Introduction to Quantum Mechanics · Problem 9.4

The WKB Approximation — David J. Griffiths

Physics — Quantum Mechanics · gamow-alpha-decay-lifetimes

Problem

Calculate the lifetimes of U

^{238}

and Po

^{212}

, using Equations 9.26 and 9.29. *Hint*: The density of nuclear matter is relatively constant (i.e. the same for all nuclei), so

(r_1)^3

is proportional to

A

(the number of neutrons plus protons). Empirically,

r_1 \cong (1.07\text{ fm})\, A^{1/3}.

The energy of the emitted alpha particle can be deduced by using Einstein's formula (

E = mc^2

E = m_p c^2 - m_d c^2 - m_\alpha c^2,

where

m_p

is the mass of the parent nucleus,

m_d

is the mass of the daughter nucleus, and

m_\alpha

is the mass of the alpha particle (which is to say, the He

^4

nucleus). To figure out what the daughter nucleus is, note that the alpha particle carries off two protons and two neutrons, so

Z

decreases by 2 and

A

by 4. Look up the relevant nuclear masses. To estimate

v

, use

E = (1/2)m_\alpha v^2

; this ignores the (negative) potential energy inside the nucleus, and surely underestimates

v

, but it's about the best we can do at this stage. Incidentally, the experimental lifetimes are

6 \times 10^9

yrs and

0.5\,\mu

s, respectively.