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In some systems we use half-life (like in radioactivity) which gives us time until a quantity changes by 50% — while in other instances (like in RC circuits) we use time constants. In both cases the rate of change of a variable over time is proportional to the instantaneous value of variable. What is a simple intuitive way to know the difference between the kind of systems where half-life is useful, versus systems where time constants are more meaningful? (Does it have anything to do with the shape of the curve representing the change in value over time, for example?)

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    $\begingroup$ I don't see why you need a general rule. $\endgroup$
    – DKNguyen
    Feb 16, 2022 at 14:19
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    $\begingroup$ @DKNguyen ... to understand why radioactivity (and innumerable other processes) use half-life rather than time constants -- and why rc-circuits use time constants rather than half-lives. What exactly is the determining factor here? $\endgroup$
    – mumtaz
    Feb 16, 2022 at 14:22
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    $\begingroup$ Time constant is more natural to calculate in electric circuits. It's just simple equations like RC or L/R and you get to use e which can simplify some math. I'm not sure how half life is determined but if its measured then 1/2 might just be more convenient to work with. $\endgroup$
    – DKNguyen
    Feb 16, 2022 at 14:23
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    $\begingroup$ unless you're calculating time to decay to a binary fraction, the time constant is pretty universally better for calculations. The half-life is simpler to explain to laypeople. That's the rule I'd use $\endgroup$
    – Tristan
    Feb 17, 2022 at 10:08
  • $\begingroup$ Half-life is a time constant. $\endgroup$
    – Myridium
    Feb 21, 2022 at 1:43

5 Answers 5

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"What is a simple intuitive way to know the difference between the kind of systems where half-life is useful , versus systems where time constants are more meaningful."

For systems obeying an exponential decay relationship, either half life or time constant can be used. I think it's largely a matter of tradition that half life ($t_{1/2}$) is used for radioactivity and time constant ($\tau$) for C–R and L–R circuits. The relationship between them is $$t_{1/2}=(\ln 2)\tau.$$ Here are some ideas on how the traditions might have arisen...

• For C-R or L-R circuits, it was marginally easier before the days of electronic calculators to calculate $\tau =\frac LR$ or $\tau =CR$ than to calculate $t_{1/2}=(\ln 2)CR$.

• and arguably there's less motivation for knowing how long a voltage across a capacitor will take to halve than how long a radioactive activity will take to halve. For circuit behaviour a general idea of a characteristic time is usually what matters, and $\tau$ is as good as $t_{1/2}$.

• The intelligent layperson is more interested in radioactivity than in capacitor discharge and it's easier to explain the idea of half life than that of time constant, or its reciprocal, decay constant. [There was huge popular interest – see for example chapters in best sellers by Jeans and Eddington – in radioactivity in the first few decades after its discovery.]

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I think the main reason why time constants are used in the first place is that this causes the least probability of nuisance constants in calculations. It is the same reason why $\exp(x)$ is more popular than $2^x=\exp(x\ln 2)$: if you derive the latter (which is done very often), you get the natural logarithm of 2 creeping into your equations.

On the other hand, for transient processes the half life is rather simply related to the period where the change is equidistant between initial and final (at least asymptotically) state. As an illustration, take an RC circuit which is loaded with a step function (turn on voltage): after the half life, the actual voltage is halfway between 0 and +V, which gives many people the feeling that they know where anything interesting starts to happen in terms of the "amount" of something: because many engineering tasks are also economical in nature (and economics is often a binary decision, i.e. "to buy or not to buy"), the half life seems to be kind-of a break-even point between effort and gain. Which is, of course, not a good case for the RC circuit, because people usually use the time constant for it. BTW, it is almost the same situation as in the frequency range (filters) where halving the "amount" is of foremost interest (given by 3dB or 6dB limits, depending on whether amplitude or power is considered). But these arguments are, of course, pretty subjective.

Hence, I shall reiterate Surprised Seagull's argument, to use generally what others in the field consider appropriate.

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  • $\begingroup$ When you say "if you derive the latter", you mean differentiate $2^x$? $\endgroup$ Feb 17, 2022 at 20:53
  • $\begingroup$ yes, sorry if that was ambiguous, (and deriving exponentials in general is done very often) $\endgroup$
    – oliver
    Feb 18, 2022 at 7:43
  • $\begingroup$ I was actually more interested in discouraging abuse of terminology. $\endgroup$ Feb 18, 2022 at 12:54
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Wrong but slightly useful heuristic is to use time constant for events that are repetitive and half life for events that are one-off

More useful is to use whatever is used by others in that particular field. Half life for radioactivity, time constant for electronic filters, time till failure for reliability calculation, annual percents for money, birth rate in demographics, R value for deseases, bushels per acre in agriculture.

Using values that are not common in that field is unwelcomed in general. Like negative half life for population growth or R value for money growth. Probably they will get what you mean, but they wont like it for sure.

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It is just a matter of taste whether you prefer to write an exponential decay with the time constant $\tau$ and powers of $e$ $$N(t)=N_0\ e^{-t/\tau} \tag{1}$$ or with the half-life $t_{1/2}$ and powers of $2$ $$N(t)=N_0\ 2^{-t/t_{1/2}}. \tag{2}$$

Both ways are equivalent and you can switch between them by using $$t_{1/2}=\tau \ln(2).$$

(1) appears more natural from a mathematical point of view, because it directly appears as the solution of the differential equation $$\frac{dN}{dt}=-\frac{N}{\tau}.$$

And (2) is easier to grasp even for a mathematical layman, who doesn't know the meaning of $e$.

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Even for radioactive systems the usage can be mixed. Isotopes are reported as half-lives, but individual nucleons or fundamental particles are often reported as lifetimes.

See for example https://pdg.lbl.gov/2021/web/viewer.html?file=../tables/rpp2021-sum-leptons.pdf where the muon and the tau leptons have their decay quoted as mean lifetimes, despite the decay being similar in quality to radioactive nuclei!

You'll note that many particles have a width measurement instead; this is used for very short-lived systems. The same notation is occasionally used for systems with extremely short lifetimes, for example 8Be, which has units of eV to describe its decay: https://www.nndc.bnl.gov/nudat3/reCenter.jsp?z=4&n=4

Given this I think it's often just historical reasons why one is used vs. the other. But in the case of extremely small lifetimes it's often neither.

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