-decay theory
description
Transcript of -decay theory
-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