Nuclear de-excitation

Outline of approach… Source of radiation Propagation of radiation field Detection of radiation ?? nucleus

Propagation of radiation field Electromagnetic radiation (quantized); v = c Power radiated P( ,  ) depends on the nature of the source (e.g., electric dipole, magnetic dipole, electric quadrupole, magnetic quadrupole, …) (May be simultaneously more than one; one will dominate.) Source uniquely describes P( ,  ) and, conversely, P( ,  ) allows the determination of the source of the radiation. P( ,  ) comes from E-M radiation theory (Maxwell)

Properties of radiation field Multipole order of the radiation field: 2 L L = 1 (dipole field); L = 2 (quadrupole); L = 3… “EL”  electric multipole of order L “ML”  magnetic multipole of order L Angular distribution of intensity of radiation field: dipole field quadrupole field Legendre polynomials (from Maxwell’s Eq)

Properties of radiation field The parity of the radiation field:  (EL) = (-1) L  (ML) = (-1) L+1 Parity of E or M multipoles of same order is opposite. The total (integrated) radiated power in the classical radiation field: radiation frequency oscillation amplitude (10.8)  (10.5) if L=1

In quantum mechanics… The total (integrated) radiated power in the classical radiation field: oscillation amplitude nuclear matrix element Transition between initial and final nuclear states Transition operator

In quantum mechanics… These properties remain unchanged - Multipole order of the radiation field: 2 L Angular distribution of intensity of radiation field The parity of the radiation field:  (EL) = (-1) L  (ML) = (-1) L+1 Parity of E or M multipoles of same order is opposite. In QM what is more meaningful is the decay rate - or the probability per unit time of de-excitation = …

In quantum mechanics… These properties remain unchanged - radiated power: watts = joules/sec Energy per quantum (photon) of frequency (  ) Therefore - the number of photons emitted per unit time is -- …from the nuclear source The form of the operators m fi is quite “classical” - it represents the nature of the time-dependent charge distribution in the nucleus to produce this radiation field.

In quantum mechanics… The transition matrix elements- We need to know three quantities… The initial state nuclear wave function The final state nuclear wave function The transition operator E-M matrix element Then, we can compute --

In quantum mechanics… The transition matrix elements- If we know these three quantities, we can compute -- Due to spatial coordinates Due to intrinsic magnetic moment

In quantum mechanics… The transition matrix elements- As an example, the electric and magnetic matrix elements due to spatial coordinates can be written -- Note: each is a sum of Z integrals! The and are the sum of A integrals and involve the Pauli spin martices… ?? source term

In quantum mechanics… The transition matrix elements- To get an estimate, try something “simple” -- Consider only a single proton transition: Assume: ~constant for 0  r  R In general: average over initial states m’ and sum over final states m”.

In quantum mechanics… The transition matrix elements- Results for single proton transitions:

Transition selection rules Angular momentum & parity conservation For E-M interactions: Conserve Consider a transition: Example: