We present U-band CCD surface brightness profiles for 15 of the 21 glo
bular clusters that have been identified as having collapsed cores by
Djorgovski & King (1986). Fourteen of the clusters were observed with
the Cerro Tololo 4 m telescope; NGC 7078 was observed with the Canada-
France-Hawaii 3.6 m telescope. We have fitted the profiles with seeing
-convolved power laws, both with and without cores, to assess the evid
ence for central power-law structure and to place upper limits on core
radius r(c). We find nine of the clusters (NGC 5946, NGC 6284, NGC 62
93, NGC 6325, NGC 6342, NGC 6558, NGC 6624, NGC 6681, and NGC 7078) to
have unresolved cores, with upper limits r(c) less than or equal to 1
''.9. Three of the clusters (NGC 6453, NGC 6522, and NGC 7099) have ma
rginally resolved cores, with upper limits in the range 2''.7 less tha
n or equal to r(c) less than or equal to 3''.4. The remaining three cl
usters (NGC 6355, NGC 6397, and NGC 6752) have resolved cores. Of the
latter three clusters, NGC 6355 and NGC 6752 are consistent with singl
e-mass King model structure. The median cluster distances are 9.2 kpc
for those with unresolved cores, 7.2 kpc for those with marginally res
olved cores, and 4.1 kpc for those with resolved cores. The 13 cluster
s that do not resemble single-mass King models have central power-law
structure with surface brightness slopes in the range of d1nS/d1nr = -
0.6 to -0.8. These slopes are consistent with the models of Grabhorn e
t al. (1992) for clusters evolving beyond core collapse. The models in
clude a centrally concentrated population of nonluminous remnants with
masses in the range 1.2-1.4 M., thus providing evidence for significa
nt neutron star populations in most of our cluster sample. This findin
g is consistent with the observation of centrally concentrated low-mas
s X-ray binary and millisecond pulsar populations in several clusters.