Calculating Activity and Count Rate of an Unstable Radioactive Isotope

Radioactive isotopes decay spontaneously, emitting particles or photons that can be measured with a detector. Two quantities are central to any quantitative radiation measurement: activity (how many decays occur per unit time) and count rate (how many signals the detector records per unit time). This page explains the physical basis of these concepts, presents the key equations, and shows how to carry out realistic calculations for an unstable isotope.

1. Basic decay law

If a sample contains N atoms of a radionuclide, the number of atoms that survive after a time t is given by the exponential decay law:

N(t) = Ne^t

where:

• N initial number of atoms at t=0
• decay constant (s)

The decay constant is related to the halflife T through the expression:

= \frac{\ln 2}{T_{}}

2. Activity (A)

Activity is defined as the number of decays per second, measured in becquerels (Bq):

A = N

Since activity changes with time, the instantaneous activity at time t is:

A(t) = N(t) = Ne^t

If the activity at the start of an experiment is known, the later activity can be obtained without calculating N explicitly:

A(t) = Ae^t

where A is the initial activity.

3. From mass to number of atoms

Often the sample is specified by its mass m (g) rather than by N. Convert mass to number of atoms using Avogadros number (N_A = 6.02210mol) and the isotopes atomic mass M (gmol):

N = \frac{m}{M}\,N_A

Putting this into the activity formula gives a convenient relation:

A = \frac{m}{M}\,N_A

4. Count rate (C)

A detector does not see every decay. The measurable count rate depends on several factors:

• Detection efficiency () probability that a decay event produces a detectable pulse.
• Geometrical factor (G) fraction of emitted radiation that reaches the detector (solidangle considerations).
• Branching ratio (b) fraction of decays that emit the specific radiation the detector is tuned to (e.g., a specific gamma line).

The expected count rate is then:

C = GbA

All three terms are dimensionless, so C has the same units as A (counts per second). In practice, the quantity Gb is often combined into a single overall efficiency term ():

C = A

5. Example calculation

Problem statement: A laboratory has a 2.5Ci sample of ^99mTc (halflife = 6.01h). The detectors overall efficiency for the 140keV gamma photon is 12% ( = 0.12). What is the count rate 3h after the start of the measurement?

Solution:

1. Convert the activity from microcuries to becquerels.
   • 1Ci = 3.710Bq, therefore 2.5Ci = 2.510Ci 3.710BqCi = 9.2510Bq.
2. Calculate the decay constant.
   • = ln2 / T = 0.693 / (6.01h 3600sh) = 0.693 / 21636s 3.2010s.
3. Find the activity after 3h.
   • t = 3h = 10800s.
   • A(t) = Ae^t = 9.2510Bqe^{(3.2010s10800s)} 9.2510Bqe^0.346 9.2510Bq0.708 6.5510Bq.
4. Convert activity to count rate using = 0.12.
   • C = A = 0.126.5510Bq 7.8610countss.

Thus the detector will record roughly 7.9kcps (kilocounts per second) after three hours.

6. Background and net count rate

Real measurements contain a background count rate B arising from cosmic rays, environmental radioactivity, and electronic noise. The net rate (C_net) attributable to the sample is obtained by subtraction:

C_{net} = C_{meas} - B

When background is significant, the statistical uncertainty of the net rate is:

\sigma_{C_{net}} = \sqrt{C_{meas} + B}

Both terms are expressed in counts, so the squareroot gives the standard deviation for a Poisson process.

7. Uncertainty propagation

If the activity is derived from a measured mass, uncertainties from the mass, atomic weight, and decay constant contribute. For a simple product A = mN_A / M the relative uncertainty is:

\frac{\sigma_A}{A} = \sqrt{\left(\frac{\sigma_}{}\right)^2 + \left(\frac{\sigma_m}{m}\right)^2 + \left(\frac{\sigma_M}{M}\right)^2}

When a count rate is calculated, the dominant uncertainty often comes from counting statistics, especially for lowactivity samples. Adding the efficiency uncertainty (_) leads to:

\frac{\sigma_C}{C} = \sqrt{\left(\frac{\sigma_{C_{net}}}{C_{net}}\right)^2 + \left(\frac{\sigma_}{}\right)^2}

8. Practical tips for reliable measurements

• Calibrate the detector with a source of known activity to determine accurately.
• Minimize geometry changes between calibration and measurement; keep the sourcetodetector distance constant.
• Allow sufficient counting time so that statistical uncertainties are low (ideally < 1% for routine work).
• Record background before and after the sample measurement to track any drift in the detector.
• Correct for dead time if the count rate approaches the detectors maximum processing capability.

9. Summary

The activity of an unstable radioisotope is governed by the decay constant and the number of atoms present. By converting mass to atom count, one can compute the initial activity, and by applying the exponential law, the activity at any later time. The measured count rate depends on the detectors overall efficiency, geometry, and the branching ratio of the observed radiation. Subtracting background and accounting for statistical and systematic uncertainties yields the net count rate, which is the quantity used in quantitative radioactivity work.

Understanding and applying these relationships enables scientists and engineers to design experiments, verify source strengths, and interpret radiation measurements with confidence.