  A system of confined charged electrons interacting via the long-range Coulomb
force can form a Wigner crystal due to their mutual repulsion. This happens
when the potential energy of the system dominates over its kinetic energy,
i.e., at low temperatures for a classical system and at low densities for a
quantum one. At $T=0$, the system is governed by quantum mechanics, and hence,
the spatial density peaks associated with crystalline charge localization are
sharpened for a lower average density. Conversely, in the classical limit of
high temperatures, the crystalline spatial density peaks are suppressed
(recovered) at a lower (higher) average density. In this paper, we study those
two limits separately using an exact diagonalization of small one-dimensional
(1D) systems containing few ($<10$) electrons and propose an approximate method
to connect them into a unified effective phase diagram for Wigner few-electron
crystallization. The result is a qualitative quantum-classical crossover phase
diagram of an effective 1D Wigner crystal. We show that the spatial density
peaks associated with the quasi-crystallization should be experimentally
observable in a few-electron 1D system. We find that the effective crystalline
structure slowly disappears with both the crossover average density and
crossover temperature for crystallization decreasing with increasing particle
number, consistent with the absence of any true long-range 1D order. In fact,
one peculiar aspect of the effective finite-size nature of 1D Wigner
crystallization we find is that even a short-range interaction would lead to a
finite-size 1D crystal, except that the crystalline order vanishes much faster
with increasing system size in the short-range interacting system compared with
the long-range interacting one.
