  We investigate the strain state of ensembles of thin and nearly
coalescence-free self-assembled GaN nanowires prepared by plasma-assisted
molecular beam epitaxy on Ti/Al$_{2}$O$_{3}(0001)$ substrates. The shifts of
Bragg peaks in high-resolution X-ray diffraction profiles reveal the presence
of a homogeneous tensile strain in the out-of-plane direction. This strain is
inversely proportional to the average nanowire radius and results from the
surface stress acting on the nanowire sidewalls. The superposition of strain
from nanowires with different radii in the same ensemble results in a
broadening of the Bragg peaks that mimics an inhomogeneous strain on a
macroscopic scale. The nanowire ensembles show a small blueshift of the
bound-exciton transitions in photoluminescence spectra, reflecting the
existence of a compensating in-plane compressive strain, as further supported
by grazing incidence x-ray diffraction measurements carried out at a
synchrotron. By combining X-ray diffraction and photoluminescence spectroscopy,
the surface stress components $f_{x}$ and $f_{z}$ of the air-exposed
GaN$\{1\bar100\}$ planes that constitute the nanowire sidewalls are determined
experimentally to be 2.25 and $-0.7$~N/m, respectively.
