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 = −𝑔푆 휇퐵 푆 (transition metals) 휇퐵 = 2푚푒 푚 = −𝑔퐽 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 component) 푎
푐 ퟏ 풌ퟐ = ퟎ, ퟎ, 풌ퟏ = ퟎ, ퟎ, ퟎ ퟐ +
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 20 Banyuls, Feb. 2018 4. Propagation vectors formalism: 풌 inside Brillouin zone
−2푖휋(푘.푅푙) 푚푙푗 = 푆푘푗 푒 푘 푘 and −푘 should be considered (푘 and −푘 are not equivalent)
−2푖휋(푘.푅푙) −2푖휋(−푘.푅푙) 푚푙푗 = 푆푘푗 푒 + 푆−푘푗 푒 1 −2푖휋휑 푆 = 푅푒 + 푖 퐼푚 푒 푘푗 푘푗 2 푘푗 푘푗 six parameters are independent
Necessary condition for real 푚푙푗: 푆−푘푗 = 푆푘푗* 2 simple cases:
1 −2푖휋휑 1) Real 푆 : 푆 = 푅푒 푒 푘푗 푘푗 푘푗 2 푘푗
2) Imaginary component 퐼푚푘푗 perpendicular to the real one (푅푒푘푗)
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 21 Banyuls, Feb. 2018 Sinusoidal magnetic structures
−2푖휋(푘.푅 ) −2푖휋(−푘.푅 ) 1 −2푖휋휑 푚 = 푆 푒 푙 + 푆 푒 푙 푆 = 푅푒 푒 푘푗 푙푗 푘푗 −푘푗 푘푗 2 푘푗 1 −2푖휋휑 푆 = 푚 푢 푒 푘푗 푘푗 2 푗 푗 푚푙푗 = 푚푗푢푗 cos 2휋 푘. 푅푙 + 휑푘푗
푏 1 푘 = , 0, 0 , 푚 ∥ 푏 12 푎 transverse
푏 1 푘 = , 0, 0 , 푚 ∥ 푎 12 푎 Example: BaCo2V2O8, see later longitudinal GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 22 Banyuls, Feb. 2018 Helical magnetic structures
−2푖휋(푘.푅푙) −2푖휋(−푘.푅푙) 푚푙푗 = 푆푘푗 푒 + 푆−푘푗 푒
1 −2푖휋휑 푆 = 푚 푢 + 푖 푚 푣 푒 푘푗 푤푖푡ℎ 푢 ⊥ 푣 푘푗 2 푢푗 푗 푣푗 푗 푗 푗 1 +2푖휋휑 푆 = 푆 ∗ = 푚 푢 − 푖 푚 푣 푒 푘푗 −푘푗 푘푗 2 푢푗 푗 푣푗 푗
1 −2푖휋(푘.푅 +휑 ) 2푖휋(푘.푅 +휑 ) 1 −2푖휋(푘.푅 +휑 ) 2푖휋(푘.푅 +휑 ) 푚 = 푚 푢 푒 푙 푘푗 +푒 푙 푘푗 + 푚 푣 푖 푒 푙 푘푗 − 푒 푙 푘푗 푙푗 2 푢푗 푗 2 푣푗 푗
푚푙푗 = 푚푢푗푢푗 cos 2휋 푘. 푅푙 + 휑푘푗 + 푚푣푗푣푗 sin 2휋 푘. 푅푙 + 휑푘푗
푘 ∥ 푎 푚 ∈ (푏, 푐 ) 푐 푏 푎
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 23 Banyuls, Feb. 2018 Helix (풌 ∥ 풂풙풊풔) Cycloid (풌 ⊥ 풂풙풊풔)
axis
k k
k axis k
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 24 Banyuls, Feb. 2018 Global outline II- Magnetic structures Description in terms of propagation vector: the various orderings, examples
• 푘 = 0, 0, 0 Magnetic cell = nuclear cell Shubnikov groups • 푘 = (1, 0, 0) Centered cells 1 • 푘 = 퐻 2 Magnetic cell 2 x, 4x, 8x bigger than nuclear cell • Pair (푘, −푘) Complex magnetic structures (helical, sinusoidal and more)
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
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 25 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 푘 → the most general (any 푘 vectors, incommensurate ones included)
tools based on this approach presented at the end of this lecture but theory not explained at all ("black box")
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 main purpose of the following Magnetic point groups Magnetic lattices Magnetic space groups
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 26 Banyuls, Feb. 2018 Magnetic point groups: Axial vs polar vectors Effect of the crystallographic point group symmetries 휶 on polar and axial vectors
Proper symmetry operations (det 훼 = +1) Electrical dipole 푝 Magnetic moment 푚 Polar vector Axial vector (current loop symmetry)
2-axis
+ +
− −
−
+
− +
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 27 Banyuls, Feb. 2018 Magnetic point groups: Axial vs polar vectors Effect of the crystallographic point group symmetries 휶 on polar and axial vectors
Improper symmetry operations (det 훼 = −1) Electrical dipole 푝 Magnetic moment 푚 Polar vector Axial vector
+ − Inversion
1
− +
+ + Mirror plane
− −
푚
−
+
− +
