83^212 Bi can alpha-decay to the ground state of 81^208 Tl, or to any of the four excited stat

step 1 I've drawn over here the different excited states for titanium 208, the four different excited s

83^212 Bi can alpha-decay to the ground state of 81^208...

${ }_{83}^{212} \mathrm{Bi}$ can alpha-decay to the ground state of ${ }_{81}^{208} \mathrm{Tl}$, or to any of the four excited states of ${ }_{81}^{208} \mathrm{Tl}$ shown in Fig. 29.7. The maximum kinetic energy of the alpha particles emitted by ${ }_{83}^{212} \mathrm{Bi}$ is $6.090 \mathrm{MeV}$. What other alpha-particle kinetic energies are possible? [Hint: Estimate the atomic mass of ${ }_{81}^{208} \mathrm{Tl}$.]

${ }_{83}^{212} \mathrm{Bi}$ can alpha-decay to the ground state of ${ }_{81}^{208} \mathrm{Tl}$, or to any of the four excited states of ${ }_{81}^{208} \mathrm{Tl}$ shown in Fig. 29.7. The maximum kinetic energy of the alpha particles emitted by ${ }_{83}^{212} \mathrm{Bi}$ is $6.090 \mathrm{MeV}$. What other alpha-particle kinetic energies are possible? [Hint:
Estimate the atomic mass of ${ }_{81}^{208} \mathrm{Tl}$.]

Key Concepts

Energy Partitioning in Two-Body Decays

In any two-body decay such as alpha decay, conservation of momentum ensures that the released energy is shared between the two fragments. Because the alpha particle is much lighter relative to the daughter nucleus, it generally carries away the majority of the kinetic energy. The exact kinetic energy of the alpha particle, however, depends on the total Q-value for that particular decay channel, which varies if the daughter nucleus is left in an excited state.

Nuclear Excited States

Nuclear excited states occur when a nucleus exists in an energy level higher than its ground state. In decay processes, if the daughter nucleus is produced in an excited state, some of the Q-value energy goes into exciting the nucleus, thereby lowering the kinetic energy available to the emitted particle. This concept is crucial for understanding different kinetic energies observed for alpha particles emitted in decays to various energy levels.

Alpha Decay

Alpha decay is a radioactive process in which an unstable nucleus emits an alpha particle (comprised of two protons and two neutrons) resulting in the formation of a daughter nucleus with a lower atomic number. The process is driven by differences in nuclear binding energy and is subject to the laws of conservation of energy and momentum.

Decay Q-value

The Q-value of a nuclear decay is the total amount of energy released when the parent nucleus transforms into daughter products. This energy is determined by the mass difference between the initial and final states and is partitioned as kinetic energy among the decay products. In alpha decay, the Q-value is reduced if the daughter nucleus is produced in an excited state, since part of the energy is ‘used’ to populate that state.

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