Magnetic moment

In physics, the magnetic moment of an object is a vector property, denoted here as m, that determines the torque, denoted here by τ, it experiences in a magnetic flux density B, namely τ = m × B (where × denotes the vector cross product). As such, it also determines the change in potential energy of the object, denoted here by U, when it is introduced to this flux, namely U = −m·B.

Origin

A magnetic moment may have a macroscopic origin in a bar magnet or a current loop, for example, or microscopic origin in the spin of an elementary particle like an electron, or in the angular momentum of an atom.

Macroscopic examples

The electric motor is based upon the torque experienced by a current loop in a magnetic field. The basic idea is that the current in the loop is made up of moving electrons, which are subect to the Lorentz force F in a magnetic field:

$\mathbf {F} =-e\left(\mathbf {v\times B} \right)\ ,$

where e is the electron charge and v is the electron velocity. This force upon the electrons is communicated to the wire loop because the electrons cannot escape the wire, and so exert a force upon it. The electrons at the top of the loop move oppositely to those at the bottom, so the force at the top is opposite in direction to that at the bottom. If the magnetic field is in the plane of the loop, the forces are normal to this plane, causing a torque upon the loop tending to turn the loop about an axis along the direction of the field.

The torque exerted upon a current loop of radius a carrying a current I, placed in a uniform magnetic flux density B at an angle to the unit normal û_n to the loop is:

${\boldsymbol {\tau }}={\mathit {I}}\mathbf {S\times B} \ ,$

where the vector S is:

$\mathbf {S} =\pi a^{2}\ {\hat {\mathbf {u} }}_{n}\ .$

Consequently the magnetic moment of this loop is:

$\mathbf {m} ={\mathit {I}}\ \mathbf {S} \ .$

Microscopic examples

Apart from macroscopic currents, at a fundamental level, magnetic moment is related to the angular momentum of particles: for example, electrons, nucleii, and so forth. In this discussion, focus is upon the electron and the atom.

The discussion splits naturally into two parts: kinematics and dynamics.

Kinematics

The kinematical discussion, which does not enter upon the physical origins of magnetism and its effects upon mechanics, deals with the classification of atomic states based upon symmetry. To emphasize this distinction, spin and orbital motion are considered here as distinct from spin angular momentum and orbital angular momentum. Although these ideas apply to nucleii and other particles, here attention is focused on electrons in atoms. The symmetry analysis leads to the identification of spin S and orbital motion L and its combination J = L + S.

The electron has a spin. The resultant total spin S of an ensemble of electrons in an atom is the vector sum of the constituent spins s_j:

$\mathbf {S} =\sum _{j=1}^{N}\ \mathbf {s_{j}} \ .$

Likewise, the orbital motions of an ensemble of electrons in an atom add as vectors.

Where both spin and orbital motion are present, they combine by vector addition:

$\mathbf {J=L+S} \ .$

Atoms are symmetrical under rotations, and J, L and S are connected to this symmetry. The mathematical basis is the infinitesimal rotation from which finite rotations can be generated. For example, a rotation by angle α about the z-axis is described by the matrix:

${\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}\ {\underset {\overset {\alpha \rightarrow 0}{}}{\to }}\ {\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}+i\alpha {\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}+i\alpha R_{z}\ ,$

where the form following the arrow applies for very small angles α. The matrix R_z is called the generator of the z-rotation. The factor i is introduced so the finite rotation can be expressed in terms of this generator as a simple exponential:

${\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}=e^{i\alpha R_{z}}\ ,$

as can be verified using the Taylor series:

$e^{i\alpha R_{z}}=1+i\alpha R_{z}+{\frac {1}{2}}\left(i\alpha R_{z}\right)^{2}+...\ .$

If the three coordinate axes are labeled {i, j, k } and the infinitesimal rotations about each of these axes are labeled {R_i, R_j, R_k}, then these generators of infinitesimal rotations obey the commutation relations:

$R_{i}R_{j}-R_{j}R_{i}=i\varepsilon _{ijk}R_{k}\ ,$

for any choices of subscripts. Here ε_ijk is the Levi-Civita symbol which equals one if ijk = xyz or any permutation that keeps the same cyclic order, or minus one if the order is different, or zero if any two of the indices are the same. The commutation relations express the fact that the order (sequence) of rotations matters.

These commutation relations now are viewed as applying in general, and while still considered as connected to rotations in three dimensional space, the question is opened as to what general mathematical objects might satisfy these rules. In particular, one can construct sets of square matrices of various dimensions that satisfy these commutation rules; each set is a so-called representation of the rules. One finds that there are many such sets, but they can be sorted into two kinds: irreducible and reducible. The reducible sets of matrices can be shown to be equivalent to matrices with smaller irreducible matrices down the diagonal, so that the rules are satisfied within these smaller constituent matrices, and the entire matrix is not essential. The irreducible sets cannot be arranged this way.

