Neutron Activation Analysis Calculator

Compute induced radioactivity or back-calculate elemental concentration from neutron activation data.

⚛️ Neutron Activation Analysis Calculator
Target Nuclide
Molar Mass (g/mol)
g/mol
Isotopic Abundance (%)
%
Activation Cross-Section
barn
Sample Mass
g
Neutron Flux (n/cm²/s)
× 10^ n/cm²/s
Product Half-Life
Irradiation Time
Target Nuclide
Molar Mass (g/mol)
g/mol
Isotopic Abundance (%)
%
Activation Cross-Section
barn
Product Half-Life
Measured Activity at End of Irradiation (Bq)
× 10^ Bq
Neutron Flux (n/cm²/s)
× 10^ n/cm²/s
Irradiation Time
Total Sample Mass
g
End-of-Irradiation Activity
—
Saturation Activity
—
Saturation Fraction
—
Target Atoms (N)
—
Decay Constant (λ)
—
Element Concentration
—
Element Mass
—
Target Atoms (N)
—
Mass Fraction
—

⚛️ What is Neutron Activation Analysis?

Neutron activation analysis (NAA) is a nuclear analytical technique that identifies and quantifies elements in a sample by bombarding it with neutrons and measuring the characteristic gamma radiation emitted by the resulting radioactive isotopes. Developed in the 1930s and refined over subsequent decades, NAA remains one of the most sensitive and accurate multi-element analysis methods available in analytical chemistry.

When a stable nucleus captures a thermal neutron, it forms a heavier, unstable isotope that subsequently decays by emitting gamma rays at energies unique to that nuclide. A germanium gamma spectrometer measures these energies, and the peak areas reveal both the identity and quantity of each element present. A single irradiation can simultaneously determine 30 or more elements, making NAA extremely efficient for multi-element surveys.

NAA is widely applied across many fields. In geochemistry and mineralogy, it determines rare-earth elements and platinum-group metals at parts-per-billion concentrations in rocks and ores. In environmental science, it traces heavy-metal contamination in soils, sediments, and biological tissue. In forensics, it has been used to match glass fragments, hair, and gunshot residue at crime scenes. In archaeology, it fingerprints ancient pottery, obsidian tools, and metalwork to identify their sources. In nutrition and biomedicine, it measures trace elements in hair, blood, and tissue samples with great accuracy.

One major advantage of NAA is its near non-destructive nature. Instruments NAA (INAA) irradiates the sample intact, with no dissolution or digestion required. The sample structure is preserved and can sometimes be returned to the owner after the radioactivity decays. Detection limits commonly reach parts per million for most elements and parts per billion for high-cross-section nuclides such as europium and gold, at fluxes available from medium-power research reactors. This calculator implements both directions of the NAA equation: forward (activity from mass) and inverse (mass from measured activity), covering the two most common laboratory workflows.

📐 Formula

A(t)  =  N  ×  σ  ×  Φ  ×  (1 − e−λt)
A(t) = activity at end of irradiation (Bq)
N = number of target atoms = (m × NA × θ) / M
m = sample mass (g); NA = 6.022 × 1023 mol−1
θ = isotopic abundance of target nuclide (fraction)
M = molar mass of the element (g/mol)
σ = thermal-neutron activation cross-section (cm²; 1 barn = 10−24 cm²)
Φ = neutron flux density (n/cm²/s)
λ = decay constant = ln(2) / T½ (s−1)
t = irradiation time (s)
Example: 0.1 g Au at flux 1013 n/cm²/s for 1 h gives A(t) ≈ 3.22 GBq

The saturation activity Asat = N × σ × Φ is the maximum achievable activity (reached as t → ∞). The saturation fraction (1 − e−λt) shows what fraction of saturation is reached at time t: 50% at t = T½, 75% at t = 2T½, 93.75% at t = 4T½.

For the concentration back-calculation, the calculator inverts the equation. Given measured activity Aeoi at end of irradiation, the number of target atoms is N = Aeoi / (σ × Φ × saturation fraction), and the element mass follows from N.

