  The concept of chirality makes ubiquitous appearance in nature. Particularly,
both a structure and its collective excitations may acquire well defined
chiralities. In this work, we reveal an intrinsic connection between the
chiralities of a crystal structure and its phonon excitations. We show that the
phonon chirality and its propagation direction are strongly coupled with the
lattice chirality, which will be reversed when a chiral lattice is switched to
its enantiomorph. In addition, distinct from achiral lattices, propagating
chiral phonons exist for chiral crystals also on the principal axis through the
$\Gamma$ point, which strengthens its relevance to various physical processes.
We demonstrate our theory with a 1D helix-chain model and with a concrete and
important 3D material, the $\alpha$-quartz. We predict a chirality diode effect
in these systems, namely, at certain frequency window, a chiral signal can only
pass the system in one way but not the other, specified by the system
chirality. Experimental setups to test our theory are proposed. Our work
discovers fundamental physics of chirality coupling between different levels of
a system, and the predicted effects will provide a new way to control thermal
transport and design information devices.
