Frequency swept pulses are used to invert population over a broad band of transition frequencies without the need for a precise calibration of the pulse amplitude. As long as the adiabatic approximation is valid, population is adiabatically inverted at each individual transition frequency of the swept range. Even though this picture fails to be true when the adiabatic approximation breaks down, population inversion can still be achieved for an appreciable range of transition frequencies and field amplitudes. We discuss population inversion in an ensemble of two-level systems by a frequency-swept pulse, the so-called constant adiabaticity pulse, without invoking the adiabatic approximation. The equations of motion are integrated by a seminumerical method to analyze population inversion in the regime where usually the adiabatic approximation is applied. The effects of resonance offset and variable pulse amplitude are described by an average Hamiltonian expansion to discuss pulse performance beyond the validity of the adiabatic approximation. As a function of the adiabaticity parameter (reciprocal of the pulse area), the inversion bandwidth increases in a stepwise fashion due to the consecutive cancellation of average Hamiltonians. The first inversion over a finite range of transition frequencies and pulse amplitudes is shown to occur for an adiabaticity parameter of 1/root15. (C) 2003 American Institute of Physics.