### Monday, November 10, 2008

## Musing about Radioactive Decay, Probability, etc.

The rate of decay of a radioactive substance follows an exponential law:

Decay rate at time t = C Exp(-t/T)

where C is the initial decay rate (at time t = 0) and T is the mean lifetime of the substance. Exp is the exponential function.

What I've just written is something that is very well-known. What I muse about is this: how does an individual atom know when its time to decay has arrived? Radioactive materials such as uranium, thorium, and potassium are created in stars, particularly when they explode. Every radioactive atom has a moment in time when it decays. The probability that an atom will decay during a time interval t is Exp(-t/T). If t is very much less than T, the atom has little probability of decaying. However, some atoms decay almost immediately after they are formed. Others decay a bit later. Some stay as formed for billions of years before they decay.

You can see where I'm going. I imagine that the time of death of each atom is specified in some manner at the moment it is formed in the stellar explosion. The formation process is, in my imagination, analogous to an automobile factory in which seemingly identical automobiles are made in large numbers. These automobiles have a design flaw that leads them to break down at some time after manufacture. In use, the automobiles fail according to an exponential law.

What is the design flaw in an atom of uranium that leads to its eventual demise?

I should know better than to waste my time musing on such questions. When I studied quantum mechanics I was taught that one can not know enough about the workings of an atom or a nucleus to predict when it will decay. In order to learn such information would require one to destroy it. The mathematics we use to describe such processes, such as the simple law of radioactive decay, simply provide the probability of a certain event, or the frequency of such events in a large population. The flaw, if there is one, that causes one uranium atom to decay after one hour, another to decay after a day, and another to decay after five billion years can not be known to us.

Decay rate at time t = C Exp(-t/T)

where C is the initial decay rate (at time t = 0) and T is the mean lifetime of the substance. Exp is the exponential function.

What I've just written is something that is very well-known. What I muse about is this: how does an individual atom know when its time to decay has arrived? Radioactive materials such as uranium, thorium, and potassium are created in stars, particularly when they explode. Every radioactive atom has a moment in time when it decays. The probability that an atom will decay during a time interval t is Exp(-t/T). If t is very much less than T, the atom has little probability of decaying. However, some atoms decay almost immediately after they are formed. Others decay a bit later. Some stay as formed for billions of years before they decay.

You can see where I'm going. I imagine that the time of death of each atom is specified in some manner at the moment it is formed in the stellar explosion. The formation process is, in my imagination, analogous to an automobile factory in which seemingly identical automobiles are made in large numbers. These automobiles have a design flaw that leads them to break down at some time after manufacture. In use, the automobiles fail according to an exponential law.

What is the design flaw in an atom of uranium that leads to its eventual demise?

I should know better than to waste my time musing on such questions. When I studied quantum mechanics I was taught that one can not know enough about the workings of an atom or a nucleus to predict when it will decay. In order to learn such information would require one to destroy it. The mathematics we use to describe such processes, such as the simple law of radioactive decay, simply provide the probability of a certain event, or the frequency of such events in a large population. The flaw, if there is one, that causes one uranium atom to decay after one hour, another to decay after a day, and another to decay after five billion years can not be known to us.

Labels: exponential function, Quantum Mechanics, radioactive decay