Electron magnetic moment

"Electron spin" redirects here. See also Electron spin resonance and Spin (physics).

In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron caused by its intrinsic properties of spin and electric charge. The electron magnetic moment has recently been measured to an accuracy of 7.6 parts in 10^-13^[1]

Magnetic moment of an electron

The electron is a charged particle of charge (−1e), where e is the unit of elementary charge. Its angular momentum comes from two types of rotation: spin and orbital motion. From classical electrodynamics, a rotating electrically charged body creates a magnetic dipole with magnetic poles of equal magnitude but opposite polarity. This analogy holds as an electron indeed behaves like a tiny bar magnet. One consequence is that an external magnetic field exerts a torque on the electron magnetic moment depending on its orientation with respect to the field.

If the electron is visualized as a classical charged particle literally rotating about an axis with angular momentum L, its magnetic dipole moment μ is given by:

\boldsymbol{\mu} = \frac{-e}{2m_\text{e}}\, \mathbf{L}.

where m_e is the electron rest mass. Note that the angular momentum L in this equation may be the spin angular momentum, the orbital angular momentum, or the total angular momentum. It turns out the classical result is off by a proportional factor for the spin magnetic moment. As a result, the classical result is corrected by multiplying it with a dimensionless correction factor g, known as the g-factor;

\boldsymbol{\mu} = g \frac{-e}{2m_\text{e}} \mathbf{L}.

It is usual to express the magnetic moment in terms of the reduced Planck constant ħ and the Bohr magneton μ_B:

\boldsymbol{\mu} = -g \mu_\text{B} \frac{\mathbf{L}}{\hbar}.

Since the magnetic moment is quantized in units of μ_B, correspondingly the angular momentum is quantized in units of ħ.

Spin magnetic dipole moment

The spin magnetic moment is intrinsic for an electron.^[2] It is:

\boldsymbol{\mu}_\text{s}=- g_\text{s} \mu_\text{B} \frac{\mathbf{S}}{\hbar}.

Here S is the electron spin angular momentum. The spin g-factor is approximately two: g_s ≈ 2. The magnetic moment of an electron is approximately twice what it should be in classical mechanics. The factor of two implies that the electron appears to be twice as effective in producing a magnetic moment as the corresponding classical charged body.

The spin magnetic dipole moment is approximately one μ_B because g ≈ 2 and the electron is a spin one-half particle: S = ħ/2.

\mu_\text{S}\approx 2\frac{e \hbar}{2m_\text{e}}\frac{\frac{\hbar}{2}}{\hbar}=\mu_\text{B}.

The z-component of the electron magnetic moment is:

(\boldsymbol{\mu}_\text{s})_z=-g_\text{s} \mu_\text{B} m_\text{s}

where m_s is the spin quantum number. Note that μ is a negative constant multiplied by the spin, so the magnetic moment is antiparallel to the spin angular momentum.

The spin g-factor g_s = 2 comes from the Dirac equation, a fundamental equation connecting the electron's spin with its electromagnetic properties. Reduction of the Dirac equation for an electron in a magnetic field to its non-relativistic limit yields the Schrödinger equation with a correction term which takes account of the interaction of the electron's intrinsic magnetic moment with the magnetic field giving the correct energy.

For the electron spin, the most accurate value for the spin g-factor has been experimentally determined to have the value

2.00231930419922 ± (1.5 × 10⁻¹²).^[3]

Note that it is only two thousandths larger than the value from Dirac equation. The small correction is known as the anomalous magnetic dipole moment of the electron; it arises from the electron's interaction with virtual photons in quantum electrodynamics. In fact, one famous triumph of the Quantum Electrodynamics theory is the accurate prediction of the electron g-factor. The most accurate value for the electron magnetic moment is

(−928.476377±0.000023)×10⁻²⁶ J⋅T⁻¹.^[4]

The classical theory of the g-factor

The Dirac theory is not necessary to explain the g-factor for the electron. The deviation of the electron g-factor from that of the rigid sphere can be readily explained assuming that the charge distribution inside electron is different from the mass distribution. The electron can still be assumed a rigid body. Assuming for example the simplest and the most physical spherical Gaussian distributions for the charge and the mass separately:

