  Using the density matrix renormalization group technique, we study the ground
state phase diagram and other low-energy properties of an isotropic
antiferromagnetic spin-half chain with both dimerization and frustration, i.e.,
an alternation $\delta$ of the nearest neighbor exchanges and a
next-nearest-neighbor exchange $J_2~$. For $\delta = 0$, the system is gapless
for $J_2 < J_{2c} ~$ and has a gap for $J_2 > J_{2c} ~$ where $J_{2c}~$ is
about $0.241$. For $J_2 = J_{2c}$, the gap above the ground state grows as
$\delta$ to the power $0.667 \pm 0.001$. In the $J_2 - \delta$ plane, there is
a disorder line $2 J_2 + \delta = 1$. To the left of this line, the peak in the
static structure factor $S(q)$ is at $q_{max} = \pi$ (Neel phase), while to the
right of the line, $q_{max}$ decreases from $\pi$ to $\pi / 2$ as $J_2$ is
increased to large values (spiral phase). For $\delta = 1$, the system is
equivalent to two coupled chains as on a ladder and it is gapped for all values
of the interchain coupling.