The matrices of dimension 2 are found from observation to be connected to the spin of the electron. One set of these matrices is based upon the Pauli spin matrices:

$\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\ ;\ \sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\ ;\ \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\ ,$

which satisfy:

${\frac {1}{2}}\sigma _{\alpha }{\frac {1}{2}}\sigma _{\beta }-{\frac {1}{2}}\sigma _{\beta }{\frac {1}{2}}\sigma _{\alpha }=i\ \varepsilon _{\alpha \beta \gamma }{\frac {1}{2}}\sigma _{\gamma }\ ,$

with αβγ any combination of xyz.

The Pauli matrices are connected to infinitesimal rotations in three-dimensional space. Finite rotations are generated by the Pauli spin matrices, with a finite rotation of angle θ about the axis aligned with unit vector û described as:

$R_{\hat {\mathbf {u} }}(\theta )=e^{i\theta {\hat {\mathbf {u} }}\mathbf {\cdot \sigma } /2}\ .$

Given a set of Euler angles α, β, γ describing orientation of an object in ordinary three-dimensional space, the general rotation about these angles is described as:

${\begin{pmatrix}e^{i\gamma /2}&0\\0&e^{-i\gamma /2}\end{pmatrix}}{\begin{pmatrix}\cos(\beta /2)&\sin(\beta /2)\\-\sin(\beta /2)&\cos(\beta /2)\end{pmatrix}}{\begin{pmatrix}e^{i\alpha /2}&0\\0&e^{-i\alpha /2}\end{pmatrix}}={\begin{pmatrix}e^{i(\alpha +\gamma )/2}\cos(\beta /2)&e^{-i(\alpha -\gamma )/2}\sin(\beta /2)\\-e^{i(\alpha -\gamma )/2}\sin(\beta /2)&e^{-i(\alpha +\gamma )/2}\cos(\beta /2)\end{pmatrix}}\ ,$

Higher dimensional irreducible sets of matrices are found to correspond to the spin of assemblies of electrons, or to the orbital motion of electrons in atoms, or a combination of both.

The matrices can be viewed as acting upon vectors in an abstract space. For example, a space with an odd number of dimensions (2ℓ+1) can be constructed from the spherical harmonics Y_ℓ^m, and their transformations under infinitesimal rotations. The Y_ℓ^m depend upon the angles θ,φ describing orientation in ordinary three-dimensional space, but infinitesimal rotations of these arguments mix up the Y_ℓ^m in a fashion described by irreducible matrices of dimension (2ℓ+1) that satisfy the commutation relations.

The construction of irreducible matrices of any dimension at all is done as follows. If the generator of an infinitesimal rotation is labeled J where J = S or L or L + S, for example, then the basis vectors in this space can be labeled by the integers j and m where m is restricted to the values { −j, −j+1, ... , j−1, j }. Denoting a basis vector by |j, m>, one finds:

$J^{2}|j,\ m>=j(j+1)|j,\ m>\ ,$

$J_{z}|j,\ m>=m|j,\ m>\ .$

Here J_z generates an infinitesimal rotation about a direction chosen as the z-axis, and J² = J_x² + J_y² + J_z². In particular, these equations recover the Pauli matrices in two dimensions and the infinitesimal transformations of the Y_ℓ^m in (2ℓ+1) dimensions.

Of course, the formalism has application to other elementary particles as well.

Dynamics

The dynamic aspect introduces the proportionality between magnetic moment and angular momentum using the gyromagnetic ratio, and attempts to explain its origin based upon quantum electrodynamics.

Angular momentum is introduced as proportional to the generator of an infinitesimal rotation, and is related to the same commutation relations, but with a proportionality factor of ℏ. Thus, in general ℏJ is an angular momentum, which clearly extends the idea of angular momentum far beyond the intuitive classical concept that applies in only three-dimensional space.

The magnetic moment m_S of a system of electrons with spin S is:

$\mathbf {m_{S}} =2m_{B}\mathbf {S} \ ,$

and the magnetic moment m_L of an electronic orbital motion L is:

$\mathbf {m_{L}} =m_{B}\mathbf {L} \ .$

Here the factor m_B refers to the Bohr magneton, defined by:

$m_{B}={\frac {e\hbar }{2m_{e}}}\ ,$

with e = the electron charge, ℏ = Planck's constant divided by 2π, and m_e = the electron mass. These relations are generalized using the g-factor:

$\mathbf {m_{J}} =gm_{B}\ \mathbf {J} \ ,$

with g=2 for spin (J = S) and g=1 for orbital motion (J = L). As mentioned earlier, where both spin and orbital motion are present, they combine by vector addition:

$\mathbf {J=L+S} \ .$

The magnetic moment of an atom of angular momentum ℏJ is

$\mathbf {m_{J}} =gm_{B}\mathbf {J} \ ,$

with g now the Landé g-factor or spectroscopic splitting factor:

$g={\frac {3}{2}}+{\frac {S(S+1)-L(L+1)}{2J(J+1)}}\ .$

If an atom with this associated magnetic moment now is subjected to a magnetic flux, it will experience a torque due to the applied field.