Retarded electromagnetic interaction between electrons

Now we consider the results of applying a static uniform electric field E. The electrostatic potential pi (see (3.134)) will now contain the extra term 

This term was encountered previously in the Hamiltonian for a single electron (3.101) (E a 7F,) is essentially equivalent to a magnetic field, which interacts with the spin magnetic moment. The term is, however, usually negligible in laboratory experiments. Summarising, the electric field, or Stark, Hamiltonian may be written 

In the derivation of the many-electron Hamiltonian (section 3.6) the interactions between electrons were introduced in the expressions for the magnetic and electric 

We commented that the second term in (3.133) is incorrect, and gave for the orbit-orbit interaction the correct form in (3.145),but without justification. We now examine the interaction between electrons more careftilly, both to justify the earlier assumption and also to prepare the ground for our later discussion of the Breit equation. 

Now we consider the resnlts of applying a static uniform electric field E. The 

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The first term, Hj, is the spin-orbit (one electron term) and spin-other-orbit (two electron term) couplings, which are the topic of the following subsection. The second term Hf contains the spin-spin coupling term and Fermi contact interaction. Both the Hj and f/ can lift degeneracy in multiplets. The parameter Hf is the Dirac correction term for electron spin and Ff is the classical relativistic correction to the interaction between electrons due to retardation of the electromagnetic field produced by an electron. The parameter H is the so-called mass-velocity effect, due to the variation of electron mass with velocity. Finally, H is the effect of external electric and magnetic fields. 

A final interesting observation is the existence of a frequency scale, 3x10 see in Eq. (2-39). This is the frequency at which the electronic cloud around an atom fluctuates it is therefore the rate at which the spontaneous dipoles fluctuate. Since the electromagnetic field created by these dipoles propagates at the speed of light c = 3 x lO cm/sec, only a finite distance c/v 100 nm is traversed before the dipole has shifted. Since the dispersion interaction is only operative when these dipoles are correlated with each other, and this correlation is dismpted by the time lag between the fluctuation and the effect it produces a distance r away, the dispersion interaction actually falls off more steeply than r when molecules or surfaces become widely separated. This effect is called the retardation of the van der Waals force. The effective Hamaker constant is therefore distance dependent at separations greater than 5-10 nm or so. 

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