📖 How to Use This Calculator

Steps for Induced Activity Mode

1
Select a nuclide preset: Choose the target nuclide from the dropdown. Molar mass, isotopic abundance, cross-section, and product half-life are filled automatically. Choose Custom to enter your own values for an unlisted nuclide.
2
Enter sample mass and neutron flux: Type the mass of the element or compound in grams. Enter the neutron flux as a mantissa (1.0 to 9.9) times a power of 10. For example, a reactor flux of 5 × 1013 n/cm²/s uses mantissa 5 and exponent 13.
3
Set irradiation time: Enter the irradiation duration and pick the time unit. One half-life gives 50% of saturation activity. For very long-lived products like Co-60, practical irradiation times reach only a small fraction of saturation.
4
Read the end-of-irradiation activity: Click Calculate. The primary result shows activity at end of irradiation in auto-scaled SI units. Auxiliary boxes show saturation activity, saturation fraction, total target atoms, and the decay constant.
5
Switch to Concentration mode for trace analysis: Click the Elemental Concentration tab. Enter the gamma-spectrometer-measured activity at end of irradiation, the irradiation flux, duration, and the total sample mass. The calculator returns the element mass and its concentration in µg/g (ppm).

💡 Example Calculations

Example 1: Gold Standard Irradiation (Au-197)

0.1 g gold foil, flux 1.0 × 1013 n/cm²/s, irradiation 1 hour

1
Number of Au-197 target atoms: N = (0.1 × 6.022 × 1023 × 1.0) / 196.97 = 3.057 × 1020 atoms.
2
Saturation activity: Asat = 3.057 × 1020 × 98.65 × 10−24 × 1013 = 3.017 × 1011 Bq = 301.7 MBq.
3
Decay constant: λ = ln(2) / (2.694 × 86400) = 2.978 × 10−6 s−1. Saturation fraction after 1 h (3600 s): 1 − e−0.01072 = 1.07%.
4
End-of-irradiation activity: 3.017 × 1011 × 0.01072 = 3.23 × 109 Bq = 3.23 GBq.
Result: 3.23 GBq end-of-irradiation activity (1.07% of saturation)
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Example 2: Manganese in a Food Sample (Mn-55)

0.5 g food sample, flux 2.0 × 1012 n/cm²/s, irradiation 30 min

1
N = (0.5 × 6.022 × 1023 × 1.0) / 54.938 = 5.477 × 1021 Mn-55 atoms.
2
Saturation activity: Asat = 5.481 × 1021 × 13.30 × 10−24 × 2 × 1012 = 1.458 × 1011 Bq = 145.8 GBq.
3
T½ of Mn-56 = 2.579 h = 9284 s; λ = 7.466 × 10−5 s−1. Saturation fraction at 30 min (1800 s): 1 − e−0.1344 = 12.57%.
4
End-of-irradiation activity: 1.458 × 1011 × 0.1257 = 1.833 × 1010 Bq = 18.33 GBq.
Result: 18.33 GBq of Mn-56 at end of irradiation
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Example 3: Trace Gold in a Geological Rock Sample (Concentration Mode)

10 g rock sample, flux 1.0 × 1012 n/cm²/s, 24 h irradiation, measured Au-198 activity = 5.0 × 104 Bq

1
Saturation fraction after 24 h: λ = 2.978 × 10−6 s−1; 1 − e−λ×86400 = 1 − e−0.2573 = 22.73%.
2
Number of Au-197 atoms: N = 50000 / (9.865 × 10−23 × 1012 × 0.2273) = 2.23 × 1015 atoms.
3
Element mass: m = N × M / (NA × 1.0) = 2.23 × 1015 × 196.97 / 6.022 × 1023 = 7.29 × 10−7 g = 0.729 µg.
4
Concentration = 0.729 µg / 10 g = 0.073 µg/g (73 ppb gold in the rock).
Result: 0.073 µg/g gold (73 parts per billion by mass)
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❓ Frequently Asked Questions

