A supplied program tracks the number of undecayed nuclei in a radioactive sample by…

PHYSICS · Subjective · Fermi University Physics · Computational Physics · Nuclear Physics · Research software practice, verification and validation

Computational Physics, Ch A workspace that can be rerun next year

Problem

A supplied program tracks the number of undecayed nuclei

N(t)

in a radioactive sample by integrating

\mathrm{d}N/\mathrm{d}t = -\lambda N

. Line 4 of the listing should read lam = np.log(2)/thalf, converting the stated half-life into the decay constant

\lambda = \ln 2/t_{1/2}

. Take the half-life

t_{1/2}=12.0\,\mathrm{s}

and

N_0 = 1.0\times10^6

nuclei at

t=0

. Decay is taken as a pure exponential process with no daughter products, so the exact solution is

N(t) = N_0\,e^{-\lambda t}

, and exactly one half-life must halve the population. (a) State the exact fraction

N(t_{1/2})/N_0

remaining after one half-life, and use it to write a known-answer test as a single assert statement, testing the fraction against a tolerance of

0.005

(that is,

0.5\%

). (b) Run at a step of

\Delta t = 0.001\,\mathrm{s}

, the correct program returns

N(12.0\,\mathrm{s})/N_0 = 0.4996

. Show that the test in part (a) passes, quoting the margin. (c) Line 4 is changed to lam = 1.0/thalf (the

\ln 2

factor is dropped). Assuming the numerical integration remains as accurate relative to its own analytic solution as in part (b), find analytically the fraction the corrupted program returns at

t=12.0\,\mathrm{s}

, and determine whether the known-answer test of part (a) passes or fails, quoting the printed difference.

Related theory

Nuclear Physics

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