  We study properties of jammed packings of frictionless spheres over a wide
range of volume fractions. There exists a crossover volume fraction which
separates deeply jammed solids from marginally jammed solids. In deeply jammed
solids, all the scalings presented in marginally jammed solids are replaced
with remarkably different ones with potential independent exponents.
Correspondingly, there are structural changes in the pair distribution function
associated with the crossover. The normal modes of vibration of deeply jammed
solids also exhibit some anomalies, e.g. strengthened quasi-localization and
absence of Debye-like density of states at low frequencies. Deeply jammed
systems may thus be cataloged to a new class of amorphous solids.
