Electrons can experience superadiabatic heating at a collisionless shock due to the presence of a cross-shock electric potential. This heating can greatly exceed the temperature rise associated with the conservation of the magnetic moment of the electrons as they move into the stronger magnetic field downstream of the shock. The criterion for the onset of superadiabatic heating and the effects of the detailed structure of the shock are investigated for the special case where the magnetic field is perpendicular to the shock normal. A comparison of the natural length scale of the electron orbits and the shock thickness is used to derive a new criterion for the onset of superadiabatic heating. This new heating criterion facilitates an improved understanding of the role of various physical parameters in determining the degree of electron heating. It has the same functional form as the condition for the electron equations of motion to permit exponential divergence of electron orbits (an important feature of dynamical chaos) which it has previously been argued determines the onset of superadiabatic heating. However, we show that the orbits of individual electrons undergoing superadiabatic heating do not diverge significantly because of the very short time they spend within the shock, the only region in which the equations of motion are nonlinear. The techniques used to analyze dynamically chaotic systems are therefore of little assistance in understanding this potentially important process. ¿ 1998 American Geophysical Union