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title page
Contents
Abstract 19
Acknowledgements 21
Chapter 1. Introduction 23
Chapter 2. Experimental Procedures 27
2.1. Sample Preparation 27
2.1.1. Polycrystal Samples : Hexagonal Manganite 27
2.2. Review of Neutron Scattering 29
2.2.1. Introduction of Neutron Scattering 29
2.2.2. Principles of Neutron Scattering 30
2.2.3. Elastic Magnetic Neutron Scattering 33
2.2.4. Elastic Diffuse Magnetic Scattering due to Paramagnets 34
2.2.5. Elastic Coherent Magnetic Neutron Scattering by Magnetically Ordered Materials 35
2.2.6. Elastic Magnetic Neutron Scattering Cross-Section by Antiferromagnets 36
2.2.7. Elastic Magnetic Neutron Scattering by Incommensurate, Antiferronmagnetic Spiral Configurations 36
2.3. Magnetic Structure Refinement by Symmetry Analysis 37
2.3.1. Polar and Axial vector properties 38
2.4. Neutron powder Diffractometers 43
Chapter 3. Crystal and Magnetic Structure of Hexagonal Manganites 48
3.1. Introduction 48
3.2. Experimental Details 49
3.3. Results and Discussion 50
3.3.1. Crystal Structure of Hexagonal Manganites 50
3.3.2. Magnetic Structure of Hexagonal Manganites 51
3.3.3. Magnetic Symmetry Analysis of Hexagonal Manganites 55
3.3.4. Magnetic Structures of YMnO₃, ErMnO₃ and LuMnO₃ 56
Chapter 4. Doping Effects of Hexagonal Manganites Er1-xYxMnO₃(이미지참조) 67
4.1. Introduction 67
4.2. Experimental Details 69
4.3. Results and Discussions 70
Chapter 5. Spin-Lattice Coupling of Lu1-xYxMnO₃(이미지참조) 90
5.1. Introduction 90
5.2. Experimental details 92
5.3. Results and Discussions 93
5.3.1. Bulk Properties and Magnetic structure of Lu1-xYxMnO₃ (x=1.0, 0.9, 0.8, 0.5, 0.3, and 0.0)(이미지참조) 93
5.3.2. Direct observation of a coupling among spin, lattice, and electric dipole moment in multiferroic YMnO₃ 98
5.3.3. Displacement of Hexagonal Manganites 110
5.3.4. Doping Effects of Spin-Lattice Coupling in Lu1-xYxMnO₃ (x=1.0, 0.5, and 0.0)(이미지참조) 115
Chapter 6. Incommensurate magnetic structure of TbNi5(이미지참조) 133
6.1. Introduction 133
6.2. Experimental details 135
6.3. Results and Discussion 136
6.3.1. Neutron diffraction studies using polycrystal samples 136
6.3.2. Neutron diffraction studies using single crystal samples 150
Chapter 7. Summary 160
Appendix 163
Appendix A. Growth of Single Crystal Samples : Rare-Earth Hexaboride and Tetraboride 163
A.1. Travelling Solvent Floating Zone Method 163
A.1.1. Thermal Reactions and Growing Single Crystals 164
Appendix B. FullProf code file for the magnetic structure refinement of HANARO data of YMnO₃ 170
Appendix C. RIETAN 2001-T code file for the crystal structure refinement of SIRIUS data of YMnO₃ 173
References 177
〈국문초록〉 183
Table 2.1: Physical properties of hexagonal manganite. The lattice parameters were obtained by neutron diffraction taken at 300 K. The antiferromagnetic ordering temperature, and ferroelectric ordering temperature are indicated by... 28
Table 2.2: The properties of SIRIUS, the neutron scattering facility (KEKS) of High Energy Accelerator Research Organization (KEK), Japan. 45
Table 2.3: The technical specifications of C2 Spectrometer, NRC, Chalk River Laboratoires, Canada 46
Table 2.4: The technical specifications of HRPD diffractometer, Korea Atomic Energy Research Institute, Korea 47
