-decay theory
DESCRIPTION
-decay theory. The decay rate. Fermi’s Golden Rule. transition (decay) rate (c). density of final states (b). transition matrix element (a). Turn off any Coulomb interactions. The decay rate (a). Fermi’s Golden Rule. V = weak interaction potential. u = nuclear states. - PowerPoint PPT PresentationTRANSCRIPT
-decay theory
The decay rate
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λ =2πh
V fi2
ρ E f( )
Fermi’s Golden Rule
density of final states(b)
transition (decay) rate(c)
transition matrix element(a)
Turn off any Coulomb interactions
The decay rate (a)
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λ =2πh
V fi2
ρ E f( )
V fi = ψ f*∫ Vβ ψ i dV
ψi = uPψ f = uDϕ β ην
V fi = uD* ϕ β
* ην*∫ Vβ uP dV
Fermi’s Golden Rule
V = weak interaction potential
u = nuclear states
= lepton () states
Integral over nuclear volume
The decay rate (a)
uPuD
“Four-fermion” (contact) interaction
uP
W
uD
(W) Intermediate vector boson
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Δt ≈ hΔE
Δt ≈ hmW c2
c Δt ≈ hcmW c2 = 197MeVfm
90 GeV
δ ≈ 2• 10−3fm Interaction range
The decay rate (a)
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Vβ ≈ gδr r i −
r r f( )
V fi = uD* ϕ β
* ην*∫ gδ
r r i −
r r f( )uP dV
V fi = g uD* uPϕ β
* ην*∫ dV
Assume: Short range interaction contact interaction
g = weak interaction coupling constant
Assume: , are weakly interacting “free particles” in nucleus
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ϕ ≈ei
r k e⋅
r r
V1/2 ; ην ≈ eir k ν ⋅
r r
V1/2Approximate leptons as plane waves
The decay rate (a)Assume: We can expand lepton wave functions and simplify
And similarly for the neutrino wave function.
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ϕ ≈ei
r k e⋅
r r
V1/2 ≈ 1V1/2 1+ i
r k e ⋅
r r +
r k e ⋅
r r ( )
2
2+ ⋅⋅⋅
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ≈ 1
V1/2
Test the approximation ---
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Tβ ≈1MeV → 1ke
≈ 2 ⋅10−13m = D ;
r ≤ R ≈10−14 m ; kr ≤ 0.1
deBroglie λ >> Rtherefore, lepton , constant over nuclear volume. (We will revisit this assumption later!)
The decay rate (a)
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V fi = g uD* uPϕ β
* ην*∫ dV ; ϕ β ≈ 1
V1/2 ; ην ≈ 1V1/2
V fi ≈ gV
uD* uP∫ dV ≡ g
VM fi
Therefore -- the matrix element simplifies to --
Mfi is the nuclear matrix element; overlap of uD and uP
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λ =2πh
V fi2
ρ E f( ) → λ ≈ 2πh
g2 M fiV
2ρ E f( )
Remember the assumptions we have made!!
The decay rate
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λ =2πh
V fi2
ρ E f( )
Fermi’s Golden Rule
density of final states(b)
transition (decay) rate(c)
transition matrix element(a)
The decay rate (b)
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λ =2πh
V fi2
ρ E f( )
ρ E f( ) = dNdE f
Fermi’s Golden Rule
Quantization of particles in a fixed volume (V) discrete momentum/energy states (phase space) --
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dN = 4π
2πh( )3 p2dpV Number of states dN in space-volume V, and momentum-volume 4p2dp
The decay rate (b)
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dN tot = dNe dNν
dNe dNν = 4π2πh( )3
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
p e2dpe pν
2 dpν V 2
Do not observe ; therefore remove -dependence --
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E f = Ee + Eν = Ee + pν c ; TD ≈ 0
pν =E f − Ee
c ; dpν =
dE f
c
dNe dNν = 16π 2
2πh( )6 p e2dpe
E f − Ee
c
⎛ ⎝ ⎜
⎞ ⎠ ⎟2 dE f
cV 2
At fixed Ee
Assume
The decay rate (b)
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dNe dNν = 16π 2
2πh( )6 p e2dpe
E f − Ee
c
⎛ ⎝ ⎜
⎞ ⎠ ⎟2 dE f
cV 2
ρ = dN totdE f
= 16π 2
2πh( )6 c3p e
2dpe E f − Ee( )2V 2
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λ =2πh
V fi2
ρ E f( )
dλ pe( )≡ λ pe( )dpe ; λ = λ pe( ) dpe0
pe−max∫
dλ pe( ) ≈ g2
2h7π 3c3 M fi2
E f − Ee( )2
pe2dpe
Fermi’s Golden Rule
Differential rate
Density of final states
The decay rate
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dλ pe( ) ≈ g2
2h7π 3c3 M fi2
E f − Ee( )2
pe2dpe
Fundamental (uniform) interaction strength
Differential decay rate
Overlap of initial and final nuclear wave functions; largest when uP uD a number
Determines spectral shape!
