 Solution

The partial decay width ${\Gamma }_x$ is the energy width associated with decay of the compound nucleus back into the entrance channel $x+X$; by time reversal it also controls formation from that channel. The partial decay width ${\Gamma }_y$ describes the selected exit channel $y+Y$. The total decay width sums all energetically accessible channels, including these: $\boxed{\Gamma =\sum _i{\Gamma }_i=\hslash /\tau }$. The branching fraction for channel $i$ is ${\Gamma }_i/\Gamma$. Widths have energy units, whereas ${\Gamma }_i/\hslash$ is a decay probability per unit time. The isolated-resonance formula assumes approximately constant widths over the resonance and neglects a nonresonant interfering background.