The Magnetic Excitation Spectrum and Thermodynamics of High-T$_c$ Superconductors

Inelastic neutron scattering was used to study the wave vector- and frequency-dependent magnetic fluctuations in single crystals of superconducting YBa$_2$Cu$_3$O$_{6+x}$. The spectra contain several important features, including a gap in the superconducting state, a pseudogap in the normal state, and the much-discussed resonance peak. The appearance of the pseudogap determined from transport and nuclear resonance coincides with formation of the resonance in the magnetic excitations. The exchange energy associated with the resonance has the temperature and doping dependences as well as the magnitude to describe approximately the electronic specific heat near the superconducting transition temperature (T$_c$).