Why Does Uranium Alpha Decay? Consider the alpha decay shown below - - PowerPoint PPT Presentation

Why Does Uranium Alpha Decay?

Consider the alpha decay shown below where a uranium nucleus spontaneously breaks apart into a 4He or alpha particle and 234

90 Th. 238 92 U → 4He + 234 90 Th

E(4He) = 4.2 MeV To study this reaction we first map out the 4He −234

90 Th potential

• energy. We reverse the decay above and use a beam of 4He nuclei

striking a 234

90 Th target. The 4He nuclei come from the radioactive

decay of another nucleus 210

84 Po.

• 1. What is the distance of closest approach of the 4He to the

234 90 Th target if the Coulomb force is the only one that matters?

• 2. Is the Coulomb force the only one that matters?
• 3. What is the lifetime of the 238

92 U?

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What Do We Know?

The 234

90Th − α Potential

r fm Vr MeV

attractive nuclear part V = Z1Z2e2

r

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Mapping the Potential Energy

Rutherford Scattering

What is the distance of closest approach of the 4He to the

234 90Th target if only the Coulomb force is active? Is the

Coulomb force the only one active? The energy of the 4He emitted by the 210

84Po is E(4He) = 5.407 MeV.

Alpha source

84 210 2 4

Po He + Pb

82 206

Microscope Alpha beam Collimator ZnS Scattered helium Thorium target

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Mapping the Potential Energy

The Total Cross Section

dA

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Areal or Surface Density of Nuclear Targets

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Mapping the Potential Energy

The Differential Cross Section

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Solid Angle

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Solid Angle

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Solid Angle

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Solid Angle

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Solid Angle

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Solid Angle

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Rutherford Scattering Results

90 180 Scattering Angle in degrees 0.5 1 Measured

• Predicted

What does this say about the 4

2He −234 90 Th potential energy?

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Actual Rutherford Scattering Results

(deg) θ 20 40 60 80 100 120 140 160 180 /2) θ (

4

Counts/sin 5 10 15 20 25 30 35 40 45 50

H.Geiger and E.Marsden, Phil. Mag., 25, p. 604 (1913) alphas on gold

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The 4He − 234

90Th Potential

ΑTh Potential Blue known Red a guess 10 20 30 40 50 60 70 40 20 20 40 rfm VMeV

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Measuring the Size of the Nucleus

θcm(deg)

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Measuring the Size of the Nucleus

θcm(deg)

Ecm = 28.3 MeV Ecm = 23.1 MeV PRL 109, 262701 (2012)

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Rutherford Trajectories

x y

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The 4He − 234

90Th Potential

4.2 MeV 20 40 60 80 10 20 30 rfm VMeV

Gamow, Condon and Gurney

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The α-Decay Puzzle

• 1. α-decay of uranium 238U → 4He + 234

90 Th ejects a 4.2-MeV 4He.

• 2. Used a 5.4 − MeV 4He beam

to probe the 4He+234

90 Th force.

• 3. It was all Coulomb down to

48 fm.

• 4. The

decay

4He

with en- ergy 4.2 MeV is, apparently, ejected at a distance of 62 fm.

• 5. How can that happen?

ΑTh Potential Blue known Red a guess 10 20 30 40 50 60 70 40 20 20 40 rfm VMeV α Decay – p. 19/4

The α-Decay Puzzle

• 1. α-decay of uranium 238U → 4He + 234

90 Th ejects a 4.2-MeV 4He.

• 2. Used a 5.4 − MeV 4He beam

to probe the 4He+234

90 Th force.

• 3. It was all Coulomb down to

48 fm.

• 4. The

decay

4He

with en- ergy 4.2 MeV is, apparently, ejected at a distance of 62 fm.

• 5. How can that happen?

ΑTh Potential Blue known Red a guess 10 20 30 40 50 60 70 40 20 20 40 rfm VMeV

QUANTUM TUNNELING!

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The Plan for Solving the Alpha Decay Puzzle

• 1. Develop the notion of particle flux or flow.
• 2. Solve the Schroedinger equation for the rectangular barrier

potential.

• 3. Determine the flux penetrating the barrier.
• 4. Develop the transfer-matrix method using the rectangular

barrier results as the starting point.

• 5. Build a model of what happens in a uranium nucleus and

predict the lifetime for α-decay.

• 6. Compare with data!

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The Rectangular Barrier

2a x V Vx Region 1 Region 2 Region 3 Incident Wave Reflected Wave Transmitted Wave Transmitted and Reflected Waves

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Particle Flux in a Beam

Particles of

Observation Point v t

0 ∆

velocity v 0 Cross−sectional area of A average

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The Rectangular Barrier

2a x V Vx Region 1 Region 2 Region 3 Incident Wave Reflected Wave Transmitted Wave Transmitted and Reflected Waves

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The Postulates

1. The state of a particle is represented by a wave function |ψ(t) in a Hilbert space. 2. The independent variables x and p are represented by Hermitian operators ˆ X and ˆ P with the following matrix elements in the eigenbasis of ˆ X x| ˆ X |x′ = xδ(x − x′) x| ˆ P |x′ = xδ′(x − x′) The operators corresponding to dependent variables ω(x, p) are given Hermitian

• perators Ω( ˆ

X , ˆ P ) = ω(x → ˆ X , p → ˆ P ). 3. If the particle is in a state |ψ measurement of the variable Ω will yield one of the eigenvalues ω with probability P(ω) = |ω|ψ|2. The state of the system will change from |ψ to |ω. 4. The state vector |ψ(t) obeys the Schroedinger equation i d dtψ(t) = ˆ H |ψ(t) where ˆ H ( ˆ X , ˆ P ) = H(x → ˆ X , p → ˆ P ) is the quantum Hamiltonian operator and H is the corresponding classical problem.

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Quantum Tunneling

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Quantum Tunneling

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Quantum Tunneling

Red Curve - Quantum Result = 1 MeV a = 1 fm Green Curve - Classical Result 5 1 15 2 25 3 1-7 1-4 1-1 Energy (MeV) T

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Hint for Shankar 5.4.2.a

The fundamental property of the Dirac delta function is the following. ∞

−∞

f(x)δ(x − a)dx = f(a) = a+ǫ

a−ǫ

f(x)δ(x − a)dx ǫ > 0 The Dirac delta function can be ‘represented’ by test functions that have the property defined above in the appropriate limit. δ(x) = lim

σ→0

1 √ 2πσ e−x2/2σ2

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The Transfer-Matrix Solution

4.2 MeV 20 40 60 80 10 20 30 rfm VMeV

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The Transfer-Matrix Solution

4.2 MeV 20 40 60 80 10 20 30 rfm VMeV

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The Transfer-Matrix Solution

4.2 MeV 20 40 60 80 10 20 30 rfm VMeV

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Results

5 6 7 8 108 0.001 100 107 1012 1017 EΑMeV Lifetime s PointsData, LineTheory

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Coordinates

ΑTh Potential Blue known Red a guess 10 20 30 40 50 60 70 40 20 20 40 rfm VMeV

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Choosing the Sign

x y Initial Target Position Negative Positive

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Center-of-Mass Rutherford Trajectories

x y Scattering Angle vi1

’

vi2

’

CM

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The Inverse Cosine

1 1 x 1 2 3 Inverse Cosine

• f

x

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The Differential Cross Section

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Predicted Differential Cross Section for 4He − Au

90 180 Scattering Angle in degrees 10 2 10 4 10 6 Cross Section in fm2

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Measured Differential Cross Section for 4He − Au

90 180 Scattering Angle in degrees 0.5 1 Measured

• Predicted

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The Evidence

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The Evidence

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