Crystallography & Symmetry
patterns derived from their groups
Concept

Magnetoelectric effect — where it appears

The linear coupling in which an electric field induces a magnetisation and a magnetic field induces a polarisation. Its tensor is axial in one index and polar in the other, so it is odd under both inversion and time reversal: it needs magnetic order, an ordinary centre forbids it, and fifty-eight magnetic classes permit it.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The classes that permit a spontaneous magnetisation. Every magnetic point group permitting a spontaneous magnetisation — 31 of the 122 — with the number of independent components each allows. an axial vector, reversed by time reversal — a ferromagnet has one and nothing else does. The count comes from averaging the character over the group, with a primed operation's contribution multiplied by −1 because the property reverses when time does. That is Neumann's principle with one extra sign in it, and it reproduces the numbers the literature records without being given them.

Which magnetism a class permits

Neumann's principle with one extra sign in it decides which of the hundred and twenty-two magnetic classes may have a spontaneous magnetisation and which may show the magnetoelectric effect. The answers are thirty-one and fifty-eight, and they come out of the same average that counts elastic constants.

point-groups · Magnetic The ninety magnetic classes that are not grey, sorted by their centre. The ninety magnetic point groups that are not grey, in three columns by whether they contain the inversion, the inversion combined with time reversal, or no centre at all, each tile shaded by what a stress can do: magnetise a crystal that is already magnetised (31 classes), magnetise one that has no moment at rest (35), or nothing. A dot marks the classes that also permit the magnetoelectric effect. No class with the primed inversion is piezomagnetic and no class with the plain one is magnetoelectric.

Thirty-five classes a squeeze can magnetise

A magnetisation proportional to a stress is permitted in sixty-six of the hundred and twenty-two magnetic classes. Thirty-one are the classes that may be magnetised already; the other thirty-five may have no moment at rest and acquire one when squeezed, and in them a stress tells apart two domains that differ only by the direction of time. Among the ninety classes that are not grey, it is forbidden only by a primed centre and in the three ordinary cubic classes with four-fold operations.

point-groups · Magnetic

Named alongside it

The objects these essays reach for when they reach for this one.

CharacterMagnetic point groupTime reversalAxial vectorDomain stateFerromagnetismInversion centreNeumanns principlePiezomagnetism

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