A New Spin on Photoemission Spectroscopy Page: 40 of 259
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separated by the vector r as
Ui - [mi-m2 - 3(mi-i)(m2-r)]. (1.39)
Without being too precise, we can estimate that each dipole has a magnitude of about iB.
Disregarding the angular dependence of the dot products, this gives us on the order of
p2 t 1 ao3
r3 (137)2 r(1.40)
where ao is the Bohr radius and 1 Ry is about 13.6 eV. With a dipole separation distance
of about 2 A ~ 4a0, this gives
Udip ~ 10-GeV. (1.41)
As only 1K ~ 10-4 eV, thermal fluctuations will clearly outweigh any dipolar interactions
well before the typical Tc of 1000K used above. Looking at it in another way, the magnetic
field of a pB dipole at a distance of 2 A is only about 103 Gauss, far smaller than the
necessary Weiss field calculated above from equation 1.38. Therefore it is clear that direct
dipole-dipole interactions are not the source of ferromagnetic ordering (although they play
a key role in domain formation, well described in chapter 33 of ref. 13).
In 1928, Heisenberg showed the Weiss field originates from the exchange interaction. 14
To illustrate the nature of the exchange interaction, it is useful to investigate examples of
two-electron system. The simplest system is the case of two free non-interacting electrons,
H = - 2h (V12 - V22) (1.42)
where the subscripts 1 and 2 refer to the two electrons. The eigenvalue equation
HP(x1, x2) = EP(xi, x2) is separable into a product of two single electron functions, and
the solution can be written
q(x1, x2) = 4'a(x1L) b(x2) = eik"x~eikbx2 (1.43)
Since the total state must be antisymmetric with respect to particle exchange (Sym-
metrization Postulate), the spatial part must be symmetrized/antisymmetrized versions
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Jozwiak, Chris. A New Spin on Photoemission Spectroscopy, thesis or dissertation, December 1, 2008; United States. (https://digital.library.unt.edu/ark:/67531/metadc1014237/m1/40/: accessed April 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.