-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

€

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 --

€

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 --

€

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

€

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