  Many theories beyond the Standard Model postulate short-range modifications
to gravity which produce deviations of Newton's gravitational potential from a
strict $1/r$ dependence. It is common to analyze experiments searching for
these modifications using a potential of the form $V^{\prime}(r)=-\frac{GMm}{r}
[1+\alpha \exp{(-r/\lambda)}]$. The best present constraints on $\alpha$ for
$\lambda <100$\,nm come from neutron scattering and often employ comparisons of
different measurements of the coherent neutron scattering amplitudes $b$. We
analyze the internal consistency of existing data from two different types of
measurements of low energy neutron scattering amplitudes: neutron
interferometry, which involves squared momentum transfers $q^{2}=0$, and
neutron gravity reflectometry, which involves squared momentum transfers
$q^{2}=8mV_{opt}$ where $m$ is the neutron mass and $V_{opt}$ is the neutron
optical potential of the medium. We show that the fractional difference
$\frac{\Delta b}{|b|}$ averaged over the 7 elements where high precision data
exists on the same material from both measurement methods is $[2.2 \pm 1.4]
\times 10^{-4}$. We also show that $\frac{\Delta b}{|b|}$ for this data is
insensitive both to exotic Yukawa interactions and also to the electromagnetic
neutron-atom interactions proportional to the neutron-electron scattering
length $b_{ne}$ and the neutron polarizability scattering amplitude $b_{pol}$.
This result will be useful in any future global analyses of neutron scattering
data to determine $b_{ne}$ and bound $\alpha$ and $\lambda$. We also discuss
how various neutron interferometric and scattering techniques with cold and
ultracold neutrons can be used to improve the precision of $b$ measurements and
make some specific proposals.
