  High-luminosity particle collider experiments such as the ones planned at the
High-Luminosity Large Hadron Collider require ever-greater vertexing precision
of the tracking detectors, necessitating also reductions in the material budget
of the detectors. Traditionally, the fractional radiation length ($x/X_0$) of
detectors is either estimated using known properties of the constituent
materials, or measured in dedicated runs of the final detector. In this paper,
we present a method of direct measurement of the material budget of a CMS
prototype module designed for the Phase-2 upgrade of the CMS detector using a
40-65 MeV positron beam. A total of 630 million events were collected at the
Paul Scherrer Institut PiE1 experimental area using a three-plane telescope
consisting of the prototype module as the central plane, surrounded by two
MALTA monolithic pixel detectors. Fractional radiation lengths were extracted
from scattering angle distributions using the Highland approximation for
multiple scattering. A statistical technique recovered runs suffering from
trigger desynchronisation, and several corrections were introduced to
compensate for local inefficiencies related to geometric and beam shape
constraints. An overall average $x/X_0$ of (0.84 $\pm$ 0.10)% across the
surveyed regions was measured, which is compatible with an empirical estimate
of 0.825% computed from known material properties. Higher-granularity maps of
the fractional radiation length were produced for both rectangular regions and
regions of uniform material composition. The results bode well for the CMS
Phase-2 upgrade modules, which will play a key role in the minimisation of the
material budget of the upgraded detector.
