  Strain engineering is used to obtain desirable materials properties in a
range of modern technologies. Direct nanoscale measurement of the
three-dimensional strain tensor field within these materials has however been
limited by a lack of suitable experimental techniques and data analysis tools.
Scanning electron diffraction has emerged as a powerful tool for obtaining
two-dimensional maps of strain components perpendicular to the incident
electron beam direction. Extension of this method to recover the full
three-dimensional strain tensor field has been restricted though by the absence
of a formal framework for tensor tomography using such data. Here, we show that
it is possible to reconstruct the full non-symmetric strain tensor field as the
solution to an ill-posed tensor tomography inverse problem. We then demonstrate
the properties of this tomography problem both analytically and
computationally, highlighting why incorporating precession to perform scanning
precession electron diffraction may be important. We establish a general
framework for non-symmetric tensor tomography and demonstrate computationally
its applicability for achieving strain tomography with scanning precession
electron diffraction data.
