"Anomalous magnetic moment of electron"의 두 판 사이의 차이

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** magnetic dipole moment, <math>\mu= -(3/10)eR^2\Omega</math>, where $R$ and $\Omega$ are the electron's classical radius and rotating frequency
 
** magnetic dipole moment, <math>\mu= -(3/10)eR^2\Omega</math>, where $R$ and $\Omega$ are the electron's classical radius and rotating frequency
 
* we get <math>\gamma = \mu/L=-e/2m_e</math>
 
* we get <math>\gamma = \mu/L=-e/2m_e</math>
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==Dirac theory==
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* if we use Dirac equation, we can get the electron spin g factor 2 "naturally"
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* http://lnu.se/polopoly_fs/1.96725!PresentationGfactor.pdf
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* http://www7b.biglobe.ne.jp/~kcy05t/diragfac.html
  
  

2015년 3월 19일 (목) 10:02 판

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

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


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 (perpendicular to the page) induce a curvature in the electron’s path.
  • In quantum electrodynamics (right) the electron interacts with the field by emitting or absorbing a photon.

[/pages/3589069/attachments/4562673 2004329152457_150.gif]

  • vertex correction

$$g = 2 ( 1 + \alpha/2\pi)$$ where $$\alpha= e^2/(4\pi \hbar c) \sim 1/137.036$$ is the fine structure constant

$$g/2=1+c_1\frac{\alpha}{2\pi}+c_2(\frac{\alpha}{2\pi})^2+c_3(\frac{\alpha}{2\pi})^3+\cdots=1.00115965219+\cdots$$

Feynmann diagrams

  • amplitude = sum of integrals = \(\sum_{n\text{ loops}}\) sum of integrals
  • as n grows, number of Feynman diagrams grows exponentially
  • integrals are becoming difficult 


tree level and one-loop diagrams

  • 1 one-loop diagram
    [/pages/7141159/attachments/4563145 2004329152921_150.gif]
  • Feynman, Julian Schwinger, Sin-Itiro Tomonaga and Freeman Dyson
  • Schwinger showed that the one-loop contribution to the "anomalous magnetic moment" of the electron is \(\alpha/{2\pi}=0.00116\cdots\)
  • Schwinger, Julian. 1948. On Quantum-Electrodynamics and the Magnetic Moment of the Electron. Physical Review 73, no. 4 (February 15): 416. doi:10.1103/PhysRev.73.416. 
     
  • http://www.wolframalpha.com/input/?i=fine+structure+constant%2F%282pi%29

 

 

two-loop diagrams

  • 7 two-loop diagrams
    [/pages/3589069/attachments/4562669 2004329153354_150.gif]
    [/pages/7141159/attachments/4562733 I15-62-g2c.jpg]

 

 

three-loop diagrams

  • 72 three-loop diagrams
  • [/pages/3589069/attachments/4562671 200432915395_150.gif]
  • Kinoshita, Toichiro. 1995. New Value of the alpha^{3} Electron Anomalous Magnetic Moment. Physical Review Letters 75, no. 26 (December 25): 4728. doi:10.1103/PhysRevLett.75.4728. 
     


four-loop diagrams

  •  891 diagrams

 

 

five-loop Feynman diagrams


anaomalous muon magnetic dipole moment


 

memo

 

 

related items

 

computational resource


encyclopedia

expositions


articles

  • Broadhurst, D. J, and D. Kreimer. 1996. Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops. hep-th/9609128 (September 16). doi:doi:10.1016/S0370-2693(96)01623-1. http://arxiv.org/abs/hep-th/9609128.
  • Bloch, F. “Experiments on the G-Factor of the Electron.” Physica 19, no. 1–12 (1953): 821–31. doi:10.1016/S0031-8914(53)80091-1.
  • Mann, A. K., and P. Kusch. “Further Data on the Spin Gyromagnetic Ratio of the Electron.” Physical Review 77, no. 4 (February 15, 1950): 435–38. doi:10.1103/PhysRev.77.435.
  • 1948 슈와잉거 (J. Schwinger, Phys.Rev. 73(1948) 416L)

books

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