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 28 Banyuls, Feb. 2018 Magnetic point groups: Spin reversal and primed symmetries
To describe magnetic point symmetries, we need to introduce a new operator …
"spin reversal" = time-reversal or anti-identity: 1′ → changes the sense of the current and thus flips the magnetic moment
1′
→ Θ = {1,1′} time reversal group
1′ does not change nuclear positions and changes the sign of all magnetic moments ⇒ always present in non magnetic structures but absent in magnetically ordered ones !
N.B.: 1′ does not change neither the direction of a polar vector (i.e., an electrical dipole)
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 29 Banyuls, Feb. 2018 Magnetic point groups: Spin reversal and primed symmetries
We can define new symmetry operations = combination of a crystallographic point group sym. with 1′ = "primed" symmetry
푚 (mirror) 푚′ (primed mirror)
When a magnetic ordering occurs: some point symmetries may be lost and become primed
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 30 Banyuls, Feb. 2018 Magnetic point groups: Classification
퐺: crystallographic point group 푀: magnetic point group → subgroup of the direct product of 퐺 with Θ = {1,1′} 푀 퐺 Θ
3 types of magnetic point groups: 1/ 32 colorless groups: 푀 = 퐺 (Fedorov groups) 2/ 32 gray groups: 푀 = 퐺 ∪ 퐺1' (paramagnetic groups) 3/ 58 black-white groups: 푀 = 퐻 ∪ 퐺 − H 1′ with 퐻: subgroup of index 2 of 퐺 (halving group) and 퐺 − 퐻: the remaining operators, i.e. those not in 퐻
⇒ 122 magnetic point groups
N.B.: Colorless groups are also called monochrome groups Analogy spin-reversal / color change
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 31 Banyuls, Feb. 2018 Magnetic point groups: Classification Maximal subgroups and minimal supergroups of point groups 6 cubic black-white groups
3 23 hexagonal/trigonal 26 tetragonal/orthorhombic black-white groups black-white groups 3 5 5 13
13 5
8
3 monoclinic/triclinic black-white groups TOTAL = 58 black-white groups
Taken from: International Tables for Crystallography, volume A, p. 796 GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 32 Banyuls, Feb. 2018 Magnetic point groups: Classification
Example: black-white magnetic point groups derived from ퟒ/풎
퐺 = 4/푚 has 3 subgroups of index 2: + − 퐻1 = 4 = {1, 4푧 , 2푧, 4푧 } + − 퐻2 = 4 = {1, 4푧 , 2푧, 4푧 } 퐻3 = 2/푚 = {1, 2푧, 1, 푚푧}
⇒ There are 4 possible magnetic groups:
+ − + − 푀0 = 퐺 = 4/푚 = 1, 4푧 , 2푧, 4푧 , 1, 4푧 , 푚푧, 4푧 colorless group
′ + − + − 푀1 = 퐻1 + 퐺 − 퐻1 1 = 1, 4푧 , 2푧, 4푧 , 1′, 4푧 ′, 푚푧′, 4푧 ′ = 4/푚′ ′ black-white 푀 = 퐻 + 퐺 − 퐻 1′ = 1, 4+ , 2 , 4−′, 1′, 4+, 푚 ′, 4− = 4′/푚′ 2 2 2 푧 푧 푧 푧 푧 푧 groups ′ + − + − 푀3 = 퐻2 + 퐺 − 퐻2 1 = 1, 4푧 ′, 2푧, 4푧 ′, 1, 4푧 ′, 푚푧, 4푧 ′ = 4′/푚
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 33 Banyuls, Feb. 2018 Magnetic point groups: Classification
+ − + − 푀1 = 1, 4푧 , 2푧, 4푧 , 1′, 4푧 ′, 푚푧′, 4푧 ′ = 4/푚′
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 34 Banyuls, Feb. 2018 Magnetic point groups: Classification
Not all of the magnetic point groups can be realized in a magnetically ordered system → Admissible magnetic point groups: (for a magnetic atom placed at the origin) all the operators leave at least one spin component invariant - none of the gray groups is admissible, - many of the colorless and black-white groups are not admissible.
