= Solution
A \b[<decahedral nanoparticle>] is a <multiply twinned nanoparticle>: five fcc-like tetrahedral domains share a common fivefold axis and meet on coherent twin planes. The angle of an ideal fcc tetrahedral wedge is $\arccos(1/3)=70.53^\circ$, so five wedges leave about \b[$7.36^\circ$ of angular deficit]. Elastic distortion and defects accommodate closure. The external shape can be a pentagonal bipyramid or a truncated variant; it is not one unstrained, periodically repeated fcc crystal.
An \b[<icosahedral nanoparticle>] comprises twenty tetrahedral domains arranged around a common centre, with twenty approximately triangular outer facets, twelve vertices and thirty edges. Five domains meet around each radial fivefold direction. Coherent twinning exposes many low-energy close-packed facets, but fivefold symmetry cannot tile ordinary three-dimensional space periodically without strain. The competition between favourable surface/twin energies and volume-dependent elastic strain explains why such morphologies are especially plausible for small particles.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2017/paper-2-twinned-particles.png]
{title=Five tetrahedral sectors in a decahedral cross-section and the twenty outer facets of an icosahedron}
{source=paper-2-twinned-particles.py}
These labels describe the observed domain geometry; cobalt can adopt different crystal phases, so identifying fcc-like twinning also requires diffraction rather than assuming every cobalt particle is bulk fcc.
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