  We have constructed an analytic formula to treat the perpendicular magnetic
anisotropy energy in ferromagnetic metals with low symmetry, such as C4V and
C3V. We find that the anisotropy energy is proportional to a part of the
expectation values of the orbital angular momentum and magnetic dipole
operator. Although the result is similar to the model proposed by Laan [J.
Phys.: Condens. Matter 10, 3239 (1998)], we have derived a concrete expression
for the spin-flip virtual excitation process term, which can be dominant in
atoms with small magnetic moments and/or small exchange splitting. Pt monatomic
layer with proximity-induced spin polarization grown on Fe is an example of
this. Other multilayer systems such as Co/Pd and Co/Ni and bilayer systems such
as Fe(CoB)/MgO can be discussed similarly. Moreover, the relation between
perpendicular magnetic anisotropy energy and measurable physical parameters is
discussed based on X-ray magnetic circular dichroism spectroscopy.
