Decay constant

Also called Lambda, Disintegration constant

The decay constant, written lambda, is the probability per unit time that a given unstable nucleus decays. It has units of inverse seconds, and a larger value means a faster-decaying material.

Two sheet equations carry it, both in the Modern Physics group of the AP Physics 2 sheet:

N = N_0 e^{-\lambda t} \qquad \lambda = \frac{\ln 2}{t_{1/2}}

The CED introduces the first at 15.7.B.2, saying a material's decay constant may be used to predict the number of nuclei remaining in a sample after a period of time, or the age of a material if the initial amount is known. The second is 15.7.B.1.iii. The logarithmic rearrangement $\ln(N/N_0) = -\lambda t$ is labelled a derived equation, so it is not printed.

Units first, because they settle what the symbol means. The exponent $\lambda t$ has to be dimensionless, so $\lambda$ is an inverse time, normally $\text{s}^{-1}$ . It is a probability per unit time for one nucleus, not a number of decays per second for the sample. AP Physics 2 never introduces activity or the becquerel, and neither word appears in the CED.

The exponential only counts elapsed time. The CED's own sample multiple-choice question gives a sample with $N_1$ nuclei at $t_1$ and asks for $N_2$ at a later $t_2$ ; the answer is $N_2 = N_1 e^{-\lambda(t_2 - t_1)}$ . Nothing depends on when the clock was started or on $N_0$ , which is why a decay problem can begin at any moment you like.

Watch the letter. Everywhere else in Unit 15 $\lambda$ is a wavelength, and the sheet's own symbol key for the Modern Physics group reads "wavelength or decay constant" on a single line. Half-life is the same information in units of time.

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