Table 3.1: The structure parameters after refinement of neutron diffraction for RMnO₃(Y, Er, and Lu) at room temperature. The crystal symmetry is hexagonal P6₃cm with the following atomic positions : R(1) and O(3) at 2a... 52
Table 3.2: 1-Dimensional irreducible representation basis vectors of the space group P6₃cm at the 6c position(x,0,0) for k=(0,0,0). 52
Table 3.3: 2-Dimensional Basis vector(Γ5 and Γ6) of irreducible representations of the space group P6₃cm on the 6c position(x,0,0) for k=(0,0,0).(이미지참조) 54
Table 3.4: The magnetic structure of hexagonal manganites. The mixing magnetic structure means a combination of 1-D irreducible representation Γ₁, Γ₂, Γ₃, and Γ₄ without the z components. 63
Table 4.1: Summary of the refinement results of neutron powder diffraction patterns taken at 10 K for Er0.5Y0.5MnO₃. The space group is P6₃cm and the model of the magnetic structure is mixed (Γ₁+Γ₂).(이미지참조) 84
Table 5.1: The atomic parameters of YMnO₃ determined from high resolution neutron diffraction patterns(KEK) and synchrotron x-ray diffraction(PLS and PSI).... 102
Table 5.2: Basis functions for the atomic displacement vector of the space group P6₃cm on the 2a and 4b-sites for k=(0,0,0) 112
Table 5.3: 1-dimensional basis functions that describe the displacement vector of the space group P6₃cm on the 6c-site for k=(0,0,0) 113
Table 5.4: 2-dimensional basis functionsthat describe the displacement vector of the space group P6₃cm on the 6c-site for k=(0,0,0) 114
Table 5.5: Atomic parameters of YMnO₃, Lu0.5Y0.5MnO₃, and LuMnO₃ determined from high-resolution neutron diffraction patterns at 10 and 300 K. The crystal symmetry is hexagonal P6₃cm and the wave vector is K=0....(이미지참조) 121
Table 5.6: Bond distances and bond angles obtained by neutron diffraction refinement of YMnO₃, Lu0.5Y0.5MnO₃, and LuMnO₃(이미지참조) 121
Table 6.1: Basis functions of the irreducible representations of the group P6/mmm on the 1a-, 2c- and 3g- positions for k₂= 2π/c (0,0, q). (u₁=3-√3,u₁=3-2√3,u₂=2+√3,u₂=1+√3,t=2-√3,ω=√3-1)(이미지참조) 138
Table 6.2: The calculated and observed integrated neutron intensities of TbNi5 at 2.2 K. The left part of the table represents the intensities associated with Kι=0....(이미지참조) 148
Table A.1: The high quality single crystals of hexaboride and tetraboride. According to our XRD measurements, expect for PrB6, most polycrystal samples of hexaboride and tetraboride have more than two phases....(이미지참조) 169
Figure 2.1: The x-ray patterns of YMnO₃ at each sintering temperature. The phases below 1300 ℃ contain the impurities such as Y₂O₃ and Mn₂O₃. The most bottom data indicates the x-ray refinement result of YMnO₃ formed at... 28
Figure 2.2: Visualization of the momentum conservation in reciprocal space for (a) elastic and (b) inelastic neutron scattering. 31
Figure 2.3: Sequence of obtaining a magnetic basis function using Mody and Baslreps programs 39
Figure 2.4: Information of input file to obtain magnetic basis function of hexagonal mangnites which has magnetic atom at 6c-site and k=(0,0,0) using Baslreps program 44
Figure 3.1: (a) Schematic representation of the crystal structure (b) The configuration of Mn atoms at ab-plane. 51
Figure 3.2: The synchrotron X-ray diffraction of YMnO₃ at (a) 8C2 beam line at Pohang Accelerator Laboratory (b) X04SA-Powder beam line at Swiss light source(SLS), Paul Scherrer Instut(PSI).... 53