\rho_\text{e}(r) = e N_\text{e} e^{-\frac{r^2}{r_\text{e}^2}}

and

\rho_\text{m}(r) = m_\text{e} N_\text{m} e^{-\frac{r^2}{r_\text{m}^2}}

where $r_\text{m}$ is the mass radius of the electron and $r_\text{e}$ is the charge radius we can obtain the tunable g-factor

as the ratio

g = \left( \frac{r_\text{e}}{r_\text{m}} \right)^8

For the electron $g=2$ they differ therefore very slightly, namely

\left( \frac{r_\text{e}}{r_\text{m}} \right) \approx 1.09051

Orbital magnetic dipole moment

The revolution of an electron around an axis through another object, such as the nucleus, gives rise to the orbital magnetic dipole moment. Suppose that the angular momentum for the orbital motion is L. Then the orbital magnetic dipole moment is:

\boldsymbol{\mu}_\text{L} = -g_\text{L}\mu_\text{B} \frac{\mathbf{L}}{\hbar}.

Here g_L is the electron orbital g-factor and μ_B is the Bohr magneton. The value of g_L is exactly equal to one, by a quantum-mechanical argument analogous to the derivation of the classical gyromagnetic ratio

Total magnetic dipole moment

The total magnetic dipole moment resulting from both spin and orbital angular momenta of an electron is related to the total angular momentum J by a similar equation:

\boldsymbol{\mu}_\text{J} =g_\text{J} \mu_\text{B} \frac{\mathbf{J}}{\hbar}.

The g-factor g_J is known as the Landé g-factor, which can be related to g_L and g_S by quantum mechanics. See Landé g-factor for details.

Example: hydrogen atom

For a hydrogen atom, an electron occupying the atomic orbital Ψ_{n, ℓ, m}, the magnetic dipole moment is given by:

\mu_\text{L}=-g_\text{L}\frac{\mu_\text{B}}{\hbar}\langle\Psi_{n,\ell,m}|L|\Psi_{n,\ell,m}\rangle=-\mu_\text{B}\sqrt{\ell(\ell+1)}.

Here L is the orbital angular momentum, n, ℓ and m are the principal, azimuthal and magnetic quantum numbers respectively. The z-component of the orbital magnetic dipole moment for an electron with a magnetic quantum number m_ℓ is given by:

(\mathbf{\mu_\text{L}})_z=-\mu_\text{B} m_\ell.

Electron spin in the Pauli and Dirac theories

Main articles: Pauli equation and Dirac equation

The necessity of introducing half-integral spin goes back experimentally to the results of the Stern–Gerlach experiment. A beam of atoms is run through a strong non-uniform magnetic field, which then splits into N parts depending on the intrinsic angular momentum of the atoms. It was found that for silver atoms, the beam was split in two—the ground state therefore could not be integral, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into 3 parts, corresponding to atoms with L_z = −1, 0, and +1. The conclusion is that silver atoms have net intrinsic angular momentum of ¹⁄₂. Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so:

H = \frac{1}{2m} \left [ \boldsymbol{\sigma}\cdot \left ( \mathbf{p} - \frac{e}{c}\mathbf{A} \right ) \right ]^2 + e\phi.

Here A is the magnetic potential and ϕ the electric potential representing the electromagnetic field, and σ = (σ_x, σ_y, σ_z) are the Pauli matrices. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual classical Hamiltonian of a charged particle interacting with an applied field:

H = \frac{1}{2m}\left ( \mathbf{p} - \frac{e}{c}\mathbf{A} \right )^2 + e\phi - \frac{e\hbar}{2mc}\boldsymbol{\sigma}\cdot \mathbf{B}.

This Hamiltonian is now a 2 × 2 matrix, so the Schrödinger equation based on it must use a two-component wave function. Pauli had introduced the 2 × 2 sigma matrices as pure phenomenology— Dirac now had a theoretical argument that implied that spin was somehow the consequence of incorporating relativity into quantum mechanics. On introducing the external electromagnetic 4-potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form (in natural units ħ = c = 1)

