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Magnetic Point Groups

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Magnetic Point Groups GDR MEETICC Matériaux, Etats ElecTroniques, Interaction et Couplages non Conventionnels Winter school 4 – 10 February 2018, Banyuls-sur-Mer, France CRYSTALLOGRAPHIC and MAGNETIC STRUCTURES from NEUTRON DIFFRACTION: the POWER of SYMMETRIES (Lecture II) Béatrice GRENIER & Gwenaëlle ROUSSE UGA & CEA, INAC/MEM/MDN UPMC & Collège de France, Grenoble, France Paris, France GDR MEETICC Banyuls, Feb. 2018 Global outline (Lectures II, and III) II- Magnetic structures Description in terms of propagation vector: the various orderings, examples Description in terms of symmetry: Magnetic point groups: time reversal, the 122 magnetic point groups Magnetic lattices: translations and anti-translations, the 36 magnetic lattices Magnetic space groups = Shubnikov groups III- Determination of nucl. and mag. structures from neutron diffraction Nuclear and magnetic neutron diffraction: structure factors, extinction rules Examples in powder neutron diffraction Examples in single-crystal neutron diffraction Interest of magnetic structure determination ? Some material from: J. Rodriguez-Carvajal, L. Chapon and M. Perez-Mato was used to prepare Lectures II and III GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 1 Banyuls, Feb. 2018 Interest of magnetic structure determination Methods and Computing Programs Multiferroics Superconductors GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 2 Banyuls, Feb. 2018 Interest of magnetic structure determination Nano particles Multiferroics Computing Methods Manganites, charge ordering orbital ordering Heavy Fermions 3 GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 3 Banyuls, Feb. 2018 1. What is a magnetic structure ? A crystallographic structure consists in a long-range order of atoms, described by a unit cell, a space group, and atomic positions of the asymmetry unit. A magnetic structure corresponds to the long range ordering of “magnetic moments” or “spins”. These “magnetic moments” or “spins” correspond to the spin of the unpaired electrons GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 4 Banyuls, Feb. 2018 1. What is a magnetic structure ? These “magnetic moments” or “spins” correspond to the spin of the unpaired electrons Example Ni2+ 3d8 in an octahedral environment eg Hund’s and Pauli’s t2g rules This is represented as a magnetic moment carried by Ni2+ 2 unpaired electrons ⇒ 푆 = 1 푚 = 2 휇퐵 푞ħ 푚 = −푆 휇퐵 푆 (transition metals) 휇퐵 = 2푚푒 푚 = −퐽 휇퐵 퐽 (rare earths) GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 5 Banyuls, Feb. 2018 1. What is a magnetic structure ? paramagnetic state ferromagnetic state antiferromagnetic state 푪 < 푆푖 > = 0 Curie-Weiss: 흌 = 푻−휽푪푾 T > 0 Néel CW < 0 CW = 0 CW TCurie Magnetic susceptibility Magnetic Magnetic susceptibility Magnetic Magnetic susceptibility Magnetic Temperature Temperature Temperature GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 6 Banyuls, Feb. 2018 What is a magnetic structure ? There are also plenty of more complex magnetic structures, arising e. g. from frustration : Helical Sinusoidal Incommensurate … ferrimagnetic state When different magnetic atoms (or different oxidation states for the same atom) => Non-zero total magnetic moment Knowing a magnetic structure means being able to say, in whatever magnetic atom of whatever unit cell, what is the direction and value of the magnetic moment GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 7 Banyuls, Feb. 2018 Tools to describe a magnetic structure ? There exist 2 approaches: Group representation theory applied to conventional crystallographic space groups and using the concept of propagation vector 풌 → the most general (any 푘 vectors, incommensurate ones included) Magnetic symmetry approach: symmetry invariance of magnetic configurations (Magnetic Space Groups, often called Shubnikov groups) 1 → only 푘 = 0, 푘 = 퐻, or 푘 = 퐻 2 Béatrice Grenier in the second part of this talk GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 8 Banyuls, Feb. 2018 2. Propagation vectors formalism to describe a magnetic structure The position of atom 푗 in unit-cell 푙 is given by: 풓풋 풎퐥퐣 푅푙푗 = 푅푙 + 푟 푗 푹풍풋 푹풍 Where 푅푙 is a pure lattice translation Arbitrary origin of the lattice 푅푙푗 = 푅푙 + 푟 푗 = 푙1 푎 + 푙2 푏 + 푙3 푐 + 푥푗푎 + 푦푗 푏 + 푧푗 푐 Whatever kind of magnetic structure in a crystal can be described mathematically by using a Fourier series −2푖휋(푘.