  Images of flux vortices in superconductors acquired by transmission electron
microscopy should allow a quantitative determination of their magnetic
structure but so far, only visual comparisons have been made between
experimental images and simulations. Here, we make a quantitative comparison
between Fresnel images and simulations based on the modified London equation to
investigate the magnetic structure of flux vortices in MgB2. This technique
gives an absolute, low-field (~30 Oe) measurement of the penetration depth from
images of single vortices. We found that these simulations gave a good fit to
the experimental images and that if all the other parameters in the fit were
known, the penetration depth for individual vortices could be measured with an
accuracy of +/- 5 nm. Averaging over 17 vortices gave a penetration depth in
the ab plane of 113 +/- 2 at 10.8 K assuming that the entire thickness of the
sample was superconducting. The main uncertainty in this measurement was the
proportion of the specimen which was superconducting. Allowing for a
non-superconducting layer of up to 50 nm thickness on the specimen surfaces
gave a penetration depth in the range 100-115 nm, close to values of 90 +/- 2
nm obtained by small-angle neutron scattering and 118-138 nm obtained by
radio-frequency measurements. We also discuss the use of the transport of
intensity equation which should, in principle, give a model-independent measure
of the magnetic structure of flux vortices.
