  We discuss the spectrum arising from synchrotron emission by fast cooling
(FC) electrons, when fresh electrons are continually accelerated by a strong
blast wave, into a power law distribution of energies. The FC spectrum was so
far described by four power law segments divided by three break frequencies
$\nu_{sa} < \nu_{c} < \nu_{m}$. This is valid for a homogeneous electron
distribution. However, hot electrons are located right after the shock, while
most electrons are farther down stream and have cooled. This spatial
distribution changes the optically thick part of the spectrum, introducing a
new break frequency, $\nu_{ac} < \nu_{sa}$, and a new spectral slope, $F_{\nu
}\propto \nu^{11/8}$ for $\nu_{ac} < \nu < \nu_{sa}$. The familiar $F_{\nu
}\propto \nu ^{2}$ holds only for $\nu < \nu_{ac}$. This ordering of the break
frequencies is relevant for typical gamma-ray burst (GRB) afterglows in an ISM
environment. Other possibilities arise for internal shocks or afterglows in
dense circumstellar winds. We discuss possible implications of this spectrum
for GRBs and their afterglows, in the context of the internal-external shock
model. Observations of $F_{\nu} \propto \nu^{11/8}$ would enable us to probe
scales much smaller than the typical size of the system, and constrain the
amount of turbulent mixing behind the shock.
