superlattice reflections n2 n2 n2, where n ¼ 1; 3; 5; 7; 9, and 11, observed at temperature T ¼ 1:6 and 2:5 K and magnetic field H ¼ 0 T of CeB6 , together with the F ðqÞ of lattice reflections hhh measured at T ¼ 1:6 K and H ¼ 0 T, where h ¼ 1; 2; 3; 4, and 5, in Fig. 1. All intensities are corrected for Lorentz factor, polarization factor and the Debye–Waller factor. Superlattice intensities are about 106 of the lattice reflections, and have a maximum between 52 52 52 and 72 72 72. The solid and broken lines show the result of a fit to a formula ahj 2 i þ bhj 4 i, which is deduced from Eq. (1) for the CeB6 crystal [6]. Here, hj 2 i and hj 4 i are the Bessel function transform of the radial density for the quadrupole (QP) and hexadecapole (HDP) moments of Ce 4f valence shell, respectively, and the coefficients a and b are the magnitudes of the QP and HDP components, respectively, that contribute at reflections n2 n2 n2. A direct evidence of multipole ordering in CeB6 is found in the good fitting curves with a ¼ 0:137 0:002 and b ¼ 0:283 0:005 in phase III, and a ¼ 0:036 0:002 and b ¼ 0:132 0:005 in phase II. Note that diffraction at reflections n2 n2 n2 sets in at the T Q ¼ 3:2 K, as a consequence of a reduction in the symmetry of sites occupied by Ce atoms, and it is absent in the structure adopted by CeB6 at room temperature. One might argue that these superlattice reflections could be due to a lattice distortion. For RE compounds, the q value at which the structure factor for a lattice distortion is a maximum is much larger than that where F s ðqÞ for a QP and HDP ordering shows a peak. We also note that the intensity of the superlattice reflection 52 52 52 at phase II depends strongly on H. It decreases with increasing H and disappears at H ¼ 0:2 T when H ? q [2], while it increases with increasing H when Hkq [3]. This means that the octupole moment induced by the external field rotates the QP and HDP moments due to the spin–orbit interaction. The alloy, Cex Pr1x B6 is an excellent candidate material for a possible complex interplay of f-electron multipoles arising
0
n for reflection nnn 2 3 4
1
5
6
0.06
60
731
Fig. 2. F ðqÞ for incommensurate reflections in Ce0:7 Pr0:3 B6 measured at T ¼ 1:7 K and zero applied magnetic field. Triangles: 3 2d 2.5 L, squares: 3 þ 2d 3.5 L, inversed triangles: 4 2d 3.5 L, diamonds: 4 þ 2d 4.5 L and solid circles: 5 2d 4.5 L. Lines are drawn as a guide to eyes.
from Kramers (Ce3þ ) and non-Kramers (Pr3þ ) ions. PrB6 and CeB6 have the same CsCl-type crystal structure at room temperature. PrB6 exhibits a first-order transition at T N ¼ 6:9 K to an incommensurate magnetic state with k ¼ ½d; 1=4; 1=2, where d ¼ 0:20 [7]. On decreasing the temperature further, a lock-in transition occurs at T IC ¼ 3:9 K with k ¼ ½1=4; 1=4; 1=2. The magnetic field-temperature phase diagram of Ce0:7 Pr0:3 B6 , in which Oxy -type antiferro-quadrupole ordering is believed to be dominant as in CeB6 , has been constructed from the measurement of the electrical resistivity and magnetization [8]. We observed commensurate superlattice reflections n2 n2 n2, where n ¼ 1; 3; 5; 7; 9, and 11, together with incommensurate superlattice reflections h 2dk2L with odd k, and integer h and L at T ¼ 1:7 K and H ¼ 0 T. The F s ðqÞ for incommensurate reflections shown in Fig. 2 has a maximum at a non-zero q as is also the case in the commensurate reflection shown in Fig. 1. From these results we confirm that the incommensurate reflections come from multipole moments. In conclusion, we have demonstrated that non-resonant X-ray diffraction provides a powerful tool to study the multipole ordering in CeB6 and Ce0:7 Pr0:3 B6 compounds. The multipole ordering is evidenced directly in the unit-cell structure factor of superlattice reflections.
0.05
50
0.04
40
0.03
30
0.02
20
0.01
F(q)
Fs(q)
T=1.6 K phase III
10
T=2.5 K phase II
0.00
0 0
2
4
6
8
10
12
14
16
q[Å-1]
Fig. 1. F s ðqÞ of CeB6 for superlattice reflections measured at T ¼ 1:6 K and H ¼ 0 T (circles) and at T ¼ 2:5 K and H ¼ 0 T (squares) as a function of the scattering vector q scaled by the left axis. Also shown are F ðqÞ for lattice reflection measured at T ¼ 1:6 K and H ¼ 0 T (triangles) scaled by the right axis. The top axis is scaled by an index n for the nnn reflection. The solid and dashed lines show the results of a fit to a formula discussed in the text. The dot line shows the F ðqÞ calculated for the CeB6 crystal structure.
These studies have been done in collaboration with F. Iga, T. Ishikawa, K. Katsumata, A. Kikkawa, S. Kishimoto, H. Kitamura, Y. Kuramoto, J.E. Lorenzo, S.W. Lovesey, T. Nakamura, Y. Narumi, Y. Onuki, V. Scagnoli, M. Sera, S. Shimomura, U. Staub, and Y. Tabata. References [1] [2] [3] [4] [5] [6] [7] [8]
S.W. Lovesey, et al., Phys. Rep. 411 (2005) 233. Y. Tanaka, et al., Europhys. Lett. 68 (2004) 671. Y. Tanaka, et al., J. Phys. Soc. Japan 74 (2005) 2201. Y. Tanaka, et al., J. Phys. Soc. Japan 75 (2006) 073702. M. Yabashi, et al., Nucl. Instrum. Methods A 467–468 (2001) 678. S.W. Lovesey, J. Phys. Condens. Matter 14 (2002) 4415. P. Burlet, et al., J. Phys. (Paris) 49 (1988) C8-459. S. Kishimoto, et al., J. Phys. Soc. Japan 74 (2005) 2913.