푅푙) 푚푙푗 = 푆푘푗 푒 푘 GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 9 Banyuls, Feb. 2018 풌 is a vector belonging to the Reciprocal Lattice The reciprocal lattice is defined as a network of points in the Fourier space (푄-space) which are the extremities of vectors: 퐻 = ℎ푎 ⋆ + 푘푏⋆ + 푙푐 ⋆ with 푎 ⋆, 푏⋆, and 푐 ⋆ the unit vectors of the reciprocal lattice, and ℎ, 푘, 푙 integers. 푏 × 푐 In solid state physics, 퐶 = 2휋 푎 ⋆ = 퐶 → 푎 ⋆ ⊥ 푏 and 푐 푉 In crystallography, 퐶 = 1 푐 × 푎 푏⋆ = 퐶 → 푏⋆ ⊥ 푎 and 푐 푉 푎 × 푏 푐 ⋆ = 퐶 → 푐 ⋆ ⊥ 푎 and 푏 푉 for C = 1 where 퐶 is a constant 푎 ⋆. 푎 = 푏⋆. 푏 = 푐 ⋆. 푐 = 1 푉 and is the volume of the unit cell ⋆ ⋆ in direct space: 푎 . 푏 = 푎 . 푐 = 0 푏⋆. 푎 = 푏⋆. 푐 = 0 ⋆ ⋆ 푉 = 푎 , 푏, 푐 = 푎 × 푏 . 푐 = 푎 . (푏 × 푐 ) 푐 . 푎 = 푐 . 푏 = 0 GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 10 Banyuls, Feb. 2018 Propagation vector formalism meaning of the propagation vector : analogy with plane waves −2푖휋(푘.푅푙) 푚푙푗 = 푆푘푗 푒 푘 The propagation vector 푘 of a magnetic structure reflects: - its periodicity 퐿 (푘 = 1/퐿) 푘 - the direction it propagates For a magnetic atom 푗, the magnetic moments 푚푙푗 (cell 푙) form planes of parallel moments that are perpendicular to the direction of the propagation vector GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 11 Banyuls, Feb. 2018 1. Propagation vectors formalism ; 풌 = (0, 0, 0) −2푖휋(푘.푅푙) 푚푙푗 = 푆푘푗 푒 푘 Let us examine the simple case 푘= (0, 0, 0) 푚푙푗 = 푆푘푗 ; both are real In this case the magnetic cell is the same as the nuclear cell GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 12 Banyuls, Feb. 2018 Propagation vectors formalism ; 풌 = (0, 0, 0) Remark : any ferromagnetic structure However the reverse is not true. has 푘 = (0,0,0) Many AF structures have 푘 = (0,0,0) GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 13 Banyuls, Feb. 2018 A notation useful in perovskites Moussa et al., 10.1103/PhysRevB.54.15149 푃푛푚푎, position 4푏 F = + + + + A = + − − + 3 4 C = + + − − G = + − + − 1 2 Constraints on the directions and couplings given by symmetry analysis (see later) GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 14 Banyuls, Feb. 2018 2. Note on centered cells: C-centered cell 2 2 −2푖휋(푘.푅푙) 푚푙푗 = 푆푘푗 푒 푎 푘 1 1 푏 푐 푘 = (0, 0, 0) ? 푘 = (1, 0, 0) 푚1 = +푚 푢 푚 = +푚 푢 1 1 1 −2푖휋( 1,0,0 .( , ,0)) 푚 = +푚 푢 푚2 = +푚푒 2 2 푢 2 1 −2푖휋 For centered cells, we sum over atoms 푗 of 푚2 = +푚푒 2 푢 1 the primitive cell and values of 푘 may be > 푚 = −푚 푢 2 2 GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 15 Banyuls, Feb. 2018 3. Propagation vector formalism ; 풌 = ½ 푯 The propagation vector is a special point of the Brillouin Zone surface and 푘 = ½ 퐻, where 퐻 is a 푐 reciprocal lattice vector. 푥 푏 푅 = 푙 푎 + 푙 푏 + 푙 푐 푎 푙 1 2 3 −2푖휋(푘.푅푙) −푖휋(퐻.푅푙) 푛푙 푚푙푗 = 푆푘푗 푒 = 푆푘푗 푒 = 푆푘푗 (−1) 푘 푛푙 푚푙푗 = 푚0푗 (−1) The structure is antiferromagnetic The magnetic symmetry may also be described using Shubnikov magnetic space groups 1 Example (see Figure): 푘 = 0, , 0 ⇒ 푚 = 푚 −1 푙2 2 푙푗 0푗 GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 16 Banyuls, Feb. 2018 First magnetic neutron diffraction experiments: MnO MnO structure 푎 = 4.45 Ǻ 퐹푚3 푚 c a b c 1 1 1 푘 = , , a b 2 2 2 GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 17 Banyuls, Feb. 2018 First magnetic neutron diffraction experiments: MnO 1 1 1 1 1 1 1 1 1 1 1 1 푘1 = , , Note: what about 푘 = , , , 푘 = , , and 푘 = , , ? 2 2 2 2 2 2 2 3 2 2 2 4 2 2 2 Are they equivalent ? 푘1 and 푘2 are equivalent if 푘2 − 푘1 is a reciprocal lattice vector 퐻 1 1 1 1 1 1 푘 − 푘 = , , − , , = 1 , 0, 0 2 1 2 2 2 2 2 2 This is not a reciprocal lattice vector (lattice F), therefore these 4 propagation vectors are not equivalent => they constitute the star of k-vectors. c Single-crystal: 4 different 푘 −domains a b GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 18 Banyuls, Feb. 2018 Note on multi-풌 structures Example 1: SrHo2O4 푘1 = 0, 0, 0 1 푘 = 0, 0, 2 2 Holmium distributed on 2 sites Ho1, Ho2 풌ퟏ = ퟎ, ퟎ, ퟎ ퟏ 풌 = ퟎ, ퟎ, ퟐ ퟐ GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 19 Banyuls, Feb. 2018 Note on multi-풌 structures Example 2: TbMg Tb is the magnetic ion This allows a canting (net ferromagnetic
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