Curium-244 decays entirely by emission to the ground state of plutonium-240

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Approximation methods · Approximation methods

The Nucleus Resolved, Ch Decay rates from the golden rule: kinetics closed and alpha decay finished

Problem

Curium-244 decays entirely by

\alpha

emission to the ground state of plutonium-240, with

Q_\alpha = 5.902\,\mathrm{MeV}

. Model the decay as

\ell = 0

tunnelling through a point-charge Coulomb barrier (

Z_d = 94

A_d = 240

), with the nuclear radius given by the touching-sphere prescription

R = 1.2(A_d^{1/3} + 4^{1/3})\,\mathrm{fm}

. Take the reduced mass of the

\alpha

-daughter pair as

\mu_{\rm red}c^2 = \dfrac{4A_d}{A_d+4}\times 931.494\,\mathrm{MeV}

(mass numbers standing in for the true atomic masses, as in the barrier estimate carried over from the first course) and the

\alpha

particle's rest energy as

m_\alpha c^2 = 3727.4\,\mathrm{MeV}

. Use

\hbar c = 197.327\,\mathrm{MeV\,fm}

and

e^2/4\pi\epsilon_0 = 1.43996\,\mathrm{MeV\,fm}

. The measured half-life is

T_{1/2}^{\rm meas} = 18.1\,\mathrm{y}

(

1\,\mathrm{y} = 3.156\times10^7\,\mathrm{s}

). ((a)) With

\kappa(r) = \sqrt{2\mu_{\rm red}(V(r)-E)}/\hbar

and

V(r) = Z_dZ_\alpha e^2/4\pi\epsilon_0 r = bQ_\alpha/r

, where

b = Z_dZ_\alpha e^2/(4\pi\epsilon_0 Q_\alpha)

is the outer turning point, show that

\begin{aligned}2G &= 2\int_R^b \kappa\,\mathrm{d}r\\ &= \frac{2\sqrt{2\mu_{\rm red}c^2Q_\alpha}}{\hbar c}\,b\Big[\arccos\sqrt{R/b} - \sqrt{(R/b)(1-R/b)}\Big].\end{aligned}

Evaluate

b

2G

and the penetrability

P = e^{-2G}

. ((b)) The

\alpha

particle carries kinetic energy

T_\alpha = Q_\alpha A_d/(A_d+4)

and (non-relativistic) speed

v

from

T_\alpha = \tfrac12 m_\alpha v^2

. Using the assault frequency

\nu_{\rm knock} = v/2R

, write the golden-rule decay constant

\lambda = \nu_{\rm knock}P

, find the calculated half-life, and compare it with

T_{1/2}^{\rm meas}

. ((c)) In

\lambda = (2\pi/\hbar)|M_{fi}|^2\rho(E_f)

, state which of

\nu_{\rm knock}

and

P

is standing in for the squared matrix element and which for the density of final states, and say why the identification is only approximate. ((d)) At

r=b

the barrier is locally linear over the Airy length

\ell = (\hbar^2/2\mu_{\rm red}F)^{1/3}

with

F = |\mathrm{d}V/\mathrm{d}r|_{r=b} = Q_\alpha/b

. Evaluate

\ell

, compare it with the width

b-R

of the forbidden region, and use the size of this ratio to say, order of magnitude, how tightly the WKB approximation itself lets the calculated

T_{1/2}

be trusted.