(15, 11$)$ The Mössbauer effect
An excited nucleus of ${ }^{57}$ Fe formed by the radioactive decay of ${ }^{57}$ Co emits a gamma ray of $1.44 \times 10^{4}$ electronvolts. In the process, there is conservation of energy and $m_{0} c^{2}=m_{0} c^{2}+h v$, where $m_{0}$ is the initial mass of the nucleus and $m_{\alpha}$ is its mass after the emission of the gamma ray. There is also conservation of momentum, $h v / c=m_{\sigma} u$, where $u$ is the recoil velocity of the iron nucleus. Let $m_{\text {oil }}$ be the rest mass of the nucleus after the reaction. Then the energy released by the reaction is $E=\left(m_{0}-m_{0 a}\right) c^{2}$.
(a) Rewrite the first equation after subtracting $h v$ on both sides, and square. Then square the second equation and substitute. You should find that
$$
h v=\frac{\mathscr{E}\left(m_{0}+m_{u 11}\right)}{2 m_{0}}=\left(1-\frac{\delta}{2 m_{0} c^{2}}\right) \mathscr{E}
$$
So $h v<\mathscr{:}$ part of $\varepsilon$ goes to the photon, and the other part supplies kinetic energy to the recoiling nucleus.
(b) Set $m_{0}=57 \times 1.7 \times 10^{-23}$, and show that $\mathcal{E} /\left(2 m_{0} c^{2}\right)=1.3 \times 10^{-7}$, Thus the fraction of the available energy $\varepsilon$ that appears as recoil is small.
(c) Mossbauer discovered in 1958 that, with solid iron, a significant fraction of the atoms recoil as if they were locked rigidly to the rest of the solid. This is the Mössbauer effect. If the sample has a mass of 1 gram, by what fraction is the gamma ray energy shifted in the recoil process?
(d) A sample of normal ${ }^{57}$ Fe absorbs gamma rays of $14.4$ kiloelectronvolts by the inverse recoilless process much more strongly than it absorbs gamma rays of any nearby energy. The excited nuclei thus formed reemit 14.4-kiloelectronvolt radiation in random directions some time later. This is resonant scattering.
If a sample of activated ${ }^{57} \mathrm{Fe}$ moves in the direction of a sample of normal ${ }^{57} \mathrm{Fe}$, what must be the value of the velocity $v$ that will shift the frequency of the gamma rays, as seen by the normal nuclei, by 3 parts in $10^{13}$ ? This is one line width.
(e) A Doppler shift in the gamma ray results in a much lower absorption by a nucleus if the shift is of the order of one line width or more. What happens to the counting rate of a gamma-ray detector placed behind the sample of normal ${ }^{57}$ Fe when the source of activated ${ }^{57}$ Fe moves (i) toward the normal $^{57} \mathrm{Fe}$, (ii) away from it?
(f) If a $14.4$-kiloelectronvolt gamma ray travels $22.5$ meters vertically upward, by what fraction will its energy decrease?
(g) A normal ${ }^{57} \mathrm{Fe}$ absorber located at this height must move in what direction and at what speed in order for resonant scattering to occur?