Magnetism |

**In General** > s.a. electromagnetism / effects and phenomenology;
history of physics; magnetism in astrophysics and cosmology.

$ __Magnetic Field__:
A pseudo-vector defined, in 3D terms, or on a spacelike hypersurface
with unit normal *n*^{a}, by

**B** = ∇ × **A**, *B _{i}*
=

* __Remark__: One view of the magnetic field is that it is just a convenient device to encode the transformation law for the electric field, or the relativistic aspects of the interactions between moving charges, without using special relativity (<RMS).

* __Ampère's law__:
In the SI system, and in the differential and integral forms, respectively,

∇ × **B** = *μ*_{0} **j** + *μ*_{0}*ε*_{0} (∂**E**/∂*t*) ,
\(\int_C\)**B** ·
d**s** = *μ*_{0} *I* + *μ*_{0}*ε*_{0} (d/d*t*) \(\int_S\) **E** ·
d**A** .

* __Values__: Earth's magnetic field is about 0.5 G or 5 × 10^{–5} T; The strongest fields obtained with permanent magnets have magnitudes above 5 T; 2011, The field with the largest magnitude so far, 91.4 T, has been produced at the Helmholtz-Zentrum Dresden-Rossendorf (HZDR).

@ __General references__:
Guimarães 05 [I];
Barbieri et al EJP(13) [vector potential, pedagogical];
Nishimura a1401
[Biot-Savart law, Ampère's law, and synthetic differential geometry];
Skomski 12 [simple models, IIb].

@ __Ampère-Maxwell law__: Heller AJP(92)jan;
Monsivais AJP(04)sep [integral form, arbitrarily-moving surfaces];
Manogue et al AJP(06)apr [teaching].

> __Types of magnetic fields__:
see quantum systems [self-linking fields].

**Magnetic Dipole Moment** > s.a. angular momentum [bound on ratio?];
Gyromagnetic Ratio; solutions in general relativity.

* __Models__:
The "Gilbert" model uses separated monopoles, the standard "Ampère" model
uses a current loop; The latter constitutes an interesting manifestation of "hidden momentum".

* __History__: The electron
(or muon) magnetic moment has been used as a precise check for QED predictions.

@ __References__: Hnizdo AJP(12)jul [for a moving electric dipole];
Bezerra et al EJP(12) [in introductory courses];
in Griffiths & Hnizdo AJP(13)aug [models];
Novello & Bittencourt IJMPA(14) [anomalous, proposal for origin].

> __For specific types of particles__:
see electron; neutrino;
particles [leptons, muons]; hadrons;
supersymmetry [muons].

**Magnetism in Matter** > s.a. electricity [Faraday's law];
electromagnetism in matter; Hysteresis;
Magnon; Susceptibility;
technology [magnets].

* __Permeability__: The second-rank tensor
*μ* (often isotropic, *μ*^{ij}
= diag(*μ*,* μ*, *μ*), and identified with a scalar)
such that **B** = *μ* **H**.

* __Diamagnetism__: The phenomenon in which
the induced magnetism in a substance tends to decrease the total magnetic field, and the induced
one opposes the external field; It can happen with atoms that don't have a permanent dipole
moment (Langevin diamagnetism) and with non-localized electrons in a metal (Landau diamagnetism);
> s.a. Van Leeuwen's Theorem;
Wikipedia page.

* __Ferromagnetism__: An extreme form
of paramagnetism, occurring at *T* < *T*_{Curie},
which corresponds to a phase transition (not in the Ehrenfest classification);
Exhibited by, e.g., Fe and Ni; > s.a. coupled-spin
models [including antiferromagnetism]; ising model.

* __Paramagnetism__: The phenomenon
in which the induced magnetism in a substance tends to enhance the total magnetic
field – the atoms have a permanent magnetic
moment and try to line up with the external field.

* __Measurement__: Most sensitive
ones uses muon spin rotation; Other methods include SQUIDs [high sensitivity],
scanning Hall probe microscopes [high spatial resolution], and BECs [nT sensitivity, 3-micron resolution].

@ __General references__: Van Vleck RMP(78);
Mattis 81, 85;
De Masi et al PRL(85) [microscopic derivation];
Jiles 91;
Craik 95;
Mattis 06;
Majlis 07;
Furrer & Waldmann RMP(13) [magnetic clusters, excitations];
Subedi et al PRL(13)
+ Park & Paulsen Phy(13) [spin-reversal avalanche].

@ __Ferromagnetism__: Esposito AP(09)-a0805 [Majorana's theory];
Katsnelson et al RMP(08) [half-metallic];
Gholizade & Momeni JSP(10)-a1011 [electron gas];
Balk et al PRB(14)
+ news pt(14)nov [critical behavior of zero-field magnetic fluctuations].

@ __Paramagnetism__: Bethlem Phy(08) [and forces on atoms];
Vijayaraghavan & Gard AP(13) [Monte Carlo simulations of magnetization relaxation].

@ __Related topics__: news PT(00)may,
news pw(07)mar [*μ* < 0];
news pn(06)jul [measuring with BECs];
news pw(07)feb [phase transition near 0 K];
news pw(13)apr [high-sensitivity atomic magnetometers];
Peng et al PRL(15) [Lee-Yang zeros and observation of imaginary magnetic fields].

**Generalizations**

@ __References__: Price et al PRL(14) [Berry curvature as a momentum-space magnetic field].

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