A Beginner’s Guide to Activation Analysis
This chapter is a conceptual introduction to activation analysis — the field Curie was built to serve — for readers who are new to it. Where the User’s Guide shows how to drive Curie and the Theory & Methodology chapter gives the equations, this guide explains why the analysis looks the way it does: what is being measured, and how each number leads to the next. Each section ends with a short pointer to the Curie classes that carry out that step.
What is activation analysis?
The idea behind activation analysis is to learn about matter by making it briefly radioactive and watching it decay. A sample is exposed to a beam or flux of particles — neutrons from a reactor, or charged particles (protons, deuterons, alphas) from an accelerator. A fraction of the nuclei in the sample undergo nuclear reactions and are transformed into new, often radioactive, nuclei. Those product nuclei then decay, emitting radiation — most usefully gamma rays, whose energies are sharp and characteristic of the emitting isotope. By detecting that radiation, you can identify which isotopes were produced and, from how much radiation you see, how much of each.
That single chain of events — produce, decay, detect — underlies two kinds of measurement Curie supports:
Quantifying activity. Given a sample that has been irradiated, how active is each isotope in it? This is the everyday task of gamma-ray spectroscopy: turn a measured spectrum into a table of activities.
Measuring cross sections. How readily does a particular reaction occur, as a function of the beam energy? This is the goal of the charged-particle stacked-target technique, where the measured activities are worked backward into the underlying reaction probabilities — the data that let you predict, before an irradiation, how much of a desired radioisotope (and how much co-produced impurity) a given beam and target will make.
The rest of this guide follows the produce–decay–detect chain forward and backward: the physics of production and decay, how activity is measured, how a measured activity becomes a production rate, how charged particles behave in a target, and finally how all of this is combined to predict yields and design an experiment.
In Curie. Every class in the toolkit sits somewhere on this chain; the Quickstart maps them out.
Production, decay, and counting
Three intervals of time structure every activation measurement: the sample is irradiated, then it cools (decays) for a while, and then it is counted in a detector. Following one product isotope through these intervals gives the whole quantitative framework.
Production. During irradiation the product is created at some rate \(R\) (atoms per second) and, being radioactive, decays at the same time with decay constant \(\lambda = \ln 2 / t_{1/2}\). The number of product atoms climbs until creation and decay balance. For a constant production rate over an irradiation of duration \(t_{\mathrm{irr}}\), the activity (the number of decays per second) at the end of the irradiation — the end of bombardment, or EoB — is
The term in parentheses is the saturation factor. Irradiate for one half-life and you reach half the maximum possible activity; irradiate for several half-lives and the activity saturates at \(A = R\) — no matter how long you keep going, the isotope decays as fast as it is made. This is the single most important consequence of the half-life for planning: short-lived products saturate quickly (little is gained by a long irradiation) while long-lived products build up slowly.
Decay. After the beam is off, the activity simply decays:
with \(t\) measured from EoB. Time zero for the whole analysis is conventionally this end-of-bombardment moment.
Counting. A detector does not measure activity directly; over a counting interval it records the number of decays that occur in that window (times the fraction it manages to detect). Because the activity is itself falling during a long count, the number of decays is the integral of \(A(t)\) across the counting interval, not simply activity times time. Keeping these three intervals — and their clocks — straight is the bookkeeping that the whole analysis depends on.
In Curie. DecayChain implements exactly this, and generalizes it
from one isotope to a full decay chain (parents feeding daughters) via the
Bateman equations. See Isotopes & Decay Chains and, for the equations,
Radioactive Decay Chains.
Measuring activity: gamma-ray spectroscopy
The most common way to observe the decays is gamma-ray spectroscopy with a high-purity germanium (HPGe) detector. Each decaying isotope emits gamma rays at a set of characteristic energies, each with a known intensity — its emission probability, the fraction of decays that produce that gamma. The detector sorts detected gammas by energy into a spectrum — a histogram of counts versus energy — in which each isotope appears as peaks at its own line energies.
To turn the area of a peak into an activity, three calibrations are needed, and it is worth understanding why each is required:
Energy calibration maps detector channel to gamma energy, so that a peak can be matched to the isotope and line that produced it.
Resolution calibration describes how wide the peaks are, which the fitting routine needs to separate nearby lines.
Efficiency calibration gives the fraction of gammas emitted at a given energy that are actually recorded as full-energy counts — counts from gammas that deposited their entire energy in the detector. Without it a peak area is only a relative number; with it, the count rate becomes an absolute emission rate. Efficiency falls with distance and varies strongly with energy, so it must be measured for the specific detector and geometry, using a source of known activity.
Given these, the number of counts in a peak relates to the activity through the gamma intensity, the efficiency, and the counting-time factors of the previous section. Inverting that relation for every clean peak yields the isotope’s activity — the measured quantity that the rest of the analysis builds on.
In Curie. Spectrum fits the peaks and Calibration produces and
stores the three calibrations. See Spectroscopy, and
Gamma-ray Peak Fitting and Detector Calibration for the models.
Turning activities into production rates
Measuring an activity is a means to an end. What you usually want is the quantity that was under experimental control: the production rate during irradiation (or, for a decay-only sample, the activity at some reference time). This is the produce–decay–detect chain run backward.
For a single isotope it is a matter of algebra: rearranging the saturation relation from Production, decay, and counting,
and \(A_{\mathrm{EoB}}\) itself is obtained by decay-correcting the activity measured at count time back to EoB. In practice several complications enter at once: the isotope of interest may be fed by the decay of a parent, several spectra taken at different cooling times each constrain the same production rate, and the beam may have varied during the irradiation. The robust way to handle all of this is to fit: adjust the production rate until the decays it predicts, across the whole chain and all counting intervals, best match the measured ones.