Example: 푀1 = 4/푚′ → 4/푚′ not admissible but 4/푚 is admissible if 푚0 ∥ 4-axis 4 4
If 푚0 ∥ 4-axis: 푚0 If 푚0 ⊥ 4-axis:
푚0 푚′ 푚′
4: 푚0 → 푚0 4 × 4: 푚0 → −푚0 푚′: 푚0 → −푚0 푚′: 푚0 → 푚0
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 35 Banyuls, Feb. 2018 Magnetic point groups: Classification The 31 admissible magnetic point groups:
Admissible magnetic point groups Admissible spin direction 1 1 any direction 2′ 2′/푚′ 푚′푚2′ ⊥ 2'-axis (and ⊥ 푚-plane for 푚′푚2′) 푚′ any direction within the 푚′-plane 푚 ⊥ 푚-plane 푚′푚′푚 ⊥ 푚-plane 2′2′2 ∥ 2-axis 2 2/푚 푚′푚′2 ∥ 2-axis 4 4 4/푚 42′2′ ∥ 4 or 4 -axis 4푚′푚′ 42푚′ 4/푚푚′푚′ ∥ 4 or 4 -axis 3 3 32′ 3푚′ 3푚′ ∥ 3 or 3 -axis 6 6 6/푚 62′2′ ∥ 6 or 6 -axis 6푚′푚′ 6푚′2′ 6/푚푚′푚′ ∥ 6 or 6 -axis
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 36 Banyuls, Feb. 2018 Magnetic point groups: Prediction for macroscopic properties Example 1: Ferromagnetoelectrics (spontaneous dielectric polarization 푃 & magnetic polarization 푀) polar vector axial vector Among the 31 admissible point groups (compatible with ferromagnetism), only 13 are also compatible with ferroelectricity
International tables for crystallography (2006), Vol. D, Section 1.5.8.3, pp. 141-142
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 37 Banyuls, Feb. 2018 Magnetic point groups: Prediction for macroscopic properties Example 2: Linear magnetoelectric effect
A magnetic field 퐻 applied in a crystal can produce an electric polarization 푃: 푃푖 = 훼푖푗퐻푗 An electric field 퐸 applied in a crystal can produce a magnetic moment 푀: 푀푖 = 훼푖푗퐸푗 → possible in 58 magnetic point groups → predictions on the form of the 훼 tensor:
Taken from the ITC, Volume D, p. 138
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 38 Banyuls, Feb. 2018 Magnetic point groups: Prediction for macroscopic properties
′ ′ ′ ′ 푃푛푚 푎 → 푚푚 푚 EuZrO3 푃푛′푚 푎′ → 푚′푚 푚′
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 39 Banyuls, Feb. 2018 Magnetic Bravais lattices
To describe magnetic translation symmetries, we introduce a new operator …
Anti-translation 푡 ′ = 푡 1′ (replaces the propagation vector formalism)
→ limitation of the Shubnikov symmetry: only 푘 = 0, 푘 = 퐻/2, or 푘 = 퐻
- Gray translation groups: not considered (incompatible with magnetic order)
- Colorless translation groups: same as the 14 Bravais lattices (only translations)
- Black-white translation groups: contain translations and anti-translations
푀퐿 = 퐻퐿 ∪ 푇 − 퐻퐿 1′
with 퐻퐿: subgroup of index 2 of the translation group 푇 and 푇 − 퐻퐿: the remaining operators, i.e. those not in 퐻퐿 → 22 black-white Bravais lattices
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 40 Banyuls, Feb. 2018 The 36 magnetic Bravais lattices triclinic monoclinic
orthorhombic tetragonal
hexagonal cubic
o: translations, : anti-translations, OG(BNS) GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 41 Banyuls, Feb. 2018 Magnetic Bravais lattices: OG vs BNS notations BNS: N. V. Belov, N. N. Neronova, and T.S. Smirnova (1957) OG: W. Opechowski and R. Guccione (1965) → same lattice symbol 푋 for colorless translation groups: 푋 = 푃, 퐼, 퐹, 퐴, 퐵, 퐶, 푅 → different lattice symbol 푋푌 for black-white translation groups 퐻퐿 ∪ 푇 − 퐻퐿 1′
BNS: 푋 = symbol of subgroup 퐻퐿 (decorated by white points only) 푌 = type of colored lattice (fractional anti-translation in 퐻퐿)
Examples: 푃푎 1 푧 푎 2 푦 푥
퐶푎 1 http://stokes.byu.edu/iso/magneticspacegroupshelp.php 푎 2
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 42 Banyuls, Feb. 2018 Magnetic Bravais lattices: OG vs BNS notations BNS: N. V. Belov, N. N. Neronova, and T.S. Smirnova (1957) OG: W. Opechowski and R. Guccione (1965) → same lattice symbol 푋 for colorless translation groups: 푋 = 푃, 퐼, 퐹, 퐴, 퐵, 퐶, 푅 → different lattice symbol 푋푌 for black-white translation groups 퐻퐿 ∪ 푇 − 퐻퐿 1′ OG: 푋 = symbol of parent group 푇 (decorated by black and white points) 푌 = type of colored lattice (translations in 푇) … Examples: 푃2푎 푧 푧 (BNS: 푃푎) 2푎 푦 푥 푦 푥
푃퐶
(BNS: 퐶푎) http://stokes.byu.edu/iso/magneticspacegroupshelp.php 푎 − 푏 푎 + 푏 GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 43 Banyuls, Feb. 2018 Magnetic space groups: Primed and unprimed symmetries
As for magnetic point groups and translation groups, magnetic space group symmetries can be primed (𝑔′) or not (𝑔) Example: Glide plane 푎 ⊥ 푐 at 푧 = 1/4 in 푃푛푚푎 Glide plane 푎′ ⊥ 푐 at 푧 = 1/4 in 푃푛′푚푎′ 1 1 1 1 푚 , 0, ′ 푚푧 2 , 0, 2 푧 2 2 1 0 0 1/2 1 0 0 1/2 0 1 0 0 0 1 0 0 0 0 − 1 1/2 0 0 − 1 1/2 0 0 0 1 0 0 0 − 1 unprimed sym. 훿 = 1 spin reversal 훿 = −1 ′ Atomic coordinates: 푟 푗 = 𝑔푟 푗 = 훼 푡 훼 푟 푗 = 훼푟 푗 + 푡 훼 1 1 푥, 푦, 푧 → 푥 + , 푦, 푧 + 2 2 ′ Magnetic moment: 푚푗 = 𝑔푚푗 = det(훼) 훿 훼푚푗
푚푥, 푚푦, 푚푧 → −푚푥, −푚푦, 푚푧 푚푥, 푚푦, 푚푧 → 푚푥, 푚푦, −푚푧
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 44 Banyuls, Feb. 2018 Magnetic space groups: Classification Using the same procedure as for magnetic point groups and translation groups allows to obtain and classify the 1651 magnetic space groups = Shubnikov groups
Daniel B. Litvin (2001) Pennsylvania State University, Reading, USA Magnetic group tables electronic book
Full description of all Shubnikov groups (in a form similar to that of the ITC, volume A, for crystallographic space groups), using the OG notation.
