  Magnetic pyrochlore oxides, including the spin ice materials, have proved to
be a rich field for the study of geometrical frustration in 3 dimensions.
Recently, a new family of magnetic oxides has been synthesised in which the
half of the tetrahedra in the pyrochlore lattice are inflated relative to the
other half, making an alternating array of small and large tetrahedra. These
"breathing pyrochlore" materials such as LiGaCr4O8, LiInCr4O8 and Ba3Yb2Zn5O11
provide new opportunities in the study of frustrated magnetism. Here we provide
an analytic theory for the ground state phase diagram and spin correlations for
the minimal model of magnetism in breathing pyrochlores: a classical nearest
neighbour Heisenberg model with different exchange coefficients for the two
species of tetrahedra. We find that the phase diagram comprises a Coulombic
spin liquid phase, a conventional ferromagnetic phase and an unusual
antiferromagnetic phase with lines of soft modes in reciprocal space,
stabilised by an order-by-disorder mechanism. We obtain a theory of the spin
correlations in this model using the Self Consistent Gaussian Approximation
(SCGA) which enables us to discuss the development of correlations in breathing
pyrochlores as a function of temperature, and we quantitatively characterise
the thermal crossover from the limit of isolated tetrahedra to the strongly
correlated limit of the problem. We compare the results of our analysis with
the results of recent neutron scattering experiments on LiInCr4O8.
