PHY982 12th week
Electromagnetic field and coupling with photons
1
Maxwell’s equations
o magnetic H and electric E fields are related
Charge current and
charge density
o radiative capture reactions: coupling of photons and the nucleus
one radiative photon at a time
1
Maxwell’s equations
o vector potential and scalar potential
o gauge invariance
o transverse gauge (Coulomb gauge)
Maxwell’s equations
o vector potential and scalar potential
o gauge invariance
o transverse gauge (Coulomb gauge)
– one photon approximation (density and scalar field at time independent)
2
Coupling photons and particles
Connecting the classical EM description with quantum particle dynamics
o free field source flux
o particle current to photon production (capture)
Coupling photons and particles
o photon current to particle flux
minimal gauge coupling
o photo-disintegration equation
o initial state is bound state and we need equation coupling the incoming
photon field to the particle in the continuum
3
Photon flux and normalization
o photon outgoing wave
o The Poynting vector provide energy flux
o outgoing photon flux is energy flux per photon energy
o normalized photon wave
Coupled photon-particle equations
o photo-production
o photo-disintegration
o generalization of wavefunction to incorporate photon channel
4
Partial wave decomposition for photons
o expanding photon into vector spherical harmonics
o radial coupled channel equations
o from asymptotics obtain T-matrix or S-matrix and construct observables.
Capture cross sections
o starting from incoming particle channel
o T-matrix equivalent integral expression
o no potential in photon wave!
o All analysis ignored gauge invariance… µ=-1,0,+1
5
Photon gauge invariant plane wave
o incoming photon along z axis:
Photon gauge invariant plane wave
o incoming photon along z axis:
o splitting into magnetic and electric parts
6
EM field normalization and other properties
oThe vector field is normalized like a plane wave
oThey are related by:
Selection rules
7
Final state photons
oFor photon in exit channel – generalize k in z-axis for any direction
o vector T-matrix can be written in this general form too:
Longitudinal part of the field
o does not satisfy the transverse gauge
o differs only previous by coefficients
o long wavelength approximation
(neglect L=J+1)
8
Electric transitions – standard form
o long wavelength approximation
oSiegert’s theorem: uses continuity equation to transform current into
charge density
Integration by parts
Continuity eq+
Schrodinger eq
Electric transitions – standard form
o long wavelength approximation
o Siegert theorem gives a matrix element proportional to multipole operator
o putting it back in the T matrix
9
Extensions and limitation of framework
o T-matrix
o Ψ(R) – separation of two bodies whereas Z(r) – distance from center of mass
non-local equation!
Extensions and limitations
- coupled channel framework can describe A+B -> (AB)* to many channels
- only two bodies = photon+AB
- decay process AB -> A’+B’ needs separate formulation (Hauser-Feshbach)
Capture cross sections and B(EL)
o electric multipole operator appearing in Coulomb inelastic scattering
o transition strength function from scattering to bound
o photons in long wavelength approximation
o capture cross section
10
Photo-absorption cross section
oUsing detailed balance this can be obtained from capture
o work out specific form for E1 – homework 6
o work out specific properties for protons – homework 8
o work out long wave approximation for magnetic field - homework 9
examples
E1 only
11
examples
12