Activity & The Decay Constant (CIE A Level Physics)

• Since radioactive decay is spontaneous and random, it is useful to consider the average number of nuclei which are expected to decay per unit time
  • This is known as the average decay rate

• As a result, each radioactive element can be assigned a decay constant
• The decay constant λ is defined as:

The probability that an individual nucleus will decay per unit of time

• When a sample is highly radioactive, this means the number of decays per unit time is very high
  • This suggests it has a high level of activity

• Activity, or the number of decays per unit time can be calculated using:

• Where:
  • A = activity of the sample (Bq)
  • ΔN = number of decayed nuclei
  • Δt = time interval (s)
  • λ = decay constant (s^-1)
  • N = number of nuclei remaining in a sample

• The activity of a sample is measured in Becquerels (Bq)
  • An activity of 1 Bq is equal to one decay per second, or 1 s^-1

• This equation shows:
  • The greater the decay constant, the greater the activity of the sample
  • The activity depends on the number of undecayed nuclei remaining in the sample
  • The minus sign indicates that the number of nuclei remaining decreases with time - however, for calculations it can be omitted

• In radioactive decay, the number of nuclei falls very rapidly, without ever reaching zero
  • Such a model is known as exponential decay

• The graph of the number of undecayed nuclei against time has a very distinctive shape

Graph of undecayed nuclei against time

Radioactive decay follows an exponential pattern. The graph shows three different isotopes each with a different rate of decay

• The key features of this graph are:
  • The steeper the slope, the larger the decay constant λ (and vice versa)
  • The decay curves always start on the y-axis at the initial number of undecayed nuclei (N₀)

Equations for radioactive decay

• The number of undecayed nuclei N can be represented in exponential form by the equation:

N = N₀e^–λt

• Where:
  • N₀ = the initial number of undecayed nuclei (when t = 0)
  • N = number of undecayed nuclei at a certain time t
  • λ = decay constant (s^-1)
  • t = time interval (s)

• The number of nuclei can be substituted for other quantities.
• For example, the activity A is directly proportional to N, so it can also be represented in exponential form by the equation:

A = A₀e^–λt

• Where:
  • A = activity at a certain time t (Bq)
  • A₀ = initial activity (Bq)

• The received count rate C is related to the activity of the sample, hence it can also be represented in exponential form by the equation:

C = C₀e^–λt

• Where:
  • C = count rate at a certain time t (counts per minute or cpm)
  • C₀ = initial count rate (counts per minute or cpm)

The exponential function e

• The symbol e represents the exponential constant
  • It is approximately equal to e = 2.718

• On a calculator it is shown by the button e^x
• The inverse function of e^x is ln(y), known as the natural logarithmic function
  • This is because, if e^x = y, then x = ln(y)