This paper is concerned with the explanation of the spread F echoes often obtained on frequency-sweep records of equivalent height in the ionosphere, especially at night and in the auroral regions. It is shown in the Appendix that the production of a mountain range of echoes from a single incident pulse can be accounted for if there is a periodic structure of ionization in the F layer. There are two mechanisms of ionization in the ionosphere, namely the incidence of ultra-violet radiation from the sun, and the influx of a stream of charged particles that are deflected by the earth's magnetic field towards the geomagnetic poles. The ultra-violet light produces well defined layers, the E, F1 and F2 layers, with a single maximum of ionic density within them, and they may be satisfactorily explained by Chapman's theory, taking into account the recombination process but neglecting diffusion effects. In the case of particle ionization, however, the effect of diffusion becomes predominant. It is shown in the paper that the inclusion of the diffusion term in the differential equation of layer formation leads to the possibility of a periodic structure within the F layer of the kind needed to explain the spread echoes. This is because the differential equation has as its solution a doubly periodic elliptic function that provides the required periodicity of ionic density as a function of height. The form of the mathematics gives the essentially periodic nature of the solution without any detailed appeal to the initial conditions, but reference is made to the auroral curtains indicating the existence of sources of charged particles that maintain the periodic structure of ionization in between the curtains. Whereas the ultra-violet light causes layers to be formed that are many wavelengths thick, the particle ionization causes very thin layers only two or three wavelengths thick embedded in the background of ionization.