  Background: Microscopic calculations of heavy nuclei face considerable
difficulties due to the sizes of the matrices that need to be solved. Various
approximation schemes have been invoked, for example by truncating the spaces,
imposing seniority limits, or appealing to various symmetry schemes such as
pseudo-SU(3). This paper proposes a new symmetry scheme also based on SU(3).
This proxy-SU(3) can be applied to well-deformed nuclei, is simple to use, and
can yield analytic predictions. Purpose: To present the new scheme and its
microscopic motivation, and to test it using a Nilsson model calculation with
the original shell model orbits and with the new proxy set. Method: We invoke
an approximate, analytic, treatment of the Nilsson model, that allows the above
vetting and yet is also transparent in understanding the approximations
involved in the new proxy-SU(3). Results: It is found that the new scheme
yields a Nilsson diagram for well-deformed nuclei that is very close to the
original Nilsson diagram. The specific levels of approximation in the new
scheme are also shown, for each major shell. Conclusions: The new proxy-SU(3)
scheme is a good approximation to the full set of orbits in a major shell.
Being able to replace a complex shell model calculation with a symmetry-based
description now opens up the possibility to predict many properties of nuclei
analytically and often in a parameter-free way. The new scheme works best for
heavier nuclei, precisely where full microscopic calculations are most
challenged. Some cases in which the new scheme can be used, often analytically,
to make specific predictions, are shown in a subsequent paper.