Figure 3.3: The magnetic structure of RMnO₃ associated with the 1-dimensional irreducible representation. For (a) Γ₁, (b) Γ₂, (c) Γ₃, (d) Γ₄, the Γ₂ and Γ₃ have other components which are parallel to the z axis. 57
Figure 3.4: The magnetic structure of RMnO₃ associated with the 2-dimensional irreducible representation Γ5. For (a) V51, (b) V52, (d) V54 and V57, and (e) V55 and V58, the magnetic moments are in a plane per...(이미지참조) 58
Figure 3.5: The magnetic structure of RMnO₃ associated with the 2-dimensional irreducible representation Γ6. For (a) V61, (b) V62, (d) V64 and V67, and (e) V65 and V68, the magnetic moments are in a plane per...(이미지참조) 59
Figure 3.6: Temperature dependence of the neutron diffraction patterns of (a) YMnO₃ in temperature range from 10 K to 90 K. (b) LuMnO₃ in the temperature range from 10 K to 100 K. 61
Figure 3.7: For YMnO₃ at 10 K, the neutron diffraction refinement results with magnetic structure Γ₁ and Γ₃ are displayed in Fig. 3.3(a) and (c), respectively. Observed(open circle) and calculated(line) neutron diffraction patterns.... 62
Figure 3.8: For LuMnO₃ at 10 K, the neutron diffraction refinement results with magnetic structure Γ₂ and Γ₄ are displayed in Fig. 3.3(b) and (d), respectively. Observed(open circle) and calculated(line) neutron diffraction patterns.... 64
Figure 3.9: Thermal evaluation of the Mn+3 magnetic moment of YMnO₃(solid circle) and LuMnO₃(open circle) obtained by Γ₁ and Γ₂, respectively.(이미지참조) 65
Figure 3.10: Temperature dependence of YMnO₃(diamond) and LuMnO₃(circle) (a) lattice parameter (b) cell volume. 66
Figure 4.1: Room temperature X-ray diffraction patterns are shown for a few representative samples of Er1-xYxMnO₃ with x=0.0, 0.07, 0.20, 0.50, 0.70, 0.90, and 1.0 (from bottom to top)....(이미지참조) 71
Figure 4.2: Lattice parameters and unit cell volumes are shown as a function of Y concentration. Error bars are smaller than the symbol size. 72
Figure 4.3: The temperature dependence of the inverse susceptibility is shown for six representative samples. All our measurements were made with applied field of 5 kOe.... 75
Figure 4.4: The temperature dependence of the specific heat of Er1-xYxMnO₃. The data are shifted upwards for the sake of presentation. The curved dashed line indicates how the antiferromagnetic transition temperature evolves with...(이미지참조) 76
Figure 4.5: The temperature dependence of Cp/T of ErMnO₃(□) and YMnO₃(▼) is shown in the inset together with their respective phonon estimate : ErMnO₃(dashed line) and YMnO₃(solid line).(이미지참조) 77
Figure 4.6: We plot as function of Y concentration, (a) the Neel temperature (solid circle) and Curie-Weiss temperature (open square), (b) the frustration parameter f(=|Θcw|/TN), and (c) the μeff obtained from the bulk susceptibil...(이미지참조) 78
Figure 4.7: Neutron diffraction patterns taken at 10 K for Er1-xYxMnO₃. The data are shifted upwards for better presentation.(이미지참조) 79
Figure 4.8: Observed (symbols) and calculated (line) neutron diffraction patterns for three representative Er1-xYxMnO₃ samples, taken at 10 K. In order to refine the data, we used (a) the Γ₁ representation for YMnO₃, (b) the Γ₁ + Γ₂...(이미지참조) 82
Figure 4.9: The magnetic structure of Er1-xYxMnO₃ obtained by symmetrical analysis(이미지참조) 83
Figure 4.10: We show (a) the angle (φ) between the direction of the Mn moment at the Mn1 position and the [1 0 0] axis and (b) the ordered magnetic moment of Mn, obtained from the refinement of the 10 K data, as a function... 86
