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Radioactive decay

An unstable nucleus has no classical countdown clock that reveals its exact decay time. Quantum theory assigns a probability per unit time. For a large population of independent nuclei in the same state, that microscopic randomness produces a predictable exponential survival law.

Radioactive decay changes one nuclide into another and can emit particles or photons. The decay law predicts population and activity; it does not by itself predict detector counts, biological dose, or every member of a decay chain.

Exponential decay law

Let \(N(t)\) be the expected number of undecayed nuclei. If each nucleus has constant decay probability per unit time \(\lambda\), then

\[\frac{dN}{dt}=-\lambda N\]

The minus sign means the parent population decreases. Separating variables gives

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

where \(N_0\) is the population at \(t=0\). The equation gives an ensemble expectation. A small physical sample contains an integer number of nuclei and can differ from that expectation.

import numpy as np

initial_count, half_life, time = 10000, 5.0, 10.0
decay_constant = np.log(2) / half_life
expected = initial_count * np.exp(-decay_constant * time)
print("Expected survivors:", expected)
assert np.isclose(expected, initial_count / 4)

At zero time the result is \(N(0)=N_0\). As time increases, the expected population never rises in this one-parent model.

Half-life and mean lifetime

The half-life \(T_{1/2}\) is the time for the expected parent population to fall by one half:

\[T_{1/2}=\frac{\ln 2}{\lambda}\]

The mean lifetime of one member of the ensemble is

\[\tau=\frac{1}{\lambda}\]

so \(T_{1/2}=\tau\ln2\). Half-life and mean lifetime are not the same number.

After \(n\) half-lives,

\[N=N_0\left(\frac12\right)^n\]

This relation remains useful for noninteger

\(n=t/T_{1/2}\)

when the power is evaluated continuously.

Activity

The activity is the expected decay rate:

\[\mathcal A(t)=-\frac{dN}{dt}=\lambda N(t)=\mathcal A_0e^{-\lambda t}\]

The SI unit is the becquerel: \(1\,\mathrm{Bq}=1\,\mathrm{s}^{-1}\). Activity is not the same as absorbed dose or biological effect. Radiation type, energy, geometry, exposure time, shielding, and tissue response are also needed.

If a sample amount is given in moles, the expected nucleus count is

\[N=\frac{m}{M}N_{\mathrm A}\]

for sample mass \(m\), molar mass \(M\), and Avogadro constant \(N_{\mathrm A}\). Purity and isotopic abundance must be included when the sample is not a pure isotope.

Exact decay probability over an interval

The probability that one nucleus survives an interval \(\Delta t\) is

\[P_{\mathrm{survive}}=e^{-\lambda\Delta t}\]

and the probability that it decays during that interval is

\[P_{\mathrm{decay}}=1-e^{-\lambda\Delta t}\]

The approximation

\(P_{\mathrm{decay}}\approx\lambda\Delta t\)

is valid only when \(\lambda\Delta t\ll1\). Using \(\lambda\Delta t\) as an exact probability can exceed one for a large interval.

def interval_decay_probability(half_life, interval):
    if half_life <= 0 or interval < 0:
        raise ValueError("Use positive half-life and nonnegative interval.")
    return -np.expm1(-np.log(2) * interval / half_life)


rng = np.random.default_rng(42)
probability = interval_decay_probability(5.0, 1.0)
decays = rng.binomial(10000, probability)
print("Decays in one second:", decays, "expected:", 10000 * probability)

Common decay modes

In alpha decay, the nucleus emits a helium-4 nucleus:

\[{}^A_ZX\rightarrow{}^{A-4}_{Z-2}Y+{}^4_2\alpha\]

In beta-minus decay, a neutron is converted into a proton while an electron and electron antineutrino are emitted:

\[n\rightarrow p+e^-+\bar\nu_e\]

The mass number stays constant and nuclear charge increases by one. In beta-plus decay, a proton becomes a neutron while a positron and electron neutrino are emitted. Electron capture is a competing route in which the nucleus captures an atomic electron.

Gamma decay changes an excited nucleus to a lower nuclear state without changing \(A\) or \(Z\):

\[{}^A_ZX^*\rightarrow{}^A_ZX+\gamma\]

Every reaction must conserve electric charge, total energy, momentum, angular momentum, and lepton number where applicable. Nucleon number is useful bookkeeping for ordinary nuclear reactions, though it is not a fundamental conservation law in all particle processes.

