  Positron annihilation experiments on an laboratory scale depend on the supply
and the availability of $\beta^+$ emitters. Here we present the production of
positron sources based on the $^{27}$Al(p,x)$^{22}$Na reaction by irradiation
of Al with a 68\,MeV proton beam. We simulated the energy loss, range and
radial scattering of the protons in Al in order to design a simple target
consisting of a stack of Al discs. Our approach allows (i) the direct use of
the Al discs as positron emitters that inherently avoids wet chemical processes
as usually applied in commercial production of carrier-free $^{22}$Na, (ii) the
production of multiple positron sources at once, and (iii) the simple
measurement of the depth and lateral distribution of $^{22}$Na. We precisely
determined the cross section of the $^{27}$Al(p,x)$^{22}$Na reaction which was
found to differ from literature values particularly for proton energies between
27 and 40\,MeV. The activity of all nuclides produced (apart from $^{22}$Na)
was shown to be negligible 15 days after irradiation. The production of
radionuclides such as $^{48}$Sc, $^{54}$Mn and $^{56}$Co can be prevented by
using Al of a higher purity. The concept presented here can easily be adapted
for the production of stronger $^{22}$Na sources by increasing the proton
current or/and the irradiation time.