\left [ -i\gamma^\mu\left ( \partial_\mu + ieA_\mu \right ) + m \right ] \psi = 0\,

where $\scriptstyle \gamma^\mu$ are the gamma matrices (known as Dirac matrices) and i is the imaginary unit. A second application of the Dirac operator will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by i, have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the gyromagnetic ratio of the electron, standing in front of Pauli's new term, is explained from first principles. This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness. The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. First the equation is written in the form of coupled equations for 2-spinors with the units restored:

\begin{pmatrix} (mc^2 - E + e \phi) & c\sigma\cdot \left (\mathbf{p} - \frac{e}{c}\mathbf{A} \right ) \\ -c\boldsymbol{\sigma}\cdot \left ( \mathbf{p} - \frac{e}{c}\mathbf{A} \right ) & \left ( mc^2 + E - e \phi \right ) \end{pmatrix} \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}.

\begin{align} (E - e\phi) \psi_+ - c\boldsymbol{\sigma} \cdot \left( \mathbf{p} - \frac{e}{c}\mathbf{A} \right) \psi_- &= mc^2 \psi_+ \\ -(E - e\phi) \psi_- + c\boldsymbol{\sigma} \cdot \left( \mathbf{p} - \frac{e}{c}\mathbf{A} \right) \psi_+ &= mc^2 \psi_- \end{align}

Assuming the field is weak and the motion of the electron non-relativistic, we have the total energy of the electron approximately equal to its rest energy, and the momentum reducing to the classical value,

\begin{align} E - e\phi &\approx mc^2 \\ p &\approx m v \end{align}

and so the second equation may be written

\psi_- \approx \frac{1}{2mc} \boldsymbol{\sigma} \cdot \left( \mathbf{p} - \frac{e}{c}\mathbf{A} \right) \psi_+

which is of order v/c - thus at typical energies and velocities, the bottom components of the Dirac spinor in the standard representation are much suppressed in comparison to the top components. Substituting this expression into the first equation gives after some rearrangement

\left(E - mc^2\right) \psi_+ = \frac{1}{2m} \left[ \boldsymbol{\sigma}\cdot \left( \mathbf{p} - \frac{e}{c}\mathbf{A} \right) \right]^2 \psi_+ + e\phi \psi_+

The operator on the left represents the particle energy reduced by its rest energy, which is just the classical energy, so we recover Pauli's theory if we identify his 2-spinor with the top components of the Dirac spinor in the non-relativistic approximation. A further approximation gives the Schrödinger equation as the limit of the Pauli theory. Thus the Schrödinger equation may be seen as the far non-relativistic approximation of the Dirac equation when one may neglect spin and work only at low energies and velocities. This also was a great triumph for the new equation, as it traced the mysterious i that appears in it, and the necessity of a complex wave function, back to the geometry of space-time through the Dirac algebra. It also highlights why the Schrödinger equation, although superficially in the form of a diffusion equation, actually represents the propagation of waves.

It should be strongly emphasized that this separation of the Dirac spinor into large and small components depends explicitly on a low-energy approximation. The entire Dirac spinor represents an irreducible whole, and the components we have just neglected to arrive at the Pauli theory will bring in new phenomena in the relativistic regime - antimatter and the idea of creation and annihilation of particles.

In a general case (if a certain linear function of electromagnetic field does not vanish identically), three out of four components of the spinor function in the Dirac equation can be algebraically eliminated, yielding an equivalent fourth-order partial differential equation for just one component. Furthermore, this remaining component can be made real by a gauge transform.^[5]

Measurement

The existence of the anomalous magnetic moment of the electron has been detected exprimentally by magnetic resonance method. This allows the determination of hyperfine splitting of electron shell energy levels in atoms of protium and deuterium using the measured resonance frequency for several transitions.^[6]^[7]

References

↑ B Odom et al. 2006 Phys. Rev. Lett. 97 030801.
↑ A. Mahajan and A. Rangwala. Electricity and Magnetism, p. 419 (1989). Via Google Books.
↑ http://physics.nist.gov/cgi-bin/cuu/Value?eqae|search_for=electron+magnetic+moment
↑ http://physics.nist.gov/cgi-bin/cuu/Value?muem|search_for=magnetic+moment+electron
↑ Source: Journal of Mathematical Physics, 52, 082303 (2011) (http://jmp.aip.org/resource/1/jmapaq/v52/i8/p082303_s1 or http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf )
↑ Polykarp Kusch, H. M. Foley
↑ intrinsic moment of the electron

Bibliography

Sergei Vonsovsky (1975). Magnetism of Elemetary Particles. Mir Publishers.
Sin-Itiro Tomonaga (1997). The Story of Spin. University of Chicago Press.

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