Recoil and Line Broadening

The two main obstacles in the path of achieving nuclear resonant emission and absorption are the recoil energy shift and the thermal Doppler shift. Figure 2.1 shows an isolated atom in the gas phase undergoing a nuclear transition from an excited state, $E_{e}$, to the ground state, $E_{g}$.

Figure 2.1: Recoil in a free nucleus during gamma ray emission.

The recoil kinetic energy of the free nucleus, $E_{R}$, is proportional to the mass of the nucleus, $M$, and the energy of the emitted gamma ray, $E_{\gamma}$, and is given by

$$E_{R}=\frac{E_{\gamma}^{2}}{2Mc^{2}}$$ (2.1)

The gamma ray energy will also be broadened into a distribution by the Doppler-effect energy, $E_{D}=M\boldsymbol{v\cdot{}V_{x}}$, which is proportional to the initial velocity, $\boldsymbol{V_{x}}$, from the random thermal motion of the atom, and $\boldsymbol{v}$ from the recoil of the nucleus. This can be expressed as

$$\overline{E}_{D}=E_{\gamma}\sqrt{ \frac{2\overline{E}_{k}}{Mc^{2}} }$$ (2.2)

where $\overline{E}_{k}$ is the mean kinetic energy per translational degree of freedom of a free atom.[5]

Heisenberg Natural Linewidth also broadens the lineshape. The uncertainty in the mean lifetime of the excited state, $\Delta{}t$, is related to the uncertainty in the energy of the excited state, $\Delta{}E$, by the Heisenberg uncertainty principle

$$\Delta{}E\Delta{}t \geq \hbar$$ (2.3)

Typical values of the linewidth broadening due to this are of the order of $10^{6}$ times less than that due to $E_{R}$ and $\overline{E}_{D}$ for isolated atoms and can be neglected in this case.

The same equations apply for absorption. This leads to a distribution of emitted and absorbed gamma ray energies as shown in Figure 2.2. The resonance overlap is extremely small and so practically useless as the basis of a technique.

Figure 2.2: Gamma ray energy distributions for emission and absorption in free atoms. The overlap is shown shaded and not to scale as it is extremely small.