Why do radioactive materials decay in half-lifes?

[andyh3930]

Why do radioactive materials decay in half-lifes exponential decay and not over mean time, i.e. like a Bell Curve

 

[Ygggdrasil]

Let's start with the assumption that the probability that an atom decays is constant over time (say ~ 50% chance of decaying every second). Let's say you start with one million atoms. How many are remaining after 1 s? After 2 s? If you plot the decay curve what does it look like?

 

[jtbell]

Unstable particles, excited atoms, etc. have no "memory". They don't "remember" how long they've been "alive". All they "know" is that they have a certain constant probability of decaying per unit time: the decay constant $\lambda$, which is related to the half-life by $t_{1/2} = (\ln 2) / \lambda$.

 

[russ_watters]

Radioactive decay can be modeled as a bell curve - you just have to be aware of what, exactly, is being modeled:

http://demonstrations.wolfram.com/RadioactiveDecayAsAProbabilityDistribution/

The "mean time" is the half life.

 

[gleem]

The half life is the expected time in which half of the nuclei decay. It is not the actual time for half of the nuclei to decay for every sample. Each sample decays at a random rate with the number of decays in an interval is dictated by the Poisson probability distribution with a mean equal to the expected number of decays in that interval (N) and a standard deviation of √ N. The expected number of decays in an interval is given by the size of the sample and the decay rate λ characteristic of the radioisotope and decay mode.

 

[lychette]

They do not just decay in half lives, they decay in 1/4 lives, 1/3 lives, 7/8 lives...any fraction you want...it is perfectly natural...called natural decay (and growth)

 

[sophiecentaur]

Decay is very often measured in terms of the exponential law. The decay of the Volts on a Capacitor after time t, discharging through a Resistor is usually described in terms of e. so
V_t = V₀ e^(t/RC)
RC is the 'time constant, or the time for the value to reach 1/e of a start value. The exponential notation is used because it is 'convenient' and it is easy to see a waveform decay against a graticule on an oscilloscope screen. Using e for such things makes the Maths very convenient because the differential of e^x is still e^x.
Room acoustics uses 'Reverberation Time', which is the time taken for a loud sound to decay by 60dB. Also very convenient to measure, in a practical situation. (Sound level meter and stop watch)
Likewise, it is very convenient to measure Half life of clicks from a Geiger-Muller tube, directly, with a stop watch and a counter.