  There are three $S$ and seven $P$ fully charmed $[cc][\bar c\bar c]$
tetraquarks, the mass spectrum from $1S$ to $2P$ excitations is calculated in a
non-relativistic quark potential model. In the calculation, the interactions
among four internal quarks/antiquark are approximated as a dominant color
interaction between a diquark and an antidiquark, and a residual interaction
responsible for the diquark/antidiquark cluster effect. The color interaction
between the diquark and the antidiquark is characterized by the conventional
Cornell potential, while the residual interaction is modeled as the Yukawa-type
scalar $\sigma$ and vector $\omega$ boson exchange potentials. In the numerical
results, though the scalar $\sigma$ and vector $\omega$ boson exchange
interactions reduce the masses of $S-$wave $[cc][\bar c\bar c]$ tetraquarks
$40-50$ MeV, they contribute to the masses of other $[cc][\bar c\bar c]$
tetraquarks small. The mass splittings of $[cc][\bar c\bar c]$ tetraquarks
between different multiplets and within the same multiplet are smaller than
those in charmonium, and the splittings are affected by the scalar $\sigma$ or
vector $\omega$ boson exchange interactions small. Our calculations suggest
that the observed $X(6600)$, $X(6900)$ and $X(7300)$ should be different radial
excitations of $[cc][\bar c\bar c]$. The measurements of the $J^{PC}$ quantum
numbers and the mass splittings will be helpful to identify and understand of
the $[cc][\bar c\bar c]$ candidates.
