1. From : Introduction to Nuclear and Particle Physics A.Das and T.Ferbel
Gama decay
From : Introduction to Nuclear and Particle Physics
A.Das and T.Ferbel

2. Gama decay
As we have already seen, when a heavy nucleus disintegrates by emitting an a- particle or a ß-particle, the daughter nucleus may be left in an excited state. If the excited nucleus does not break apart or emit another particle, it can de-excite to the ground state by emitting a high energy photon or gamma ray .

4. As we saw in Example 1 in alfa decay lecture, 1
As we saw in Example 1 in alfa decay lecture, 1. the characteristic spacing of nuclear energy levels is about 50 keV in that example, and
2. typical energies of nuclear g-rays can therefore range from a fraction to several MeV. Because this kind of de-excitation is electromagnetic,
3. we expect lifetimes for such processes to be about sec. As in atomic transitions,
4. the photon carries away at least one unit of angular momentum (the photon, being described by the vector electromagnetic field, has spin angular momentum of ℏ), and
5.the process conserves parity

5. The study of the emission and absorption of nuclear g- rays, forms an essential part of the development of nuclear spectroscopy. The subject has a direct parallel in the study of atomic spectroscopy, however, there are important differences.
Consider, for example, a system initially in a state of energy Ei making a transition to a state with energy Ef through the absorption or emission of a photon of frequency n. In such processes, we can define what are known as resonant or recoilless transitions, for which

7. Where “ - ” corresponds to absorption and “ + " to emission
Where “ - ” corresponds to absorption and “ + " to emission. Thus, in principle measuring n determines the level spacing . However, in absorbing or emitting a photon, any system must, in fact, recoil to conserve momentum. If M denotes the mass of the final-state object and v the magnitude of its recoil velocity, it then follows from conservation of momentum that

8. where 𝚫 𝑬 𝑹 denotes the kinetic energy of the recoil
where 𝚫 𝑬 𝑹 denotes the kinetic energy of the recoil. Now, every unstable energy level has a "natural" width 𝜹𝑬=𝚪 and a life time t, which can be related through the uncertainty principle:

9. In other words, the exact value of an energy level is uncertain, and cannot be defined in any given transition to better than ≈𝚪 . Consequently, if the kinetic energy of the recoil is such that 𝚫 𝑬 𝑹 <<𝚪 then Eq. (4.45) is essentially equivalent to Eq. (4.43), and resonant absorption can take place. On the other hand, if 𝚫 𝑬 𝑹 , it is then impossible to excite the system to a higher level through resonant absorption within the bounds provided by the uncertainty relation.

10. To appreciate this more fully , consider an atom with A=50
To appreciate this more fully , consider an atom with A=50 . The typical spacing of atomic levels is of the order of 1 eV , and we will therefore consider absorption of a photon or energy of 𝒉𝝂=𝟏𝒆𝑽. For the atom, we have Mc2≈ 50 x 103 MeV= 5 x1010eV, and, consequently,

11. Consequently,𝚫 𝑬 𝑹 <<𝚪 , and, for atomic transitions, resonant absorption can therefore take place.
In contrast, typical nuclear spacings have hn ≥100 kev =105eV. If we consider again a nucleus with A =50, we still have Mc2 =5 x 1010 eV but now, with the higher photon energy, the nuclear recoil energy is given by
If we assume a typical lifetime of about sec for a nuclear level, then

12. It is clear, therefore, that for such nuclear transitions 𝚫 𝑬 𝑹 ≫𝚪, and resonant absorption cannot occur.
In fact, for resonant absorption to take place in nuclei, the recoil energy must somehow be reduced, and this is done beautifully through what is known as the Mössbauer effect (named after its discoverer Rudolf Mössbauer). The basic idea rests on the fact that, the heavier the recoiling system, the smaller is the recoil energy (see Eq. (4.49)). An enormous increase in the mass of the recoil can be achieved by freezing the nucleus into a rigid crystal lattice, which, of course, has a much larger mass than a single nucleus

13. . As a result, the mass of the recoiling system becomes the mass of the macroscopic crystal, thereby increasing the effective mass of the recoil by many orders of magnitude, and consequently making the recoil energy 𝚫 𝑬 𝑹 negligible relative to 𝚪 . Because of this feature, the Mössbauer technique can provide exceedingly precise estimates of widths of levels. For example, level widths in iron have been measured to an accuracy of about 10-7 eV, which leads to an accuracy of about 1 part in 1012 in level spacing. The technique is therefore extremely useful in determining hyperfine splittings of nuclear energy levels.