  The Isochronous Mass Spectrometry (IMS) based on storage rings is a powerful
technique for mass measurement of short-lived exotic nuclei. The transition
energy $\gamma_t$ of the storage ring is a vital parameter of the IMS
technique. It is difficult to measure the $\gamma_t$ and its relation to
momentum spread or circulating length, especially to monitor the variation of
$\gamma_t$ during experiments. An experimental investigation on the $\gamma_t$
has been performed for the IMS experiment at the Cooler Storage Ring of the
Heavy Ion Research Facility in Lanzhou (HIRFL-CSRe). With the velocity measured
by two time-of-flight (TOF) detectors, the $\gamma_t$ as a function of orbital
length can be determined. The influences of higher order magnetic field
components on the $\gamma_t$ function were inferred for isochronous correction.
This paper introduces and investigates the influence of dipole magnetic fields,
quadrupole magnetic fields and sextupole magnetic fields on the $\gamma_t$
function. With the quadrupole magnets and sextupole magnets corrections, a mass
resolution of 171332 (FWHM) and $\sigma(T)/T=1.34\times10^{-6}$ were reached,
which shall be compared with 31319 (FWHM) and $\sigma(T)/T=7.35\times10^{-6}$
obtained without correction.