Figure 4.11: The magnetic diffuse scattering taken at 65 K is shown for five samples of Er1-xYxMnO₃ after removing the Bragg peaks. The lines are Gaussian curve fitting results....(이미지참조) 87
Figure 5.1: The temperature dependence of the (a) susceptibility and (b) inverse susceptibility for Lu1-xYxMnO₃ (x=1.0, 0.9, 0.8, 0.5, 0.3, and 0.0)(이미지참조) 95
Figure 5.2: The temperature dependence of magnetic heat capacity △CP/T is shown for Lu1-xYxMnO₃ (x=1.0, 0.9, 0.8, 0.5, 0.3, and 0.0). The antiferromagnetic transition points(TN) are indicated by the peaks.(이미지참조) 96
Figure 5.3: The excess specific heat of Lu1-xYxMnO₃ (x=1.0, 0.9, 0.8, 0.5, 0.3, and 0.0) after subtraction of the phonon contribution. The solid line represents the magnetic entropy(△ S).(이미지참조) 97
Figure 5.4: Temperature dependence of (a) the ordered magnetic moment of Mn and (b) the angle(Ø) between the direction of the Mn moment at the Mn position and the [100] axis. 99
Figure 5.5: The doping dependence of the magnetic structure of Lu1-xYxMnO₃. The error bars are smaller than the symbol size. A schematic picture of the magnetic structure is shown in the inset.(이미지참조) 100
Figure 5.6: (a) A schematic representation of the crystal structure of hexagonal YMnO₃. The arrows indicate the displacement of each ion within the unit cell when the temperature drops below TN....(이미지참조) 101
Figure 5.7: Neutron-diffraction data (symbols) taken at 10 K for YMnO₃. The solid line represents the results of Rietiveld refinement with the hexagonal(P6₃cm) symmetry. The line at the bottom shows the difference curve.... 104
Figure 5.8: Temperature variations of (a) the unit cell volume, (b) the a- and c-axis lattice constants, and (c) four Mn-O bond distances. The error bars in (a) and (b) are smaller than the symbol size. 106
Figure 5.9: The temperature dependence of △Pz (filled circles), △dMn-o (open squares), and μord (open diamonds). ΔPz is the change in the calculated electric dipole moment with respect to that at 300 K while ΔdMn-O is the dif...(이미지참조) 109
Figure 5.10: Basis functions describing the displacement vector of the space group P6₃cm on the 2a-site(R1 and O3) for k=(0,0,0) (a) т₁, (b) т₄, (c)/(d) т5, (e)/(f) т6(이미지참조) 116
Figure 5.11: One-dimensional basis functions that describe the displacement vector of the space group P6₃cm on the 4b-site(R2 and O4) for k=(0,0,0) (a) т₁, (b) т₂, (c) т₃, and (d) т₄(이미지참조) 117
Figure 5.12: 1-dimensional basis functions describing the displacement vector of the space group P6₃cm on the 6c-site(Mn, O1, and O2) for k=(0,0,0) (a)/(b) т₁, (c) т₂, (d) т₃, and (e)/(f) т₄(이미지참조) 118
Figure 5.13: Neutron diffraction data(symbols) of Lu1-xYxMnO₃ taken at 10 K. The synchrotron X-ray diffraction result of YMnO₃ at 10 and 300K is shown in the inset....(이미지참조) 120
Figure 5.14: The lattice parameters of YMnO₃, Lu0.5Y0.5MnO₃, and LuMnO₃ obtained from the neutron diffraction. The error bars are smaller than the symbol size.(이미지참조) 122
Figure 5.15: The unit cell volume of YMnO₃, Lu0.5Y0.5MnO₃, and LuMnO₃ obtained from the neutron diffraction data. The error bars are smaller than the symbol size.(이미지참조) 123
Figure 5.16: The temperature dependence of the z-positions of R1, R2, O1, O2, O3, and O4 obtained for Lu1-xYxMnO₃ (x=1.0, 0.5, and 0.0).(이미지참조) 127
