  An exact expression for the mass-gap, the ratio of the physical particle mass
to the $\Lambda$-parameter, is found for the principal chiral sigma models
associated to all the classical Lie algebras. The calculation is based on a
comparison of the free-energy in the presence of a source coupling to a
conserved charge of the theory computed in two ways: via the thermodynamic
Bethe Ansatz from the exact scattering matrix and directly in perturbation
theory. The calculation provides a non-trivial test of the form of the exact
scattering matrix.
