Calculator Inputs
Formula Used
Centered circular detector
Ω = 2π(1 - d / √(d² + r²))
Here, Ω is solid angle, r is detector radius, and d is normal distance.
Off-axis circular detector
Ω = ∬ z / ((x - ρ)² + y² + z²)^(3/2) dA
Here, ρ is lateral offset. The tool integrates over the circular detector face.
Counting estimate
Counts = Activity × Probability × Efficiency × Time × Ω / 4π
This assumes isotropic emission and ignores absorption, shielding, scattering, and dead time.
Small angle approximation
Ω ≈ A cos(θ) / s²
This is best for distant detectors. Close geometries should use the exact or integrated result.
How to Use This Calculator
- Enter the active circular detector size.
- Select whether the size is radius or diameter.
- Enter the normal source-to-plane distance.
- Add lateral offset if the source is not centered.
- Select a unit used by all geometry fields.
- Add activity, probability, efficiency, and time if count estimates are needed.
- Choose detailed integration for off-axis precision.
- Press the calculate button and review the result above the form.
- Use CSV or PDF export for reports.
Example Data Table
| Case | Radius cm | Distance cm | Offset cm | Solid angle sr | Full sphere % |
|---|---|---|---|---|---|
| Centered small detector | 2.5 | 10 | 0 | 0.1876002 | 1.492875 |
| Off-axis laboratory setup | 5 | 20 | 3 | 0.18204069 | 1.448634 |
| Distant axial detector | 10 | 50 | 0 | 0.12201521 | 0.970966 |
| Compact close geometry | 1 | 5 | 1 | 0.11547538 | 0.918924 |
Understanding Circular Detector Solid Angle
Solid angle tells how large a detector appears from a source. It is measured in steradians. A full sphere has four pi steradians. A flat circular detector covers only part of that sphere. The covered part depends on radius, distance, and alignment.
Why Geometry Matters
When the source is on the detector axis, the calculation is exact. A larger radius increases coverage. A shorter distance also increases coverage. The angle can grow quickly when the detector is close. This is why close geometry needs a solid angle correction. Simple area over distance squared may become inaccurate.
Off Axis Sources
Real experiments are often not perfectly centered. A source may sit away from the detector axis. Then each small area of the circular face sees a different distance and angle. The calculator handles that case with numerical integration. It divides the detector into rings and angular slices. Each slice adds its contribution. Higher quality settings use more slices. They take longer but give smoother results.
Useful Measurement Outputs
The steradian value is the main result. The full sphere fraction is also useful. It estimates the part of isotropic emission reaching the detector face. Multiply that fraction by activity, emission probability, efficiency, and time to estimate counts. This assumes no absorption, shielding, dead time, or scattering losses. Those effects need separate corrections.
Small Angle Approximation
For distant sources, the detector appears small. Then the solid angle is close to projected area divided by distance squared. This approximation is fast. It helps check the exact result. It is not reliable when the detector is near the source. It also becomes weaker when the source is far off axis.
Practical Advice
Use consistent units. Measure radius from the detector center to its active edge. Use diameter only when that value is known. Enter the normal distance from source to detector plane. Enter lateral offset when the source is not centered. Compare the exact, numerical, and approximate outputs. Large differences warn that geometry is sensitive. In that case, use careful measurements and repeat the calculation. Record every setup note, because small shifts can change close range detection results during calibration.
FAQs
What is solid angle?
Solid angle measures how large an object appears from a point in three-dimensional space. It is measured in steradians. A full sphere equals 4π steradians.
When should I use the exact axial formula?
Use it when the source is centered on the detector axis. The detector face should be perpendicular to the source direction. This gives a fast and exact circular disk result.
What does lateral offset mean?
Lateral offset is the sideways distance between the source projection and detector center. Use zero when the source is perfectly centered. Use a positive value for off-axis setups.
Why is numerical integration used?
Off-axis circular geometry has changing distance and angle across the detector face. Numerical integration adds many small surface contributions to estimate the total solid angle.
Is the small angle approximation always accurate?
No. It works best when distance is much larger than detector radius. It can fail for close detectors, large detectors, or strong off-axis geometry.
What is full sphere fraction?
It is the solid angle divided by 4π. For isotropic emission, it estimates the emission fraction that reaches the detector face before other losses.
Can this estimate detected counts?
Yes. It multiplies activity, emission probability, intrinsic efficiency, time, and full sphere fraction. It does not include attenuation, dead time, pileup, or scattering.
Which unit should I choose?
Choose any listed unit, but keep all geometry entries consistent. The calculator converts the selected unit internally before applying the formulas.