-decay theory

-particle penetration through Coulomb Barrier

• Goal: estimate the parent lifetime for -decay • Assume an -particle is formed in the parent

nucleus • Parent nucleus = -particle + daughter nucleus-decay -particle must “tunnel” through

the Coulomb barrier from R (nuclear matter radius) to b (called the “turning point”).

Alpha Decay for Rn-222

-40

-30

-20

-10

0

10

20

30

40

0 9.17 11.6 14 16.4 18.8 21.2 23.6 26 28.4 30.8 33.2 35.6 38 40.4 42.8 45.2 47.6 50 52.4 54.8 57.2 59.6 62 64.4

r (fm)

Energy (MeV)

b

B

R

Q

V

-particle penetration through Coulomb Barrier

• Goal: estimate the parent lifetime for -decay • Consider the -particle moving in the potential

of the daughter nucleus. • For r > R, this is a central force problem with

€

Q = Tα + TD ; Tα =1

2mα vα

2 ; TD =1

2mDvD

2

Q = Tm =1

2mvrel

2 ; m =mα mD

mα + mD

E = Q Motion of a reduced mass -particle relative to center of daughter nucleus

-particle penetration through Coulomb Barrier

• Goal: estimate the parent lifetime for -decay • Assume that the rate of -decay per nucleus () can be obtained as • = (probability for -particle to penetrate

the barrier) (number of hits on the boundary per sec.)

• (probability for -particle to penetrate the barrier) = “transmission coefficient” T

-particle penetration through Coulomb Barrier

• (probability for -particle to penetrate the barrier) = “transmission coefficient” T

• To calculate T for the Coulomb barrier -- consider the -particle transmission through a one-dimensional rectangular barrier.

• From chapter 2, we have the transmission coefficient T for (simple) case --

1-dimensional rectangular barrier

0.0

Vo

E

2a

€

T =2kκ( )

2

k2 + κ 2( )

2sinh2 2κa + 2kκ( )

2

k =2mE

h ; κ =

2m Vo − E( )

h

sinh2 2κa = e2κa − e−2κa[ ]

2

sinh2 2κa ≈ e4κa

k2 + κ 2( )

2e4κa + 2kκ( )

2 ≈ k2 + κ 2( )

2e4κa

T ≈2kκ( )

2

k2 + κ 2( )

2e4κa

=2kκ( )

2e−4κa

k2 + κ 2( )

2

Vo >> E ; a is “large”

€

e2κa − e−2κa[ ] ≈ e2κa

e4a important!!

Coulomb barrier penetration

• Consider penetration through Coulomb barrier as a sequence of rectangular barriers

• Total transmission coefficient is

€

T = T1 ⋅T2 ⋅T3 ⋅T4 ⋅⋅⋅= Π i Ti

LnT = Lni∑ Ti

Coulomb barrier penetration

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LnTi = −2κ i 2a( ) + 2Ln2 ka( ) κ ia( )

ka( )2 + κ ia( )

2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

LnT ≈ −2κ i 2a( )i∑

LnT ≈ −2κ i Δx( )i∑ ; LimΔx→0

LnT ≈ −2barrier

∫ κ dx = −22m V (x) − E[ ]

h∫ dx

Ln term small

Coulomb barrier penetration

€

LnT ≈ −2barrier

∫ κ dx = −22m V (x) − E[ ]

h∫ dx

• We have assumed V(x) >> E

• Where in x is this approximation less good?

€

T = e−G

G = 22m

h2

⎛

⎝ ⎜

⎞

⎠ ⎟1/2

Z1Z2e2

4πεor− E

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

R

b∫

1/2

drZ1 daughter

Z2 = 2 ()

Coulomb barrier penetration

€

E =Z1Z2e

2

b

1

r−

1

b

⎛

⎝ ⎜

⎞

⎠ ⎟

R

b

∫1/ 2

= b cos−1 R

b

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

−R

b−

R2

b2

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

If E << V(R) Low Energy -particle b >> R

€

R

b

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

≈ 0 → cos−1 R

b

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

≈π

2 ;

R2

b2<<

R

b

€

1

r−

1

b

⎛

⎝ ⎜

⎞

⎠ ⎟

R

b

∫1/ 2

≈ bπ

2−

R

b

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2 ⎡

⎣ ⎢

⎤

⎦ ⎥

Coulomb barrier penetration

€

G ≈ 22mc 2Z1Z2e

2b

4πεoh2c 2

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2π

2−

R

b

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2 ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

€

b =Z1Z2e2

4πεoE; E =

1

2mvin ; m =

mα mDmα + mD

E = Q

R

b=

Q

BYou can easily show this ratio is true.

Coulomb barrier penetration

€

G ≈ 22mc2

h2c2Q

⎛

⎝ ⎜

⎞

⎠ ⎟

1/2Z1Z2e2

4πεo

π

2−

Q

B

⎛ ⎝ ⎜

⎞ ⎠ ⎟1/2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

T = e−G

T ≈ exp −22mc2

h2c2Q

⎛

⎝ ⎜

⎞

⎠ ⎟

1/2Z1Z2e2

4πεo

π

2−

Q

B

⎛ ⎝ ⎜

⎞ ⎠ ⎟

1/2 ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

Barrier hit frequency

• For r < R, -particle moves with vin

€

f =vinR

vin =2Tinm

Tin = Q −Vo

λ = f T

Frequency of hits on Coulomb barrier

Velocity of -particle inside daughter

Remember: Vo is negative!

-particle decay constant -probability per unit time for decay

-particle decay lifetime

€

= f T

λ =

2 Q −Vo( )mR

exp −22mc2

h2c2Q

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

1/2Z1Z2e2

4πεo

π

2−

Q

B

⎛

⎝ ⎜

⎞

⎠ ⎟1/2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

t1/2 =Ln2

λ

t1/2 = Ln2Rc

2 Q −Vo( )

mc2

exp +22mc2

h2c2Q

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

1/2Z1Z2e2

4πεo

π

2−

Q

B

⎛

⎝ ⎜

⎞

⎠ ⎟1/2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

-particle decay lifetime

€

t1/2 = Ln2Rc

2 Q −Vo( )

mc2

exp +22mc2

h2c2Q

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

1/2Z1Z2e2

4πεo

π

2−

Q

B

⎛

⎝ ⎜

⎞

⎠ ⎟1/2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

Q is measured for -decay

m is calculated for and daughter; Z1 & Z2 are known

R is assumed to be RoA1/3 Calculate B

Vo is taken to be ~35 MeV (from other studies)

A is for parent nucleus; may not give best radius!

-particle decay lifetime

• Assumptions:-particle preformation in nucleus; did not

estimate the rate for this from Fermi GR -- involves

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V fi = ψ f∫ Vαψ idV

Final nuclear state of -particle + daughter nucleus

Initial nuclear state of parent nucleus

V causes transition from i to f

-particle decay lifetime

• Assumptions:

• Assumed nucleus is a sphere; many large-A nuclei are deformed (ellipsoids)

• Assumed the -particle takes off zero angular momentum, i.e.,

€

l =0