Calculate half-value layer, linear attenuation coefficient, and shielding thickness from material, photon energy, or measured intensity drop.
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Half Value Layer Formula
The half-value layer (HVL) is the thickness of a material that reduces the intensity of a narrow beam of photons to half its initial value. It is tied directly to the linear attenuation coefficient μ.
HVL = ln(2) / μ
Rearranged for μ:
μ = ln(2) / HVL
To find the shielding thickness needed to reduce intensity from I₀ to I, the calculator uses the number of HVLs:
N = log₂(I₀ / I) x = N × HVL
To extract HVL from a measured beam attenuation through a known thickness x:
μ = ln(I₀ / I) / x HVL = ln(2) / μ
- HVL — half-value layer thickness (cm, mm, m, in)
- μ — linear attenuation coefficient (cm⁻¹)
- I₀ — initial beam intensity or dose rate
- I — transmitted intensity or dose rate after the absorber
- x — absorber thickness in the path of the beam
- N — number of half-value layers
- TVL — tenth-value layer = ln(10) / μ ≈ 3.32 × HVL
These equations assume a narrow, monoenergetic beam and good geometry. Real shielding usually involves a polyenergetic spectrum, scatter, and beam buildup, so a buildup factor or safety margin is normally added on top of the result.
Reference Values
Approximate narrow-beam half-value layers for common shielding materials at gamma and X-ray energies. Values in centimeters.
| Energy | Lead | Steel | Concrete | Aluminum | Water |
|---|---|---|---|---|---|
| 100 keV | 0.011 | 0.24 | 1.78 | 1.51 | 4.05 |
| 200 keV | 0.065 | 0.64 | 2.31 | 2.14 | 5.10 |
| 500 keV | 0.38 | 1.05 | 3.47 | 3.05 | 7.18 |
| 662 keV (Cs-137) | 0.57 | 1.21 | 3.96 | 3.50 | 8.08 |
| 1.25 MeV (Co-60) | 1.04 | 1.65 | 5.29 | 4.68 | 10.97 |
How transmitted fraction relates to the number of HVLs:
| Number of HVLs | Transmitted | Attenuation |
|---|---|---|
| 1 | 50% | 2× |
| 3.32 (1 TVL) | 10% | 10× |
| 5 | 3.13% | 32× |
| 6.64 (2 TVL) | 1% | 100× |
| 10 | 0.098% | 1024× |
Worked Example
A Cs-137 source produces 200 µSv/h at a workstation. You need to reduce the dose rate to 5 µSv/h using lead. From the table, HVL for lead at 662 keV is about 0.57 cm.
Number of HVLs needed:
N = log₂(200 / 5) = log₂(40) ≈ 5.32
Required lead thickness:
x = 5.32 × 0.57 cm ≈ 3.03 cm
For a real installation, add a buildup factor or safety margin since narrow-beam HVLs underestimate thickness for broad beams.
FAQ
Is HVL the same for every isotope? No. HVL depends on photon energy and the absorber material. Higher energy photons have larger HVLs in the same material.
Why does the calculator say “narrow beam”? The HVL formula assumes scattered photons leave the beam path. In a broad geometry, scattered photons reach the detector and effective attenuation is lower than predicted.
What is the difference between HVL and TVL? One HVL cuts intensity to 50%. One tenth-value layer (TVL) cuts it to 10%. TVL ≈ 3.32 × HVL.
Can I use HVL for beta or neutron shielding? Not directly. HVL is defined for exponentially attenuated photon beams. Beta particles have a finite range, and neutron attenuation depends on hydrogen content and energy spectrum.
