Electromagne,c  (gamma)  decay    Coupling between nucleons and EM field

•  Electromagnetic and weak interactions can be treated as perturbations •  Emission of a γ-ray is caused by the interaction of the nucleus with an external

electromagnetic field •  Besides γ-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.

Electromagnetic Decay Kinematics of photon emission

€

Ei = E f + Eγ + T0recoil term

€

T0 ≈Eγ2

2M0c2

Ei

Ef

For A=100 and Eγ=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  la;ce.  In  such  a  case,  the  recoil  is  taken  by  the  whole  la;ce  and  not  by  a  single  nucleus.  If  

!ωlattice >> T0then,  quantum-­‐mechanically,  the  energy  of  the  emiBed  gamma  radia,on  takes  away  the  total  energy  difference  (Mössbauer  effect  –  1958  −  or  recoilless  nuclear  resonance  fluorescence).  

�r�r0

s

✓0

P

s = |�r � �r0|

V (⇧r) =

1

4⇥⌅0

1X

n=0

1

rn+1

Z(r0)nPn(cos �0)⇤(

⇧r0)d3r0

V (⇤r) =1

4�⇥0

0

@Q

r+

⇤p · ⇤rr3

+12

X

i,j

Qijxixj

r5· · ·

1

A

Qij =Z

(3x0ix

0j � r02�ij)⇥(⇤r0)d3r0

⇥p =Z

⇥r0�(⇥r0)d3r0 =NX

k=1

qk⇥r0k

Multipole expansion of electrostatic potential

dipole

quadrupole

€

V = −1cjµA

µ

nuclear current

two polarization states

!A(!r, t) = 1

Nb!kη!εηe

i(!k!r−ωt ) + b!kη

+ !εηe−i(!k!r+ωt ){ }

!k ,η∑

external EM field

photon creation operator

!A(!r, t) =

!Aλµ (!r, t)

λµ

∑ multipole expansion

The  typical  gamma-­‐rays  in  nuclear  transi,ons  have  energies  less  than  10  MeV,    corresponding  to  wave  numbers  of  the  order  k~1/20  fm-­‐1  or  less.  The  mul,pole  operators  give  contribu,ons  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  contribu,on  of  each  mul,pole  order  to  the  transi,on  probability  from  an  ini,al  nuclear  state  to  a  final  state.  

W (λ;Jζ i → J fξ ) =8π (λ +1)k2λ+1

λ (2λ +1)!![ ]2 !B(λ;Jiζ → J fξ )

transition probability reduced transition probability

This  contains  both  electric  and  magne,c  interac,ons  

λ=1,2,3…  

But  what  about  λ=0?  

Electromagnetic Rates

B(λ;Jiζ → J fξ ) = J fM fζ Oλµ JiMiξµM f

∑2=

12Ji +1

J fζ Oλµ Jiξ2

Oλµ (Eλ) = e(i)i=1

A

∑ riλYλµ (Ωi )

Oλµ (Mλ) = gs (i)!si + gl (i)

2!li

λ +1#

$%

&

'(!∇i

i=1

A

∑ riλYλµ (Ωi )#$ &'

!ji =!si +!li

gyromagnetic factors

Selection Rules

J f − Ji ≤ λ ≤ J f + Ji

W (λ +1)W (λ)

~ (kR)2 large reduction in probability with increasing multipolarity order!

POλµ (Eλ)P-1 = (−1)λOλµ (Eλ), POλµ (Mλ)P-1 = (−1)λ+1Oλµ (Mλ)

π iπ f = (−1)λ π iπ f = (−1)

λ+1

Magnetic transitions are weaker than electric transitions of the same multipolarity

6+  

4+  E2,  (M3,  E4,…)  

5+  

4+  M1,  E2,  (M3,  E4,…)   mixed M1+E2 transition

140Gd  

A decay scheme: E1, E2 and M1 transitions

What is the radiative width corresponding to: a)  1 fs b)  1ps c)  1ns d)  1µs gamma decays?

Internal conversion

An  atomic  electron  is  ejected  instead  of  gamma-­‐ray  (atomic  Auger  effect).  Atomic  electrons,  especially  those  in  the  inner  orbits,  such  as  K-­‐  and  L-­‐orbits,  spend  a  large  frac,on  of  their  ,me  in  the  vicinity  of  the  nucleus,  the  source  of  the  EM  field.  

Te−= Ei −Ef( )−Bn

Electron kinetic energy

nucleonic wave functions electron wave functions

Electron binding energy

Virtual photon exchange. This process can occur between I=0 states!

Electron spectrum: internal conversion + beta-decay background

 Important  for  heavy  nuclei  (large  Z).  The  importance  of  internal  conversion  goes  as  Z3  !!!  

 Important  for  heavy  nuclei  (large  Z).  The  importance  of  internal  conversion  goes  as  Z3  !!!  

Important for heavy nuclei (large Z): it goes as Z3 !!!

Internal pair production

An  electron-­‐positron  pair  is  emiBed  in  the  place  of  gamma-­‐ray.  This  can  happen  if  the  energy  of  the  decay  

Eγ > 2mec2 =1.02MeV

Usually,  this  process  is  several  orders  of  magnitude  retarded  compared  to  allowed  gamma-­‐ray  decays.  The  pair  produc,on  rate  is  largest  where  the  internal  conversion  rate  is  smallest.  That  is  in  the  region  of  low  atomic  number  and  high  transi,on  energies.  

electron scattering pair creation

E0  (IC+e+e_)  

γγ = 2E1+2M1

Another interesting candidate is 137Ba. Why?

Nucl.  Phys.  A474,  412  (1987)  

2-gamma decay

Provides  important  structural  informa,on  about  nuclear  electric  polarizability  and  diamagne,c  suscep,bility.  

Best candidates: 16O, 40Ca, 90Zr

M1 and E1 transitions are similar in strength!

γ-ray laser?

hBps://str.llnl.gov/str/JulAug05/Becker.html    

178Hf  

2008  LLNL  report:  “Our  conclusion  is  that  the  u,liza,on  of  nuclear  isomers  for  energy  storage  is  imprac,cal  from  the  points  of  view  of  nuclear  structure,  nuclear  reac,ons,  and  of  prospects  for  controlled  energy  release.  We  note  that  the  cost  of  producing  the  nuclear  isomer  is  likely  to  be  extraordinarily  high,  and  that  the  technologies  that  would  be  required  to  perform  the  task  are  beyond  anything  done  before  and  are  difficult  to  cost  at  this  ,me.”  

hBp://defensetech.org/2006/06/13/superbomb-­‐or-­‐crapshoot/    hBp://www.theguardian.com/science/2008/aug/14/par,clephysics.research    

Hafnium-­‐178m  has  a  long  half-­‐life  of  31  years  and  a  high  excita,on  energy  of  2.4  MeV.  As  a  result,  1  kilogram  of  pure  178mHf  contains  approximately  1012  J  of  energy.  Some  es,mates  suggest  that,  with  accelerated  decay,1  gram  of  100-­‐percent  isomeric  178mHf  could  release  more  energy  than  the  detona,on  of  200  kilograms  of  TNT.  

hBp://physics.aps.org/synopsis-­‐for/10.1103/PhysRevA.84.053429