Dear Chenyang

THis is not terribly complicated!

Consider an orthorhombic space group with typically a 4-fold position.

The Int. Tables For Crystallography give you the siccesive positions in terms of atom coordinates

(1) x, 1/4, z, (2) -x, 3/4, -z, (3)......, (4).....

with Bertaut (who was my thesis supervisor ) notations, there are 4 basis vectors to consider

F, G, C, A, to which the attached indices x, y, or z, means that the magnetic moment directions are ALONG a, b or c respectively.

According to Group Theory, the unique irreducible vectors are

F meanings ++++ for atoms (1), (2), (3), (4) sucessively

G meanings +-+- for atoms (1), (2), (3), (4) sucessively

C meanings ++-- for atoms (1), (2), (3), (4) sucessively

A meanings +--+ for atoms (1), (2), (3), (4) sucessively

GxAy means that exists a moment component directed parallel to a (mode Gx)

obeying the sequence (1) +, (2) -, (3) +, (4) - parallely to x

 and the mode Ay means that exists a moment component directed parallel to b

obeying the sequence (1) +, (2) -, (3) -, (4) + parallely to xy..

Interesting to read is the reference https://hal.archives-ouvertes.fr/jpa-00206863/document

from E.F. Bertaut and my colleague M. Bacmann

very clear, but... in french

If you use neutron diffraction, that is the appropriated experimental approach (moments or spins act as axial vectors), might be you cannot find directly the appropriated configuration as teached by group theory. Anyway, you try to "experiment" a symmetry lowering (might be unexpected, or clearly evidenced by the diffraction pattern in terms of extinction rules or cell distortion). After consideration of a symmetry lowering, application of the group theory to the new SG should propose you a consistent spin configuration, or lower the symmetry either more or differently...

Sometimes, a splitting of one 4-fold position to two 2-fold positions must be considered. Then there should be no direct symmetry rule between the 4 atoms of the position, but between the 2 first and the 2 others ones atoms!

Good luck

With my best regards

Daniel

,

It is really hard to tell from your picture which is the one. Check Figure 1 from the attached paper where AxGy and GxAy are compared.

Here is a similar comparison but in 2D (check Fig.4 from the attached paper). 

@Sergei Sergeenkov:Thanks for your good answer. In the specified Bertaut's notation, I think G configuration is unique. While for the A configuration, the plane(ferromagnetic coupling) could be perpendicular or parallel to the axis. For the case that the plane is parallel to the axis, I think the results are more than one， because we could always find out more than one planes that is parallel to one straight line. In peroskite ferrites, the magnetic stucture is unique once the Bertaut's notation is designated. Does this mean that we could have more than one magnetic configuration in specified Bertaut's notation, but in face the  result is only and should be determined by Neutron diffraction experiment?

Yes, I think you are quite right: there is some kind of freedom in Bertaut's notation (which is based on using irreducible representations of crystal symmetries), especially as far as spin configurations are concerned. As usual, the only true (and final) judge is experiment. For magnetic systems, simple neutron diffraction (which is good for ceramic samples) might be not enough. Polarized neutrons are more suitable for this purpose (especially for high quality single crystals). 

Dear Chenyang

THis is not terribly complicated!

Consider an orthorhombic space group with typically a 4-fold position.

The Int. Tables For Crystallography give you the siccesive positions in terms of atom coordinates

(1) x, 1/4, z, (2) -x, 3/4, -z, (3)......, (4).....

with Bertaut (who was my thesis supervisor ) notations, there are 4 basis vectors to consider

F, G, C, A, to which the attached indices x, y, or z, means that the magnetic moment directions are ALONG a, b or c respectively.

According to Group Theory, the unique irreducible vectors are

F meanings ++++ for atoms (1), (2), (3), (4) sucessively

G meanings +-+- for atoms (1), (2), (3), (4) sucessively

C meanings ++-- for atoms (1), (2), (3), (4) sucessively

A meanings +--+ for atoms (1), (2), (3), (4) sucessively

GxAy means that exists a moment component directed parallel to a (mode Gx)

obeying the sequence (1) +, (2) -, (3) +, (4) - parallely to x

 and the mode Ay means that exists a moment component directed parallel to b

obeying the sequence (1) +, (2) -, (3) -, (4) + parallely to xy..

Interesting to read is the reference https://hal.archives-ouvertes.fr/jpa-00206863/document

from E.F. Bertaut and my colleague M. Bacmann

very clear, but... in french

If you use neutron diffraction, that is the appropriated experimental approach (moments or spins act as axial vectors), might be you cannot find directly the appropriated configuration as teached by group theory. Anyway, you try to "experiment" a symmetry lowering (might be unexpected, or clearly evidenced by the diffraction pattern in terms of extinction rules or cell distortion). After consideration of a symmetry lowering, application of the group theory to the new SG should propose you a consistent spin configuration, or lower the symmetry either more or differently...

Sometimes, a splitting of one 4-fold position to two 2-fold positions must be considered. Then there should be no direct symmetry rule between the 4 atoms of the position, but between the 2 first and the 2 others ones atoms!

Good luck

With my best regards

Daniel

,

@Sergei Sergeenkov: Dear Sergei Sergeenkov, according to  D. Fruchart 's answer, once the space group and position of the atom is known, following the group theory and neutron diffraction, the magnetic structure could be confirmed and the magnetic structure also could be well understood in Bertaut's notation. I think the Bertaut's notation should be understood based on the positon of the atom.

Thanks for your help again.

With my best regards.

Chenyang Zhang

@D. Fruchart: Dear D. Fruchart, thanks so much for your good answer . I think I am clear now. 

With my best regards,

Chenyang Zhang