Branching fractions

A nuclide can decay through several channels. If partial decay constants are \(\lambda_i\), then

\[\lambda_{\mathrm{total}}=\sum_i\lambda_i,\qquad b_i=\frac{\lambda_i}{\lambda_{\mathrm{total}}}\]

The branching fractions satisfy

\(\sum_i b_i=1\)

when all channels are included. The parent population follows the total decay constant, while the expected production rate into channel \(i\) is \(b_i\mathcal A\).

Decay energy

A spontaneous decay must have a positive available energy under the chosen mass convention:

\[Q=(m_{\mathrm{initial}}-m_{\mathrm{final,total}})c^2\]

The final total includes every daughter and emitted particle. Atomic masses can simplify some electron bookkeeping, but beta-plus and electron-capture formulas require especially careful treatment of atomic electrons. Use one documented mass convention throughout.

In a two-body decay of a parent initially at rest, energy and momentum conservation fix the daughter and emitted-particle kinetic energies. Beta decay has three final particles, so the electron or positron has a continuous energy spectrum; the neutrino carries variable energy and momentum. Treating all \(Q\) as electron kinetic energy would violate the measured spectrum and conservation bookkeeping.

A two-member decay chain

If parent 1 decays into radioactive daughter 2, their mean populations obey

\[\frac{dN_1}{dt}=-\lambda_1N_1,\qquad \frac{dN_2}{dt}=\lambda_1N_1-\lambda_2N_2\]

For no initial daughter and unequal decay constants,

\[N_2(t)=\frac{\lambda_1N_{1,0}}{\lambda_2-\lambda_1}\left(e^{-\lambda_1t}-e^{-\lambda_2t}\right)\]

The daughter can first grow and later decline. Therefore the rule “radioactive population always decreases” applies to an isolated parent population, not to every member of a chain.

If the parent lives much longer than the daughter, their activities can approach secular equilibrium, with daughter activity tracking parent activity. Chemical separation or an initially present daughter changes the solution.

Measurement counts and uncertainty

A detector records only some emitted radiation. In a simple count model,

\[R_{\mathrm{measured}}=\epsilon\,g\,\mathcal A+R_{\mathrm{background}}\]

where \(\epsilon\) is intrinsic detection efficiency and \(g\). Real detectors also have dead time, energy thresholds, scattering, and energy-dependent response.

For independent counts at a constant expected rate, a Poisson model has standard deviation approximately

\(\sqrt{C}\)

for expected count \(C\). Background subtraction adds uncertainty from both the sample and background measurements. A fractional count can be an expectation, never one observed event.

Dating and assumptions

Radiometric dating uses a decay clock only when initial conditions and material history are constrained. From the parent ratio,

\[t=\frac{1}{\lambda}\ln\left(\frac{N_0}{N}\right)\]

but \(N_0\) is often inferred from daughter products, reference isotopes, or a formation model. The sample must remain sufficiently closed to parent and daughter exchange, and the decay constant must apply over the interval.

Checks and scope

A reliable calculation checks:

The simple law assumes a constant decay constant and independent nuclei in one state. Environmental effects are normally tiny for nuclear decays but can matter for some electron-capture processes. This page does not calculate radiation transport, shielding, absorbed dose, or medical risk.

A worked Python notebook

Start with ten thousand independent nuclei having a half-life of five seconds. The question for this calculation is precise: how does expected survivor count change when we change elapsed time, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

The smooth result is an expectation; an actual count is integer-valued and fluctuates. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0 to 20 s. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for elapsed time, expressed in s. The output is expected survivor count, expressed in count. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 10000 e^{- \frac{q \log{\left(2 \right)}}{5}}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 10 s, the expression gives 2500 count. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("10000*exp(-log(2)*q/5)", locals={"q": q})
reference_input = 10.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print("Predicted expected survivor count:", reference_output, "count")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 10000 count. The midpoint, at 10 s, gives 2500 count. The upper endpoint gives 625 count. Read these together with the assumptions: they describe different choices of elapsed time within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0, 10.0, 20.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} s -> {output_value:.6g} count")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows elapsed time; the vertical axis shows expected survivor count. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.0, 20.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Elapsed time (s)")
ax.set_ylabel("Expected survivor count (count)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does expected survivor count change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 10 s is -346.574 in those ratio units. Locally, the output decreases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = - 2000 e^{- \frac{q \log{\left(2 \right)}}{5}} \log{\left(2 \right)}\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 10.1
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "count")
print("Full expression:", full_prediction, "count")
print("Approximation difference:", full_prediction - linear_prediction, "count")

Here the changed input is 10.1 s. The tangent estimate is 2465.34 count, while the full expression gives 2465.58 count. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 20.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of elapsed time inside the stated interval, predict whether expected survivor count will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: The smooth result is an expectation; an actual count is integer-valued and fluctuates. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of count: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(20.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 20.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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