Electromagnetic Decay Transition Matrix Element

1. Electromagnetic Decay Transition Matrix Element trensition probability or decay constant half-live lifetime The probability of finding an excited state at energy E is therefore given by a Lorentzian distribution:

2. Electromagnetic Decay Transition Probability time-dependent Hamiltonian …expansion of  in the basis of Ho As initial conditions, let us assume that at t=0 the system is in the state 0. That is, If the perturbation is weak, in the first order, we obtain: Furthermore, if the time variation of V is slow compared with exp(iwkot), we may treat the matrix element of V as a constant. In this approximation: The probability for finding the system in state k at time t if it started from state 0 at time t=0 is:

3. Electromagnetic Decay Transition Probability (cont.) The total probability to decay to a group of states within some interval labeled by f equals: The transition probability per unit time is Since the function sin(x)/x oscillates very quickly except for x~0, only small region around E0 can contribute to this integral. In this small energy region we may regard the matrix element and the state density to be constant. This finally gives: Fermi’s golden rule

4. Electromagnetic Decay Transition Probability (cont.) Emission and absorption of a boson charge carrier by a fermion. The coupling constants are indicated. Transforming a boson into another boson. Strong and weak interactions Density of states… Therefore, for the photons, the density of final states goes as

5. Electromagnetic Decay Coupling to EM field • Electromagnetic and weak interactions can be treated as perturbations • Emission of a g-ray is caused by the interaction of the nucleus with an external electromagnetic field • Besides g-decay, electromagnetic perturbation can also induce nuclear decay through internal conversion whereby one of the atomic electrons is ejected. This is particularly important for the heavy nuclei. • The decay can also proceed by creating an electron-positron pair (internal pair creation) • Since the nuclear wave function has a definite angular momentum, the external EM field has to be decomposed in spherical multipoles. The quantization and multipole expansion of EM field is straightforward by tedious. Nuclear current External EM filed Photon creation operator Two polarization states Spherical tensor rank (lm) The functions Alm are vector functions. They may be expressed in terms of vector spherical harmonics.

6. Electromagnetic Decay Coupling to EM field; EM multipole transition operators Electric multipole Magnetic multipole transition operators The normalizations used in the above definitions are such that they reduce to static multipole moments in the limit of k=0. The typical gamma-rays in nuclear transitions have energies less than 10 MeV, corresponding to wave numbers of the order k~1/20 fm-1 or less. The multipole operators give contributions only within the nuclear volume. That is, in most cases kr<<1 and the above series may be approximated by the first term in the expansion alone (“The long-wavelength limit”). We are now ready to calculate the contribution of each multipole order to the transition probability from an initial nuclear state to a final state. reduced transition probability transition probability

7. Electromagnetic Decay Kinematics of photon emission recoil term For A=100 and Eg=1 MeV, the recoil energy is about 5 eV. But the natural linewidth of the radiation is even smaller. The emission of photons without recoil is possible if one implants the nucleus in a lattice. In such a case, the recoil is taken by the whole lattice and not by a single nucleus. If then, quantum-mechanically, the energy of the emitted gamma radiation takes away the total energy difference (Mössbauer effect).

8. Electromagnetic Decay EM rates gyromagnetic factors Selection Rules large reduction in probability with increasing multipolarity order! Magnetic transitions are weaker than electric transitions of the same multipolarity 6+ 5+ mixed M1+E2 transition E2, (M3, E4,…) M1, E2, (M3, E4,…) 4+ 4+

9. Electromagnetic Decay EM rates Single-particle estimates… A decay scheme.. E2 and M1 transitions

10. Electromagnetic Decay EM rates High-multipole transitions…

11. Electromagnetic Decay Internal conversion An atomic electron is ejected instead of gamma-ray (atomic Auger effect). Atomic electrons, especially those in the intermost orbits, such as K- and L-orbits, spend a large fraction of their time in the vicinity of the nucleus, the source of the EM field. Electron kinetic energy Electron binding energy nucleonic wave functions electron wave functions Electron spectrum: internal conversion + beta-decay background Virtual photon exchange. This process can occur between I=0 states! Important for heavy nuclei (large Z). The importance of internal conversion goes as Z3 !!!

12. Electromagnetic Decay Internal conversion (cont.) Decay of 196Bi A typical electron spectrum…

13. Electromagnetic Decay Internal pair production An electron-positron pair is emitted in the place of gamma-ray. This can happen if the energy of the decay electron scattering The pair creation Usually, this process is several orders of magnitude retarded compared to allowed gamma-ray decays. Coulomb excitation Inverse of gamma emission!

14. Electromagnetic Decay 2-gamma decay Nucl. Phys. A474, 412 (1987) 1-st order (2-nd order contribution) 2-nd order gg = 2E1+2M1 M1 and E1 transitions are similar in strength! Best candidates: 16O, 40Ca, 90Zr Provides important structural information about nuclear electric polarizability and diamagnetic susceptibility.