# Can a Decay Constant be bigger than 1?

1. May 22, 2013

### Namige

Decay Constant, as it says on my revision sheet is defined as 'The probability of a nucleus decaying per unit time'.

Does it mean that it can't be greater than 1? Otherwise, doesn't that imply that a nucleus is most certainly going to decay(greater than 100% chance within a certain period.

Based on the formula λ = ln2/T where T is the half life, it seems if T < ln2 then λ > 1?

Last edited: May 22, 2013
2. May 22, 2013

### Staff: Mentor

A decay constant has the unit of 1/time, its numerical value depends on the time unit you choose.

A decay constant of 100/year implies that the nucleus will not survive for one year (with a probability extremely close to 1), but the same decay constant can be expressed as 0.000003/second (give or take a zero) - a very high probability that it will survive a second.

You cannot compare quantities with units with dimensionless quantities (like 1).

3. May 22, 2013

### Namige

But what does 100/year mean exactly? The definition 'The probability of a nucleus decaying per unit time' doesn't seem to explain this.

4. May 22, 2013

### Staff: Mentor

It is the constant in the expression for the exponential decay: $$P(t)=e^{-kt}$$ where P(t) is the probability that the particle is still there and k is the decay constant.
In the limit of small timesteps (something like 0.000003/second), it is close to the probability that the particle will decay within that timestep.

5. May 22, 2013

### Namige

Oh I see now. Thanks.

Last edited: May 22, 2013
6. May 22, 2013

### Staff: Mentor

The probability of what?

100/year = 100/(356.25*24*3600 second) = 0.000003/second
It is the same value (well, up to rounding errors), just expressed in different units.

7. May 22, 2013

### Namige

So is the definition I gave for the decay constant completely correct?

8. May 22, 2013

### Staff: Mentor

λ = ln2/T? That is fine, if T is the half-life.

9. May 22, 2013

### Namige

No, the word definition 'The probability of a nucleus decaying per unit time'

10. May 22, 2013

### Staff: Mentor

Well, in the limit of short timescales, yes.

11. May 22, 2013

### Simon Bridge

Looking at where it appears, the "decay constant" would be the inverse mean-life of the state.

Since the mean-life can be less than one in some units, the decay constant can be bigger than one.
The probability of a given particle decaying within it's mean life would be 0.68 or something wouldn't it?

12. May 22, 2013

### Staff: Mentor

Right

The number on its own does not have a physical meaning anyway.

13. May 22, 2013

### Simon Bridge

Well it has abut the same kind of physical meaning as the half-life
... the mean life is the amount of time it would have taken for the initial sample to vanish if the decay rate were constant.
The decay constant is the magnitude of the slope of the decay curve at t=0.

It's probably not terribly helpful to think of it in terms of the chance of an individual particle decaying though.

14. May 23, 2013

### Staff: Mentor

The number has a meaning together with the unit only, that was what I meant.

It is easier to see with a length: what does a length of "100" mean? 100nm? 100m? 100 light years?
"100" alone does not tell you anything.

15. May 24, 2013

### Simon Bridge

Well naturally - it is the nature of numbers, isn't it, that none of them mean anything without reference to the thing it is a number of?

They don't have to though do they? Isn't that basis for the whole field of mathematics?

I probably just misunderstood what you were replying to. :)

16. May 24, 2013

### Staff: Mentor

No. The fine-structure constant, for example, is roughly 1/137. It is a dimensionless number, it has a physical meaning to say "it is smaller than 1".
The electron to proton ratio is a dimensionless number, too - it has a physical meaning to say "it is smaller than 1".

Only dimensionless physical values and constants are really fundamental. They have the same value in all unit systems, and everyone in the universe will get the same result (maybe up to factors like 2 pi). You could transmit "1/137" (together with instructions how to read that) to any other technological species (if we would know any other), and they would be able to understand the physical meaning of the number.

17. May 24, 2013

### Simon Bridge

I did not mean anyone to get the idea that I thought that dimensionless quantities had no meaning.
Well, without knowing that the fine structure constant is what is intended - or a context suggesting that - the number "1/137" has no meaning ... it could be, say, the probability that someone in my household wears a red shirt. "fine structure constant" is the "thing it is a number of" of which I wrote.

Another technological species would get the same value for Plank's constant too - but they would express it in different actual digits - trivially because they'd use a different system of symbols, and non-trivially because they'd use different units. (They'd also get the same number for the probability of a red shirt ... but only trivially different digits.)

But I think I know what you mean:
You mean that OP needs to realise that the units are also important when it comes to figuring out what the numerical value of a measurement of a physical quantity is telling you - not just it's definition. Students/novices often forget to include the units in their thinking after all.

i.e. "the probability of an event occurring in a particular time period" may not all that helpful without the time period in question or some idea of how the probability varies with time.

Asking if the fine structure constant could be bigger than 1 is much more profound than asking if the decay constant could be bigger than 1.
The decay constant can be trivially bigger than 1 by choosing different units. But, if it is a probability (asks OP) how can this make sense?

OP did provide units in several posts though.
iirc: one post asked what a decay constant of "100 per year" could mean (inferring the context: given the definition provided in OP's fact sheet.)
Of course it's meaning is ambiguous in terms of probability - it just means that the mean-life of the state is 1/100 years.

I think the short-direct answer to OPs question is that the fact-sheet definition is, at best, incomplete, and should not be relied upon.
I'd favor that response ... but it would be valid to point out that it is a probability divided by a time period - not a probability.
I figure you mean to point out that the numerical value depends on the measure used for time.

... sooooo... I figure OPs question is well and truly answered now?

18. May 25, 2013

### Staff: Mentor

How is that related to the difference between dimensionless numbers and values with units?

Exactly.

19. May 25, 2013

### Staff: Mentor

The probability of a nucleus decaying per unit time is the same as the fraction of the nuclei present that decay per unit time. The fraction can't be greater than 100%, since you can't have more nuclei decaying than there are nuclei present.

20. May 25, 2013

### jbriggs444

If not taken as a short-time limit, the "probability of decay per unit time" is an oxymoron since the probability is not a linear function of time.

"The probability of decaying in one unit of time" is a different metric. In very much the same way that a simple interest rate is different from an APR.