The angular correlation of the photons emitted when positrons annihilate with electrons has been studied in single crystals of the rare-earth metals Y, Gd, Tb, Dy, Ho, and Er, and in a single crystal of an equiatomic alloy of Ho and Er. A comparison of the results for Y with the calculations of Loucks shows that the independent-particle model gives a good first approximation to the angular distribution, although correlation effects probably smear out some of the structure. The angular distributions from the heavy rare-earth metals are very similar to that from Y and can be understood qualitatively in terms of the relativistic augmented-plane-wave calculations by Keeton and Loucks. The angular distributions in the c direction in the paramagnetic phases are characterized by a rapid drop at low angles followed by a hump, and these features are associated with rather flat regions of Fermi surface normal to the c axis. The same aspects of the Fermi surface are believed to be important in determining the stability of the periodic magnetic structures formed in some of the metals, and there is a strong correlation between the structure in the angular distribution and the magnetic properties. In the spiral phase of Ho, the structure in the c-axis distribution is much reduced, indicating that the Fermi surface is substantially modified by the magnetic ordering, as expected. The photon distribution from the equiatomic Ho-Er alloy is very similar to those from the constituent metals, although the structure is somewhat more pronounced.