Anomalous magnetic moment of electron

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introduction

  • The electron spin g-factor \(g\) is approximately two
  • Dirac theory explains why \(g=2\), but it's not exactly 2
  • the currently accepted value is 2.00231930436153
  • this discrepancy is explained by QED
  • theoretical computation matches 11 digits with experiments
  • anomalous electron magnetic dipole moment \(g/2=1.00115965219\)


Lande's question

  • Bohr magneton : natural unit of magnetic moment due the electron's motion in orbit

\[\mu_0=e\hbar /2m_ec\] where

  1. \(e\) is the elementary charge,
  2. \(\hbar\) is the reduced Planck constant,
  3. \(m_e\) is the electron rest mass and
  4. \(c\) is the speed of light.
  • spin magnetic dipole moment \(\mu_s\)
  • Q. \(\mu_s=\mu_0\) ? (Back and Lande 1925)
  • We define \(g\) to be the gyromagnetic ratio

\[\mu_s=\frac{g\mu_0}{2}\]


history

  • \(g_c\) be the g-factor for 'core'

background

the ratio of the intrinsic magnetic moment of an electron to its intrinsic or spin angular momentum is not equal to (but very nearly twice) that ratio for the magnetic moment and angular momentum arising from the orbital motion of the electron.


Lande g-factor

Basically through the work of Lande it was known that \(g_c=2\) fitted the observed multiplets of alkalies and also earth alkalies quite well. This value clearly had to be considered anomalous, since the magnetic moment and angular momentum of the core were due to the orbital motions of the electrons inside the core, which inevitably would lead to \(g_c=1\), as explained in section 2.1. This was a great difficulty for the core model at the time, which was generally referred to as the “magneto-mechanical anomaly”. Pauli pointed out that one could either say that the physical value of the core’s gyromagnetic factor is twice the normal value, or, alternatively, that it is ob- tained by adding 1 to the normal value.


Pauli's insight

Note that this hypothesis replaces the atom’s core as carrier of angular momentum by the valence electron. This means that (17), (18), and (20) are still valid, except that the subscript c (for “core”) is now replaced by the subscript s (for “spin”, anticipating its later interpretation), so that we now have a coupling of the electron’s orbital angular momentum (subscript e) to its intrinsic angular momentum (subscript s). In (20), with \(g_c\) replaced by \(g_s\), one needs to set \(g_s=2\) in order to fit the data.

classical magnetic moment

  • read spin system first
  • gyromagnetic ratio is defined as "magnetic dipole moment"/"angular momentum"

\[\gamma = \mu/L=-e/2m_e\] [/pages/7141159/attachments/4562863 I15-62-g20.jpg]


orbital

  • Let \(e\), \(m_e\), \(v\), and \(r\) be the electron's charge, mass, velocity, and radius, respectively.
  • A classical electron moving around a nucleus in a circular orbit
    • orbital angular momentum, \(L=m_evr\)
    • magnetic dipole moment, \(\mu= -evr/2\)
  • we get \(\gamma=\mu/L=-e/2m_e\)

spin

  • A classical electron of homogeneous mass and charge density rotating about a symmetry axis
    • spin angular momentum, \(L=(3/5)m_eR^2\Omega\)
    • magnetic dipole moment, \(\mu= -(3/10)eR^2\Omega\), where \(R\) and \(\Omega\) are the electron's classical radius and rotating frequency
  • we get \(\gamma = \mu/L=-e/2m_e\)


Dirac theory


anamalous electron magnetic dipole moment

electron spin g-factor

  • there are correction terms to the spin magnetic moment of the electron as shown by experiments
  • actual spin magnetic moment of the electron involves the spin g-factor (gyromagnetic ratio)

\[\vec{\mu}_S \ = g \mu_0 \frac{\vec{S}}{\hbar}=g\frac{e}{2 m_{e}} \ \vec{S}\]

  • The g factor sets the strength of an electron’s interaction with a magnetic field

QED

  • In classical physics (left) magnetic lines of force (perp

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Spacy 패턴 목록

  • [{'LOWER': 'anomalous'}, {'LOWER': 'magnetic'}, {'LOWER': 'dipole'}, {'LEMMA': 'moment'}]
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