http://sites.psu.edu/ecsphysicslitvin/
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 45 Banyuls, Feb. 2018 Magnetic space groups: Classification
Harold T. Stokes and Branton J. Campbell (2010) Brigham Young University, Provo, Utah, USA Isotropy software suite Compiled by using the data of D. B. Litvin (OG and BNS notations) http://iso.byu.edu/iso/isotropy.php
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 46 Banyuls, Feb. 2018 Magnetic space groups: Classification
Mois I. Aroyo, J. Manuel Perez-Mato, G. de la Flor, E. S. Tasci, S. V. Gallego, … Many tools dealing with magnetic space groups (from 2010) http://www.cryst.ehu.es
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 47 Banyuls, Feb. 2018 Magnetic space groups: Classification 퐺: crystallographic space group, 퐻: subgroup of index 2 of 퐺 푀: magnetic space group
4 types of magnetic space groups: Type I: 230 colorless groups: 푀 = 퐺 (Fedorov groups) Type II: 230 gray groups: 푀 = 퐺 ∪ 퐺1' (paramagnetic groups) 1191 black-white groups: 푀 = 퐻 ∪ 퐺 − 퐻 1′ (BW groups)
Type III: Type IV: 674 BW groups such that 517 BW groups such that 퐻 is an equi-translation subgroup 퐻 is an equi-class subgroup (퐻 has the same translation group 푇 (푀 has a colored lattice: as 퐺: translations only) translations and anti-translations) → first kind, BW1 → second kind, BW2 1 푘 = 0 푘 = 퐻 (for centered cells) or 푘 = 퐻 2
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 48 Banyuls, Feb. 2018 Magnetic space groups: Classification
BNS vs OG notations for the Shubnikov groups:
→ types I, II, and III: same notation
→ type IV: different notation 푀 = 퐻 ∪ 퐺 − 퐻 1′ with translation group 푇 split between 퐻 and 퐺 − 퐻 ( anti-translations) 푀퐿 = 퐻퐿 ∪ 푇 − 퐻퐿 1′ Lattice symbol: see previous part (magnetic Bravais lattices)
Symbols for the planes and axes of symmetry: BNS notation: those belonging to subgroup 퐻 → always unprimed → can be different from those given for parent group 퐺 OG notation: those given for parent group 퐺 → can be primed or unprimed
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 49 Banyuls, Feb. 2018 Magnetic space groups: Example using the ITC – volume A The magnetic space groups can be constructed using the International tables for Crystallography, Volume A
Example: space group 퐼푚푎2 (No. 46) Colorless trivial magnetic space group: 퐼푚푎2 = 1, 푚푥, 푎푦, 2푧 푇
For simplicity, we drop off the translation group 푇 in the following
BW1 magnetic space groups: 푀 = 퐻 ∪ 퐺 − 퐻 1′ with all translations of 퐺 in 퐻 ′ ′ ′ 퐼1푎1 ∪ 퐼푚푎2 − 퐼1푎1 1 = 1, 푎푦 + {푚푥, 2푧}1 = 퐼푚 푎2′ ′ ′ 퐼푚11 ∪ 퐼푚푎2 − 퐼푚11 1 = 1, 푚푥 + {푎푦, 2푧}1 = 퐼푚푎′2′ ′ ′ ′ 퐼112 ∪ 퐼푚푎2 − 퐼112 1 = 1,2푧 + {푚푥, 푎푦}1 = 퐼푚 푎′2
BW2 magnetic space groups
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 50 Banyuls, Feb. 2018 Magnetic space groups: Example using the ITC – volume A Example: space group 퐼푚푎2 (No. 46) - continuation
BW2 magnetic space groups: ′ 푀 = 퐻 ∪ 퐺 − 퐻 1 with 퐻퐿 in 퐻 and 푇 − 퐻퐿 in 퐺 − 퐻 1 1 1 푡 = ( , , ) becomes an anti-translation 퐼 2 2 2 퐻퐿 = {푡 |푡 = 푢푎 + 푣푏 + 푤푐 ; 푢, 푣, 푤 ∈ } 푇 − 퐻퐿 = {푡 |푡 = 푢푎 + 푣푏 + 푤푐 + 푡 퐼; 푢, 푣, 푤 ∈ }=퐻′퐿
BNS OG ′ ′ 푃푛푎21: 1, 푛푥, 푎푦, 21푧 퐻퐿 + 1, 푚푥, 푐푦, 2푧 퐻퐿 = 푃퐼푛푎21 퐼푃푚 푎2′ ′ ′ 푃푛푐2: 1, 푛푥, 푐푦, 2푧 퐻퐿 + 1, 푚푥, 푎푦, 21푧 퐻퐿 = 푃퐼푛푎21 퐼푃푚 푎2′ ′ ′ 푃푚푎2: 1, 푚푥, 푎푦, 2푧 퐻퐿 + 1, 푛푥, 푐푦, 21푧 퐻퐿 = 푃퐼푛푎21 퐼푃푚 푎2′ ′ ′ 푃푚푐21: 1, 푚푥, 푐푦, 21푧 퐻퐿 + 1, 푛푥, 푎푦, 2푧 퐻퐿 = 푃퐼푛푎21 퐼푃푚 푎2′
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 51 Banyuls, Feb. 2018 Magnetic space groups: Example using the ITC – volume A
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 52 Banyuls, Feb. 2018 Magnetic space groups: Application to LaMnO3
Above 푇푁: magnetic space group = 푃푛푚푎1′ (gray group) Below 푇푁: 푘 = (0,0,0) ⇒ possible maximal subgroups (index 2)?
