  The first measurement of the dependence of $\gamma\gamma$ $\to$
$\mu^{+}\mu^{-}$ production on the multiplicity of neutrons emitted very close
to the beam direction in ultraperipheral heavy ion collisions is reported. Data
for lead-lead interactions at $\sqrt{s_\mathrm{NN}} =$ 5.02 TeV, with an
integrated luminosity of approximately 1.5 nb$^{-1}$, were collected using the
CMS detector at the LHC. The azimuthal correlations between the two muons in
the invariant mass region 8 $\lt$ $m_{\mu\mu}$ $\lt$ 60 GeV are extracted for
events including 0, 1, or at least 2 neutrons detected in the forward
pseudorapidity range $|\eta|$ $\gt$ 8.3. The back-to-back correlation structure
from leading-order photon-photon scattering is found to be significantly
broader for events with a larger number of emitted neutrons from each nucleus,
corresponding to interactions with a smaller impact parameter. This observation
provides a data-driven demonstration that the average transverse momentum of
photons emitted from relativistic heavy ions has an impact parameter
dependence. These results provide new constraints on models of photon-induced
interactions in ultraperipheral collisions. They also provide a baseline to
search for possible final-state effects on lepton pairs caused by traversing a
quark-gluon plasma produced in hadronic heavy ion collisions.
