  A low-background neutron detector array was developed to measure the cross
section of the $^{13}$C($\alpha$,n)$^{16}$O reaction, which is the neutron
source for the $s$-process in AGB stars, in the Gamow window ($E_{c.m.}$ = 190
$\pm$ 40 keV) at the China Jinping Underground Laboratory (CJPL). The detector
array consists of 24 $^{3}$He proportional counters embedded in a polyethylene
cube. Due to the deep underground location and a borated polyethylene shield
around the detector array, a low background of 4.5(2)/hour was achieved. The
$^{51}$V(p, n)$^{51}$Cr reaction was used to determine the neutron detection
efficiency of the array for neutrons with energy $E_n$ $<$ 1 MeV. Geant4
simulations, which were shown to well reproduce experimental results, were used
to extrapolate the detection efficiency to higher energies for neutrons emitted
in the $^{13}$C($\alpha$,n) $^{16}$O reaction. The theoretical angular
distributions of the $^{13}$C($\alpha$,n)$^{16}$O reaction were shown to be
important in estimating the uncertainties of the detection efficiency.