Mn: ⇒ possible space groups: those containing 1 and not 1′ 푃푛푚푎′ 푃푛푚′푎 푃푛푚′푎′ 푃푛′푚′푎 푃푛′푚′푎′ 푃푛′푚푎′ 푃푛푚푎 푃푛′푚푎
neutron diffraction experiment GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 53 Banyuls, Feb. 2018 Magnetic space groups: Application to LaMnO3
Magnetic components
푚 cannot be on a Wyckoff position whose symmetry is 1 ′ !!! (1 ′ is not an admissible magnetic point group)
Parts of pages 1 and 2 (/2) of 푃푛′푚푎 Shubnikov group, D. B. Litvin
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 54 Banyuls, Feb. 2018 Magnetic space groups: Application to LaMnO3 푷풏′풎풂′ Shubnikov group Black: unprimed sym. op. Red: primed sym. op. 푃푛′푚푎′
= {1,21푦, 1, 푚푦} ∪ 21푥, 푛푥, 21푧, 푎푧 1′
푐 푏 +푚푧 −푚푧
푎
page 1/2 of 푃푛′푚푎′ D. B. Litvin
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 55 Banyuls, Feb. 2018 Magnetic space groups: Application to LaMnO3
Mn
page 2/2 of 푃푛′푚푎′ D. B. Litvin
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 56 Banyuls, Feb. 2018 Magnetic space groups: Application to LaMnO3
Admissible magnetic point groups Admissible spin direction 1 1 any direction 2′ 2′/푚′ 푚′푚2′ ⊥ 2'-axis (and ⊥ 푚-plane for 푚′푚2′) 푚′ any direction within the 푚′-plane 푚 ⊥ 푚-plane
Mn
page 2/2 of 푃푛′푚푎′ The symmetry of this Wyckoff position imposes 푚 ∥ 푏 D. B. Litvin (푚 must be ⊥ to the mirror plane)
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 57 Banyuls, Feb. 2018 Magnetic space groups: Application to LaMnO3
0 0 0 1
0 0 0 -1
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 58 Banyuls, Feb. 2018 Magnetic space groups: Application to LaMnO3
LaMnO3: Mn located at (0,0,0), i.e. on 1 Macroscopic measurements → AF with 푚 ∥ 푎 -axis 푏 푐 푐 푏
ퟏ ퟏ ퟏ ퟐ ퟐ ퟐ 푎 푎 origin shift by (1/2, 0, 0) in this Fig. GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 59 Banyuls, Feb. 2018 Magnetic space groups: Application to LaMnO3
.mcif file for LaMnO3
훿 = ±1 whether the sym. op. is unprimed or primed
3+ 푚푥 = 3.87 3 휇퐵/Mn , 푚푦 = 0, 푚푧 = 0
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 60 Banyuls, Feb. 2018 Visualization of *.mcif files
• FpStudio (FullProf Suite) – J. Rodriguez-Carvajal and L. Chapon
• Bilbao Crystallographic Server http://webbdcrista1.ehu.es/magndata/mvisualize.php
• Vesta
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 61 Banyuls, Feb. 2018 Magnetic domains Symmetry of the ordered magnetic state lower than that of the paramagnetic state
퐺 : paramagnetic Shubnikov group of order 푛 푛 0 0 ⇒ 0 magnetic domains 퐺: ordered Shubnikov group of order 푛 푛
There exist 4 types of magnetic domains: Time-reversed domains: 180° domains Orientation domains: 푠-domains Configuration domains: 푘-domains 푘 Chiral domains (if 1 is lost) 푐 푏 푎 푘 푘 푘
푘 푘
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 62 Banyuls, Feb. 2018 Group representation theory
How to find the possible magnetic structures using group theory? Determine experimentally the propagation vector 푘 Select in the crystallographic space group 퐺 the sym. op. 𝑔 that leave 푘 invariant