Ef(Q)
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Q ≈ Te + Tν → Q ≈ Ee − mec2 + TνE f = Ee + Eν = Ee + Tν
Q ≈ E f − mec2 ; E f = Q + mec2
Q-value for decay
Definition of Ef
dλ(pe)
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dλ pe( ) ≈ g2
2h7π 3c3 M fi2
E f − Ee( )2
pe2dpe
Ee = pe2c2 + m e
2c4( )
1/ 2
dλ pe( ) ≈ g2
2h7π 3c3 M fi2
E f − pe2c2 + m e
2c4( )
1/2 ⎛
⎝ ⎜
⎞
⎠ ⎟2
pe2dpe
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Q ≈ Te + Tν → Q ≈ Ee − mec2 + TνE f = Ee + Eν = Ee + Tν
Q ≈ E f − mec2 ; E f = Q + mec2
c.f. Fig. 9.2
dλ(Ee)
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dλ pe( ) ≈ g2
2h7π 3c3 M fi2
E f − Ee( )2
pe pe dpe
pe2c2 = E e
2 − m e2c4 ; pe =
E e2 − m e
2c4( )
1/ 2
c2pe dpe = 2Ee dEe
dλ Ee( ) ≈ g2
2h7π 3c4 M fi2
E f − Ee( )2
E e2 − m e
2c4( )
1/ 2Ee dEe
dλ Ee = 0( ) = dλ Ee = E f( ) = 0
dλ(Te)
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dλ Ee( ) ≈ g2
2h7π 3c4 M fi2
E f − Ee( )2
Ee2 − me
2c4( )
1/2Ee dEe
E f = Q + mec2 ; Ee = Te + mec2 ; dEe = dTe
E f − Ee = Q − Te
Ee2 − me
2c4 = Te + mec2( )
2− me
2c4 = Te2 + 2Temec2
( )
dλ Te( ) ≈ g2
2h7π 3c4 M fi2
Q − Te( )2 Te2 + 2Temec2
( )1/2
Te + mec2( )dTe
dλ Te = 0( ) = dλ Te = Q( ) = 0 c.f. Fig. 9.2
Consider assumptions
Look at data for differential rates - c.f., Fig. 9.3
Calculate corrections for Coulomb effects on or Fermi Function F(Z’,pe) or F(Z’,Te)
Coulomb Effects --
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F Z ', pe( ) ≈ 2π ε 11− e−2επ( )
ε = Z 'e2
hve
ve velocity of electronfar from nucleus
Consider assumptionsLepton wavefunctions --
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ϕ ≈ei
r k e⋅
r r
V1/2 ≈ 1V1/2 1+ i
r k e ⋅
r r +
r k e ⋅
r r ( )
2
2+ ⋅⋅⋅
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ≈ 1
V1/2
In some cases, the lowest order term possible in the expansion is not 1, but one of the higher order terms!
More complicated matrix element; impacts rate!
Additional momentum dependence to the differential rate spectrum; changes the spectrum shape!
Consider assumptionsLepton wavefunctions --
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ϕ ≈ei
r k e⋅
r r
V1/2 ≈ 1V1/2 1+ i
r k e ⋅
r r +
r k e ⋅
r r ( )
2
2+ ⋅⋅⋅
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ≈ 1
V1/2
“Allowed term”
“First forbidden term”
“Second forbidden term”
etc….
Consider assumptionsLepton wavefunctions --
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ϕ ≈ei
r k e⋅
r r
V1/2 ≈ 1V1/2 1+ i
r k e ⋅
r r +
r k e ⋅
r r ( )
2
2+ ⋅⋅⋅
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ≈ 1
V1/2
Change in spectral shape from higher order terms “Shape Factor” S(pe,p)
The decay rate
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dλ pe( ) ≈ g2
2h7π 3c3 M fi2
E f − Ee( )2
pe2dpe
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λ pe( ) ≡dλ pe( )
dpe≈ g2
2h7π 3c3 M fi2
F Z ', pe( ) S pe, pν( ) E f − Ee( )2
pe2
Fermi function
Shape correction
Density of final states
Nuclear matrix element