# What is the purpose of the decay time distribution equation?

• tryingtolearn1
In summary: The equation ##D(t) = \lambda \exp(−\lambda t)## gives the probability of a decay occurring in the time interval between t and t+dt, which is useful for predicting the behavior of the sample.
tryingtolearn1
Homework Statement
Muons decay time distribution
Relevant Equations
##N(t) = N_0 exp(−\lambda t)## and ##D(t) = \lambda \exp(−\lambda t)##
I know for muons that the the probability that a muon decays in some small time interval ##dt## is ##\lambda dt##, where ##\lambda## is a decay rate. Thus the change in the population of muons is just ##dN/N(t) = −\lambda dt##. Integrating gives ##N(t) = N_0 \exp(−\lambda t)##. This makes sense to me but my book goes on to say the following,

By decay time distribution D(t), we mean that the time-dependent probability that a muon decays in the time interval between ##t## and ##t + dt## is given by ##D(t)dt##. If we had started with ##N_0## muons, then the fraction ##−dN/N_0## that would on average decay in the time interval between ##t## and ##t + dt## is just given by differentiating the above relation: ##−dN = N_0\lambda \exp(−\lambda t) dt## ##\therefore## ##−dN/ N_0 = \lambda \exp(−\lambda t) dt##. The left-hand side of the last equation is nothing more than the decay probability, so ##D(t) = \lambda \exp(−\lambda t)##.

What exactly is that explaining? Don't we need to know what ##\lambda## is before using the ##D(t)## equation? Because trying to find ##\lambda## using ##D(t) = \lambda \exp(−\lambda t)## will give the wrong results.

tryingtolearn1 said:
Homework Statement:: Muons decay time distribution
Relevant Equations:: ##N(t) = N_0 exp(−\lambda t)## and ##D(t) = \lambda \exp(−\lambda t)##

I know for muons that the the probability that a muon decays in some small time interval ##dt## is ##\lambda dt##, where ##\lambda## is a decay rate. Thus the change in the population of muons is just ##dN/N(t) = −\lambda dt##. Integrating gives ##N(t) = N_0 \exp(−\lambda t)##. This makes sense to me but my book goes on to say the following,
What exactly is that explaining? Don't we need to know what ##\lambda## is before using the ##D(t)## equation? Because trying to find ##\lambda## using ##D(t) = \lambda \exp(−\lambda t)## will give the wrong results.
I'm not entirely sure what you are asking, but it looks to me that D(t) is defined as ##\frac{P(decay in interval (t,t+dt))}{dt}##, whereas the ##\lambda dt## expression assumes it has not decayed at time t.
So D(t)=P(undecayed_at_time (t))λ = ##\lambda \exp(−\lambda t)##

haruspex said:
I'm not entirely sure what you are asking, but it looks to me that D(t) is defined as ##\frac{P(decay in interval (t,t+dt))}{dt}##, whereas the ##\lambda dt## expression assumes it has not decayed at time t.
So D(t)=P(undecayed_at_time (t))λ = ##\lambda \exp(−\lambda t)##
Hmm but why would that equation be relevant? Suppose you know what ##t## is and you're trying to find ##\lambda##, why would ##D(t)=\lambda\exp(-\lambda t)## be relevant?

tryingtolearn1 said:
Hmm but why would that equation be relevant? Suppose you know what ##t## is and you're trying to find ##\lambda##, why would ##D(t)=\lambda\exp(-\lambda t)## be relevant?
I see no suggestion that this is to do with finding λ. Rather, it assumes you have already determined λ and now wish to estimate the rate of decays in a sample at some future point.

tryingtolearn1

## 1. What is muon decay time distribution?

Muon decay time distribution refers to the pattern of decay times for a large number of muons. Muons are subatomic particles that have a finite lifetime, and their decay follows a specific distribution over time.

## 2. Why is studying muon decay time distribution important?

Studying muon decay time distribution can provide valuable information about the properties of muons and the fundamental forces of nature. It can also help in understanding the behavior of other subatomic particles and their interactions.

## 3. How is muon decay time distribution measured?

Muon decay time distribution is typically measured using detectors that can identify and track the decay of individual muons. These detectors record the time and energy of each decay event, which can then be analyzed to determine the overall distribution.

## 4. What factors can affect muon decay time distribution?

The decay time distribution of muons can be affected by various factors, such as the energy and velocity of the muon, the presence of other particles, and the strength of the forces involved in the decay process.

## 5. What are some applications of muon decay time distribution?

The study of muon decay time distribution has various applications in particle physics, astrophysics, and cosmology. It can also be used in medical imaging and radiation therapy, as well as in the development of new technologies and materials.

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