What is neutron activation analysis and how does it work?+
Neutron activation analysis works by placing a sample in a neutron field, typically inside a research reactor. Stable nuclei capture neutrons and become radioactive isotopes that emit gamma rays at characteristic energies. A high-purity germanium detector records these energies. Each element produces a unique gamma-ray fingerprint, and the peak area reveals its concentration. NAA can detect 30 or more elements simultaneously in a single irradiation.
What is the NAA formula for induced activity?+
The formula is A(t) = N x sigma x phi x (1 - e^-lambda*t). N is the number of target atoms, sigma is the thermal-neutron cross-section in cm², phi is the neutron flux in n/cm²/s, lambda = ln(2)/T½ is the decay constant, and t is the irradiation time. This formula assumes a constant, uniform thermal flux and a thin sample with negligible self-shielding.
What is saturation activity and when is it reached?+
Saturation activity A_sat = N x sigma x phi is the maximum activity achievable at infinite irradiation time when production equals decay. After one half-life of irradiation, 50% of saturation is reached. After two half-lives, 75%. After five half-lives, 96.9%. For practical purposes, irradiating for three to five half-lives approaches saturation closely. For very long-lived products like Co-60 (T½ = 5.27 yr), saturation is essentially unattainable in a reactor campaign.
What neutron flux is needed for trace-element NAA?+
Research reactors used for NAA provide thermal fluxes of 10^12 to 10^14 n/cm²/s. A flux of 10^13 n/cm²/s is common in medium-power reactors and can detect gold at sub-nanogram levels in a 1 g sample after a 24 h irradiation. Californium-252 neutron sources (10^6 to 10^8 n/cm²/s) are used for portable or in-situ applications where reactor access is impractical.
Which elements have the highest sensitivity in NAA?+
Elements with the highest thermal-neutron cross-sections are most sensitive. Europium (9200 barns for Eu-151), dysprosium (2650 barns for Dy-164), indium (202 barns for In-115), and gold (98.65 barns for Au-197) can be detected at sub-nanogram to picogram levels. Elements with small cross-sections, such as carbon, nitrogen, oxygen, and silicon, are poorly suited to NAA.
Is NAA destructive or non-destructive?+
Instrumental NAA (INAA) is non-destructive. No chemical treatment is needed: the sample is irradiated intact, counted by gamma spectrometry, and returned after radioactive decay. Radiochemical NAA (RNAA) dissolves the sample after irradiation to separate the analyte element, improving sensitivity but destroying the sample. Both forms leave the sample radioactive for days to years depending on the nuclides present.
How do I choose the irradiation time for best results?+
For elements with short product half-lives, such as Mn-56 (2.58 h) or In-116m (54 min), irradiating for 2 to 5 half-lives approaches saturation efficiently. For long-lived products like Fe-59 (44.5 d) or Zn-65 (244 d), even 24 h gives only a small saturation fraction. In multi-element analysis, a short irradiation is often done first to measure short-lived nuclides, then a long irradiation for long-lived ones.
What are the main sources of uncertainty in NAA?+
Key uncertainty sources include neutron flux spatial and temporal variations (typically 1 to 5%), detector efficiency calibration (1 to 2%), gamma-ray peak area integration (0.5 to 5% depending on count statistics), nuclear data (cross-sections and half-lives), and sample self-shielding for elements with very high cross-sections or thick samples. Using flux monitors (gold or cobalt wires) irradiated alongside the sample reduces flux uncertainty substantially.
What is the k0-NAA method?+
The k0-standardization method determines elemental concentrations from a single comparator element (typically gold) irradiated with the sample, using pre-tabulated k0 factors that incorporate the cross-section, isotopic abundance, gamma-ray emission probability, and detector efficiency ratio. It eliminates the need for element-specific standards for each analyte, making multi-element analysis faster and more flexible than classical comparator NAA.
What is the minimum detectable amount in NAA?+
The minimum detectable amount (MDA) depends on the nuclide cross-section, flux, irradiation time, decay time, counting time, detector efficiency, and background. At a flux of 10^13 n/cm²/s with a 24 h irradiation and a modern HPGe detector, gold can be detected at around 10 picograms and europium at sub-picogram levels. For most elements, practical MDAs range from 1 nanogram to 1 microgram in a 1 g sample.
How is NAA applied in forensic science?+
Forensic NAA has been used to compare glass fragments, soil, hair, gunshot residue, and paint chips at crime scenes. By measuring trace-element fingerprints, analysts can determine whether two samples share a common origin with high statistical confidence. NAA is particularly valued in forensics because it is non-destructive and preserves the sample for other tests. However, probability interpretations must be applied carefully to avoid overstating the significance of elemental matches.
What is the difference between INAA and RNAA?+
Instrumental NAA (INAA) counts the irradiated sample directly with no chemical treatment. It is fast, non-destructive, and suitable for routine multi-element screening. Radiochemical NAA (RNAA) dissolves the sample post-irradiation and chemically separates the analyte element, removing interfering activity and lowering the detection limit by one to two orders of magnitude. RNAA is reserved for elements with small cross-sections or severe spectral interferences.
Tags: Neutron Activation Analysis NAA Calculator Induced Radioactivity Nuclear Calculator Formula Free Online