Figure 5.17: The temperature dependence of the x-positions of O1 and O2 obtained for Lu1-xYxMnO₃ (x=1.0, 0.5, and 0.0).(이미지참조) 128
Figure 5.18: (a) The temperature dependence of the x-positions of Mn obtained from the neutron diffraction of Lu0.5Y0.5MnO₃ (b) A schematic representation of the displacement of Mn atoms in the ab plane with decreasing temperatures....(이미지참조) 129
Figure 5.19: The normalized bond distance of Lu1-xYxMnO₃(x=1.0, 0.5, and 0.0) (a) Mn-O3 and (b) Mn-O4(이미지참조) 130
Figure 5.20: The temperature dependence of the normalized polarization of LuMnO₃(●), Lu0.5Y0.5MnO₃(□), and YMnO₃(▲)(이미지참조) 131
Figure 5.21: (a) A schematic representation of the crystal structure of hexagonal YMnO₃ and LuMnO₃. The displacements of the Mn, O1, and O2 atoms for YMnO₃ and LuMnO₃ are indicated by the arrows(- - →) and (→), respectively.... 132
Figure 6.1: Crystal structure of TbNi5. The crystal symmetry is hexagonal P6/mmm with the following atomic positions : Ni(1) at 2c (1/3,2/3,0), Ni(2) at 3g (1/2, 0, 1/2) and Tb at 1a (0, 0, 0).(이미지참조) 137
Figure 6.2: Temperature dependence of the neutron diffraction patterns of TbNi5. We have diagonally shifted the patterns for better presentation.(이미지참조) 139
Figure 6.3: Observed (open circles) and calculated (line) neutron diffraction pattern For TbNi5 at (a) 30 and (b) 2.2 K. Nucleus and magnetic Bragg reflections are indicated by vertical bars....(이미지참조) 140
Figure 6.4: (a) Temperature dependence of the average value of the background at 2θ=15-16˚. (b) Temperature dependence of the peak intensity of the (101) reflection (○) and (101)- satellite (●) and heat capacity (line)....(이미지참조) 141
Figure 6.5: Schematic view of some incommensurate magnetic structures. 142
Figure 6.6: Temperature dependence of т. The line through the data points is a guide to the eye.(이미지참조) 143
Figure 6.7: Temperature dependence of the ferromagnetic (μf) and sinusoidal (μs) components and the total value (μs) of the Tb-ion. The line through the data points is a guide to the eye.(이미지참조) 146
Figure 6.8: Temperature dependence of the (a) real and (b) imaginary parts of the AC susceptibility and (c) heat-capacity, measured while heating (closed symbols) and cooling (open symbols). 151
Figure 6.9: Neutron diffraction spectra around (0 0 1) reflection scanning along [0 0 L] direction taken at 1.7(●), 10.0(△), 20.1(■) and 25.1(◇) K while cooling.(이미지참조) 152
Figure 6.10: Temperature dependence of the integrated intensities of (a) (001) and (c) (101) reflections and (b) (001)± and (d) (101)± satellites. Open and closed symbols are for data taken while cooling and heating, respectively....(이미지참조) 153
Figure 6.11: Temperature dependence of the wave vector k₂. We obtained the т value from neutron diffraction data of single crystal(○) and polycrystal samples(■).(이미지참조) 155
Figure 6.12: Neutron spectra around the (1 0 1) reflection scanning along the [0 0 L] direction at 2 K by varying magnetic fields. 157
Figure 6.13: Field dependence of the integrated intensities of the (101) reflection (circles) and (101)± satellites (up and down triangles) at (top) 2 K and (bottom) 10 K....(이미지참조) 158
Figure A.1: The sample setting for thermal reaction. 165
Figure A.2: (a) Initial and secondary scans using the travelling solvent floating zone method. (b) An overview of the growing crystal. 167
Figure A.3: The phase diagram of Gd-B. 168
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