Calculate half-life from data or decay curves from which background radiation has not been subtracted

5.2.4 Half‑life

Learning Objective

Calculate the half‑life of a radioactive sample from experimental data or from a decay curve when the background radiation has not been subtracted.

Key Concepts

• The half‑life (\$t_{1/2}\$) is the time required for the activity of a radioactive sample to decrease to half of its initial value.
• Radioactive decay follows the exponential law

  \$N(t)=N_0e^{-\lambda t}\$

  where \$N(t)\$ is the activity at time \$t\$, \$N_0\$ the initial activity and \$\lambda\$ the decay constant.

• When background radiation is present, the measured activity is

  \$N_{\text{meas}}(t)=N(t)+B\$

  where \$B\$ is the constant background count rate. Because \$B\$ is the same at all times, the half‑life can still be obtained directly from the raw data.

Method 1 – Using Tabulated Data

Follow these steps with a set of measured counts (including background) taken at regular time intervals.

1. Record the first reading as the initial activity \$N_0\$ (raw count).
2. Calculate half of this value: \$\displaystyle \frac{N_0}{2}\$.
3. Locate the time at which the measured count first falls below \$\frac{N_0}{2}\$.
4. If the exact half‑value lies between two recorded points, interpolate linearly:

   \$t{1/2}=t1+\frac{\left(\frac{N0}{2}-N1\right)}{(N2-N1)}\,(t2-t1)\$

   where \$(t1,N1)\$ and \$(t2,N2)\$ are the two successive measurements surrounding \$\frac{N_0}{2}\$.

5. The resulting \$t_{1/2}\$ is the half‑life of the sample.

Example – Raw Data (Background Not Subtracted)

Steps:

1. \$N_0 = 800\ \text{cpm}\$
2. \$\frac{N_0}{2}=400\ \text{cpm}\$
3. The count drops below 400 cpm between 90 s (380 cpm) and 60 s (490 cpm).
4. Interpolate:

   \$\$t_{1/2}=60+\frac{(400-490)}{(380-490)}\,(90-60)

   =60+\frac{-90}{-110}\times30

   =60+24.5\approx84.5\ \text{s}\$\$

5. Thus the half‑life is approximately 85 s.

Method 2 – Using a Decay Curve

When a graph of counts versus time is available, the half‑life can be read directly.

1. Plot the raw counts (including background) on the vertical axis and time on the horizontal axis.
2. Draw a horizontal line at the level \$\displaystyle \frac{N0}{2}\$, where \$N0\$ is the first plotted point.
3. The point where this line intersects the decay curve gives the half‑life \$t_{1/2}\$.
4. If the intersection falls between two plotted points, use the same linear interpolation formula as in Method 1.

Why Background Does Not Need to Be Subtracted

• The background count \$B\$ adds a constant offset to every measurement.
• When we take the ratio \$N(t)/N0\$, the background does not cancel, but the definition of half‑life uses the absolute value \$\frac{N0}{2}\$, not a ratio.
• Consequently, the time at which the measured count falls to \$\frac{N_0}{2}\$ is the same whether or not \$B\$ is removed, provided \$B\$ is truly constant.
• However, if \$B\$ is a significant fraction of \$N_0\$, the statistical uncertainty increases; in practice we still prefer to subtract \$B\$, but it is not mandatory for a correct half‑life determination.

Common Pitfalls

• Using the background‑subtracted value for \$N_0\$ while still using raw data for later points – this gives an inconsistent half‑life.
• Assuming the decay is linear; remember it is exponential, so interpolation should be linear only over a short interval near the half‑value.
• Reading the half‑life from a curve that has not been drawn to scale – always check the axes.

Practice Questions

1. Given the following raw counts, determine the half‑life:

   Time (s)	Counts (cpm)
   0	1200
   40	950
   80	720
   120	540
   160	410

2. A decay curve shows \$N_0=500\$ counts at \$t=0\$ and the curve passes through \$N=250\$ counts at \$t=75\,\$s. What is the half‑life?
3. Explain why a constant background of 50 cpm does not affect the half‑life obtained from raw data.