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What is neutron activation analysis used for?

NAA is used to determine elemental concentrations in environmental, geological, archaeological, and forensic samples. A sample is irradiated in a neutron flux and the resulting gamma rays are measured to identify and quantify elements at concentrations as low as parts per trillion.

What is the induced-radioactivity formula used in NAA?

A(t) = N x sigma x phi x (1 - e^-lambda*t). N is the number of target atoms, sigma is the thermal-neutron cross-section in cm2, phi is the neutron flux in n/cm2/s, lambda is the decay constant, and t is the irradiation time in seconds.

What is saturation activity in neutron activation?

Saturation activity A_sat = N x sigma x phi is the maximum achievable activity when production equals decay. It is approached asymptotically. Irradiating for one half-life gives 50% of saturation, five half-lives give 96.9%.

How accurate is neutron activation analysis?

NAA typically achieves 1 to 3% relative uncertainty for major elements and 5 to 10% for trace elements. Accuracy depends on neutron flux stability, detector efficiency calibration, and peak-area integration quality.

What neutron flux is needed for routine NAA?

Research reactors used for NAA operate at 10^12 to 10^14 n/cm2/s. Higher flux enables shorter irradiations and lower detection limits. Californium-252 portable sources offer 10^6 to 10^8 n/cm2/s for field screening.

Which nuclides have the highest sensitivity in NAA?

Europium-151 (9200 barns), Dysprosium-164 (2650 barns), and Indium-115 (202 barns) offer the highest sensitivity because of their large activation cross-sections. Gold-197 (98.65 barns) is the most widely used standard due to a clean 411.8 keV gamma line from Au-198.

Is neutron activation analysis destructive?

NAA is often described as non-destructive because the bulk sample is not dissolved. However, the sample becomes radioactive and requires controlled handling and eventual disposal. Instrumental NAA (INAA) needs no chemical separation, preserving the sample structure.

How does irradiation time affect NAA results?

Longer irradiation increases activity toward the saturation limit. For short-lived nuclides like Mn-56 (T1/2 = 2.58 h), near-saturation is reached in a few hours. For Co-60 (T1/2 = 5.27 yr), even weeks of irradiation give only a small saturation fraction.

📌 Quick Tips

💡Au-197 is the most common NAA standard. It has a large cross-section of 98.65 barns and a convenient product half-life of 2.694 days, making post-irradiation counting easy to schedule.
💡A saturation fraction below 1% means the irradiation time is much shorter than the half-life. Extending irradiation gives proportionally higher activity up to the saturation limit.
💡Fluxes in research reactors typically range from 10^12 to 10^14 n/cm2/s. Higher flux gives proportionally higher activity and lower detection limits.
💡For multi-element trace analysis, cool the sample for several half-lives after irradiation before gamma counting. Short-lived interferences decay away, revealing longer-lived analyte lines.
💡Eu-151 (9200 barns) and Dy-164 (2650 barns) have enormous cross-sections. Even sub-nanogram quantities produce measurable activity at moderate flux.