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Lifetimes of the $T=1$ isobaric analog ${I}^{\ensuremath{\pi}}={2}^{+}$ states in $^{50}\mathrm{Mn}$ and $^{50}\mathrm{Cr}$ were measured simultaneously by employing the recoil distance Doppler-shift (RDDS) technique. The states were populated in a fusion-evaporation reaction with a $^{12}\mathrm{C}$ beam on a $^{40}\mathrm{Ca}$ target. An analysis of the data and the calculations from the present work, together with the available $B(E2:{2}^{+}\ensuremath{\rightarrow}{0}^{+})$ data of the isobaric analog states in $^{50}\mathrm{Fe}$ and $^{50}\mathrm{Cr}$, were used to study isospin symmetry in the three $A=50$ isobaric nuclei. Shell-model calculations reproduce the magnitudes as well as the increasing trend of the $B(E2)$ data with increasing $Z$. To draw a firm conclusion on the level of isospin mixing in the triplet, a new precision measurement of the ${2}^{+}\ensuremath{\rightarrow}{0}^{+}$ transition rate in $^{50}\mathrm{Fe}$ will be required.

A new method of extracting the γ -ray intensities necessary to perform lifetime measurements using the differential decay curve method (DDCM) is presented in this work, the unresolved Doppler-shifted components method (UDCM). The UDCM allows for a DDCM analysis to be performed using a γ -ray transition for which the fully Doppler-shifted and degraded components are unresolvable in energy and so are detected as a single peak. This technique was used to measure the known lifetime of the yrast 2 1 + state in 50 Mn with a depopulating transition that does not have resolvable fully Doppler-shifted and degraded components. The lifetime measured through applying the UDCM was consistent with the standard DDCM measurement of the 2 1 + state. Use of the UDCM allows for DDCM lifetime measurements to be made using transitions of smaller γ -ray energies, smaller recoil velocities and, in some cases, with a smaller uncertainty. In contrast to a standard DDCM analysis, a UDCM analysis is also independent of the widths of the fully Doppler-shifted and degraded components and as a result they do not need to be determined.