Subshell formula for the Bethe-Born stopping approximation

It is known that the subshell contributions to the coefficient of the $O\left(\ln E\right)$ asymptotic term of the Bethe-Born stopping-number formula are not given correctly by partitioning the squared matrix element defining this coefficient into a simple sum of its subshell contributions. This interesting result is extended to the second, or $O\mathrm{}\left(1\right)\mathrm{}$, asymptotic term in the stopping number. In this case, the subshell decomposition results in a sum of two expressions. One is simply the subshell decomposition of the squared matrix element defining the $O\mathrm{}\left(1\right)\mathrm{}$ coefficient for the total system. The second term is more complex in form but sums to zero, as it must, when summed over all subshells in the target. It is pointed out that this second term and its summation requirement can be used to impose a useful restriction on the construction of theoretical subshell corrections for the Bethe-Born formula.