→ Little group 퐺푘 (subgroup of 퐺) = {𝑔 ∈ 퐺|훼푘 = 푘 + 퐻} with 𝑔 = {훼|푡 훼} Write the magnetic representation Γ = set of 3푛 × 3푛 matrices for all sym. op. of 퐺푘 describing how each magnetic component is transformed 푛 equivalent magnetic atoms, 3 magnetic components 푢, 푣, 푤 for each atom 푑 such matrices with 푑 the order of 퐺푘 Reduce Γ into Irreducible representations (Ireps) Γ휈 (i.e., block diagonalize the matrices as much as possible): Γ = 푎1Γ1 푎2Γ2 … 1 2 For each IRep Γ휈 appearing in the decomposition of Γ, find its basis vectors 휓휈 , 휓휈 , … Landau theory (for 2nd order transition): The magnetic structure that establishes at the phase transition corresponds to an IRep that persists while all other IReps cancel ⇒ the magnetic structure is described by the basis vectors of the IRep that persists, while the basis vectors associated to all the other IReps cancel.
Group representation theory (= representation analysis) developped by E. F. Bertaut
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 63 Banyuls, Feb. 2018 Group representation theory
Most frequently used softwares to determine the IReps and thus the possible magnetic orderings:
Basireps (from the Fullprof suite) – Juan Rodriguez-Carvajal Sarah – Andrew Wills MODY – W. Sikora, F. Białas, L. Pytlik ISOTROPY - Harold T. Stokes, Dorian M. Hatch, and Branton J. Campbell
Input file: crystallographic space group propagation vector atomic coordinates of the magnetic atom(s)
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 64 Banyuls, Feb. 2018 Group representation theory: Application to LaMnO3 Input: crystallographic space group 푃푛푚푎 propagation vector 푘 = (0,0,0) atomic coordinates of the magnetic atom 푥 = 0, 푦 = 0, 푧 = 0
Output: 4 irreducible representations of dimension 1, each contained 3 times in Γ Γ = 3Γ1 3Γ2 3Γ3 3Γ4
Γ1 Γ2 Γ3 Γ4 Mn (0, 0, 0) 푢, 푣, 푤 푢, 푣, 푤 푢, 푣, 푤 푢, 푣, 푤 1 1 Mn , 0, −푢, −푣, 푤 −푢, −푣, 푤 푢, 푣, −푤 푢, 푣, −푤 2 2 1 Mn 0, , 0 −푢, 푣, −푤 푢, −푣, 푤 −푢, 푣, −푤 푢, −푣, 푤 2 1 1 1 Mn , , 푢, −푣, −푤 −푢, 푣, 푤 −푢, 푣, 푤 푢, −푣, −푤 2 2 2
푃푛푚푎 푃푛′푚′푎 푃푛′푚푎′ 푃푛푚′푎′
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 65 Banyuls, Feb. 2018 Summary
Magnetic point groups: As for crystallographic point groups, magnetic point groups are very important to make predictions on macroscopic magnetic properties
Magnetic space groups vs Propagation vectors & IReps: A magnetic structure can be described in two ways: 1/ Propagation vector and irreducible representations (IReps) or 2/ Magnetic space group (like crystallographic space groups with an additional symmetry operator: 1′ = spin reversal) Both descriptions can be used for softwares refining a magnetic structure
N.B.: 1st approach: more general propagation vector: very useful for diffraction see lecture III Nevertheless, the 2nd approach can be generalized to incommensurate structures → superspace groups
GDR MEETICC Crystallographic and Magnetic Structures / Neutron Diffraction, Béatrice GRENIER & Gwenaëlle ROUSSE 66 Banyuls, Feb. 2018