  The beta-decay of the 6He halo nucleus and the M1-transition processes of the
excited 6Li(0+) state into the alpha + d continuum are studied in a three-body
model. The initial nuclear states are described as an alpha + 2N system in
hyperspherical coordinates on a Lagrange mesh. The energy dependence and
absolute values of the beta-transition probability per time and energy units of
a recent experiment can be reproduced very well with an appropriate alpha + d
potential. A total transition probability of 2.04E-6 s^(-1) is obtained in
agreement with the experiment. Due to a strong cancellation of the internal and
halo components of the matrix elements, halo effects are shown to be very
important. For the M1 transition process the halo structure of the excited
6Li(0+) isobaric analog state is confirmed. The cancellation effects, however,
are not so strong. The transition probability is strongly sensitive to the
description of the alpha + d phase shifts. Charge symmetry is analyzed through
a comparison of the two processes. The present branching ratio Gamma(0^+ -->
alpha + d)/Gamma(0^+ --> 1+) is of the order 1.0E-4, and is expected to be
observable in current experiments.
