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CHAPTER 2 Magnetic Structures Tapan Chatterji Institut Laue–Langevin, B.P. 156X, 38042 Grenoble cedex, France E-mail: chatt@ill.fr Contents 1. Introduction . . . . . . . . . . . 27 2.Determinationofmagneticstructures 28 2.1. Polycrystalline samples . 29 2.2.Singlecrystals 31 3.Ferromagneticandsimpleantiferromagneticstructures 31 3.1. Cubic Bravais lattice . . . 33 3.2. Hexagonal Bravais lattice 36 3.3. Tetragonal Bravais lattice 36 3.4. Single-k and multiple-k magneticstructures 36 4. Modulated magnetic structures 38 4.1.Sine-wavemagneticstructures 39 4.2.Helimagneticstructures 39 5. Complex modulated structures . 40 5.1.Magneticstructuresofheavyrare-earthelements 40 5.2.Magneticstructuresoflightrare-earthelements 48 5.3.Spindensitywaveinchromium 49 5.4. Modulated magnetic structures in CeSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.5. Modulated magnetic structure of CeAl 2 56 5.6. Modulated magnetic structures of EuAs 3 and Eu(As 1−x P x ) 3 57 5.7. Modulated magnetic structures in MnP: Lifshitz point . . . . . . . . . . . . . . . . . . . . . . . . 62 5.8.HelimagneticphaseinCuO 64 5.9. Modulated magnetic structures in MnSi and FeGe . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.10. Microscopic origin of modulated magnetic structures . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.Magneticstructuresofnovelelectronicmaterials 68 6.1.Magneticstructuresofcuprates 68 6.2.Magneticstructuresofmanganites 79 7.Concludingremarks 87 Acknowledgment 88 References 88 NEUTRON SCATTERING FROM MAGNETIC MATERIALS Edited by Tapan Chatterji © 2006 Elsevier B.V. All rights reserved 25 [...]... MnF2 , FeF2 , CoF2 and NiF2 with rutile type crystal structure (P 42 /mnm) illustrate this situation [10] The magnetic structures of these compounds can be described by the wave vector k = (0, 0, 0) The magnetic unit cell is the same as the nuclear unit cell The two magnetic sublattices characterized by positions (000) and ( 1 , 1 , 1 ) are related to one another by the screw tetrad of the space 2 2 2. .. has been investigated both by neutron and by more recently developed X-ray magnetic scattering [27 29 ] Metallic holmium orders at TN = 1 32 K with a simple helimagnetic structure The magnetic moments are ferromagnetically aligned within the basal planes, but rotate from plane to plane with a turn angle determined by the propagation or the wave vector k = (0, 0, τ ) of the magnetic structure Just below... [10]) For a primitive cubic lattice there are three symmetry points with k = H /2 viz., k = (0, 0, 1 ), k = ( 1 , 1 , 0) and k = ( 1 , 1 , 1 ) The corresponding three different types 2 2 2 2 2 2 of ordering are illustrated in Figure 2 along with the ferromagnetic ordering The CsCland the Cu3 Au-type crystal structures with the magnetic rare-earth atoms on a simple cubic Bravais lattice are quite common... recently [23 ] We will discuss these structures in more details in Section 6 .2 3 .2 Hexagonal Bravais lattice There exist five symmetry points in a hexagonal lattice defining three kinds of magnetic structures: (1) hexagonal (k = 0, k = (0, 0, 1 )), (2) orthohexagonal (k = ( 1 , 0, 0), 2 2 k = ( 1 , 0, 1 )) and (3) triangular (k = ( 1 , 1 , 0), k = ( 1 , 1 , 1 )) Note that the triangular 2 2 3 3 3 3 2 structures... takes place between the two commensurate phases with periods, viz., 0.1 82 = 2/ 11 and 0.167 = 2/ 12 There are indications of lock-in behavior near 0.190 = 4 /21 and 0.185 = 5 /27 Gibbs and coworkers [27 29 ] have developed the spin–slip model of the magnetic structures of Ho The model can be described briefly in the following way: The ferromagnetic basal planes in Ho, instead of being uniformly distributed,... vector δ is usually small, k0 is either the initial ferromagnetic (k0 = 0) or the antiferromagnetic wave vector (for example, k0 = ( 1 , 1 , 1 )) The modu2 2 2 lated magnetic structure is characterized by the appearance of satellite magnetic reflections close to the initial ferromagnetic (superimposed on the nuclear reflection) or antiferromag- Magnetic structures 39 netic reflections The wave vector gives... triangular 2 2 3 3 3 3 2 structures are not strictly antiferromagnetic 3.3 Tetragonal Bravais lattice For a body centered tetragonal lattice there are the following symmetry points with k = H /2: k = (0, 0, 1) (c > a), k = (1, 0, 0) (c < a), k = ( 1 , 1 , 0) and k = ( 1 , 0, 1 ) 2 2 2 2 Among the body centered tetragonal systems, K2 NiF4 - and ThCr2 Si2 -type structures have been widely investigated The former... distinguish between these magnetic structures only by applying magnetic fields or uniaxial stresses along one or several crystallographic directions The examples of the most symmetrical multiple-k structures associated with a wave vector k = 1 , 1 , 1 or k = 0, 0, 1 for the f.c.c Bra2 2 2 vais lattice is shown in Figure 4(a) For k = 1 , 1 , 1 there are four possible magnetic 2 2 2 structures: single-k,... the temperature range 17 25 K studied for decreasing temperature Apart from the sharp 46 T Chatterji magnetic satellite a broad peak develops below 23 K which has been associated with the accompanying lattice modulations As the temperature is lowered the magnetic wave vector approaches the value (5 /27 ), the scattering due to lattice modulation becomes sharper and approaches (2/ 9) A polarization analysis... vector along the b axis and magnetic moments parallel to the a axis The magnetic periodicity is twice that of the nuclear cell The magnetic structure on the hexagonal sites of Sm is antiferromagnetic At lower temperatures, Sm ions on the cubic sites also order antiferromagnetically [26 ] Eu (b.c.c.) orders in a first-order transition to a helical structure [26 ] 5 .2. 1 Multiple-k magnetic structure of neodymium . 3d n SLJGround-state term Ti 3+ ,V 4+ 3d 1 1 2 2 3 2 2 D 3 2 V 3+ 3d 2 13 2 3 F 2 Cr 3+ ,V 2+ 3d 3 3 2 3 3 2 4 F 3 2 Mn 3+ ,Cr 2+ 3d 4 22 0 5 D 0 Fe 3+ ,Mn 2+ 3d 5 5 2 0 5 2 6 S 5 2 Fe 2+ 3d 6 22 4 5 D 4 Co 2+ 3d 7 3 2 3 9 2 4 F 9 2 Ni 2+ 3d 8 13 4 3 F 4 Cu 2+ 3d 9 1 2 2 5 2 2 D 5 2 28. 0, 1 2 ) ∗ ( 1 2 , 0, 0)( 1 2 , 1 2 , 0) ∗ ( 1 2 , 0, 1 2 )( 1 2 , 1 2 , 1 2 ) ∗ I (c > a) (0, 0, 1) ∗ ( 1 2 , 1 2 , 0)( 1 2 , 0, 1 2 ) I (c < a) (1, 0, 0) ∗ ( 1 2 , 1 2 , 0)( 1 2 , 0, 1 2 ) Orthorhombic. 0) ∗ ( 1 2 , 1 2 , 1 2 ) I (c > a > b) (0, 0, 1) ∗ ( 1 2 , 1 2 , 0)( 1 2 , 0, 1 2 )(0, 1 2 , 1 2 ) Monoclinic P (0, 0, 1 2 ) ∗ (0, 1 2 , 0) ∗ ( 1 2 , 0, 1 2 ) ∗ ( 1 2 , 1 2 , 0) ∗ ( 1 2 , 1 2 , 1 2 ) ∗ B