A useful distinction: fitting a production rate applies when the sample was being made during an irradiation, whereas fitting an initial activity applies to a sample that was simply decaying from some reference time (a calibration source, say). The two answer different questions and take different starting information.
In Curie. DecayChain.get_counts reads measured decays (including
straight from fitted Spectrum peaks), and fit_R and fit_A0 fit a
production rate or an initial activity to them. See the
Decay Chain Worked Examples for a full inverse example.
Charged-particle interactions in a target
So far the beam could have been anything. Charged-particle beams behave in a way that shapes the entire stacked-target technique, so they deserve their own discussion.
Unlike neutrons, which travel until they happen to react, a charged particle interacts continuously with the electrons of the material it traverses, losing energy little by little. The rate of energy loss per unit path length is the stopping power. Two consequences follow, and both are central to activation work:
The beam energy decreases with depth. A particle that enters a foil at one energy leaves it at a lower one, and eventually — after a distance called its range — stops entirely. Because a reaction’s cross section depends on energy, the production rate is not uniform through a thick target: different depths are effectively irradiated at different energies.
The beam spreads in energy. A foil sees a range of energies, not a single value: partly because the beam loses energy continuously across the foil’s own thickness, and partly because any initial energy spread grows as the beam slows (slower particles lose energy faster). Random fluctuations in the energy loss (straggling) broaden it further still.
These facts are usually a nuisance — but they can be turned into an advantage. If a thin foil barely changes the beam energy, it measures a reaction essentially at a single energy. Stack many thin foils, with degraders between them to step the energy down, and one irradiation samples the reaction at a whole ladder of energies at once. This is the stacked-target technique, and reading the energy in each foil correctly is the crux of it.
In Curie. Element and Compound compute stopping powers and ranges;
Stack transports a beam through a stack of foils and reports the energy
distribution in each. See Stopping Power Calculations and Stopping Power and Particle Transport.
Predicting production rates and isotope yields
With the pieces in hand, the forward calculation — how much of a product will an irradiation make? — comes together. In a single foil, the production rate is the reaction cross section combined with the beam and the target:
where \(I_p\) is the beam current (particles per second),
\(n_t\) the number of target atoms per unit area, and
\(\langle\sigma\rangle\) the cross section averaged over the energy
distribution of the beam in that foil — the foil’s energy spectrum, which
Stack.get_flux supplies. (Curie also folds in the unit factor converting
mb to cm2; the Reactions page gives the fully dimensioned
form.)
That flux-averaging is where the
physics of the previous section enters: for a thin foil the average is essentially the
cross section at the beam energy, but for a thick foil, where the beam
spans a wide energy range, using the cross section at the mean energy alone
can be badly wrong — the average must be taken over the real distribution.
Once \(R\) is known, the activation equation of Production, decay, and counting turns it into an activity at end of bombardment, and the decay and counting relations turn that into the number of decays a detector would record at any later time. The chain is now closed: a cross section, a beam, and a target predict the very counts that a measurement would produce — and, run the other way, measured counts yield the cross section.
In Curie. Reaction.average (and Reaction.integrate) combine a
cross section with a beam spectrum — for a foil, Stack.get_flux supplies
that spectrum — and the result feeds DecayChain as a production rate.
See the “Averages vs. integrals” discussion on the Reactions page,
and the Stopping Power Worked Examples for the thin-versus-thick comparison.
Designing a stacked-target experiment
A stacked-target cross-section measurement is planned backward from what it needs to produce: measurable activities of the product isotopes, at a set of well-known beam energies, with the beam current under control. The main design choices are:
Energy coverage. The incident beam energy and the stack’s degraders set the range of energies sampled. Enough foils are included, with degraders sized so that consecutive target foils sit at usefully spaced energies across the region of interest.
Target foils. Each target foil must be thin enough that the beam energy is well defined across it (so the measured cross section belongs to a narrow energy), yet thick enough to produce enough activity to measure. These pull in opposite directions and are traded off against the expected cross section and beam current.
Monitor foils. Interleaved foils of well-characterized materials carry monitor reactions — reactions whose cross sections are known to high accuracy. Measuring their activities reconstructs the actual beam current and energy at each position in the stack, which is how the experiment calibrates itself rather than trusting the nominal beam parameters.
Cooling and counting schedule. The half-lives of the products dictate the timing: short-lived isotopes must be counted soon after irradiation, long-lived ones may need to cool so that short-lived activities decay away first.
Range check. The beam must actually reach the last foil of interest: the material in front of it has to stay within the particle’s range at the incident energy (degraders or catchers — passive foils that stop the beam or recoiling reaction products — further downstream may lie beyond the range without harm).
The Curie workflow mirrors the experiment. Before the beam time, define
the foils and use Stack to predict the energy in each, then combine
monitor and product cross sections with those fluxes to estimate the
activities you will produce — confirming the design will yield measurable,
well-placed data. After the irradiation, fit the counted spectra with
Spectrum and Calibration, decay-correct and fit production rates with
DecayChain, normalize to the monitor foils, and divide out the beam and
target to recover the cross sections.
In Curie. This ties together every part of the toolkit; the four topic groups of the User’s Guide cover the individual steps, and their troubleshooting pages collect the pitfalls (unit mix-ups, a beam that stops in the stack, thick-foil energy spread, monitor-reaction choice) that most often trip up a first experiment.
For a single script that runs this whole chain on a real dataset, see examples/stacked_target_analysis.py: it builds the foil stack of a published natLa(p,x) measurement (Morrell et al., arXiv:1907.04431), transports the beam, flux-averages an evaluated excitation function (a cross section as a function of energy, taken from a nuclear-data library) over each foil, and compares the result against the published cross sections.