Moliere Radius Calculator
Find the Moliere radius Rm, the transverse radius that contains 90 percent of an electromagnetic shower's energy, used to size calorimeter cells.
🌊 What is the Moliere Radius Calculator?
This Moliere radius calculator finds Rm, the transverse (lateral) distance from an electromagnetic shower's axis that contains about 90% of its deposited energy, computed from the radiation length X0 and the Rossi critical energy Ec. It reports Rm in g/cm2 (density-independent) and in cm for the chosen material's density, plus a step-by-step breakdown of X0 and Ec.
Calorimeter designers use the Moliere radius to choose transverse cell sizes: a calorimeter segmented finer than Rm can separate two nearby showers (for example, the two photons from a boosted neutral pion decay), while segmentation coarser than Rm blurs them together. Detector groups routinely quote Rm alongside radiation length when comparing absorber choices for a new calorimeter design.
An interesting and easy-to-miss fact: unlike radiation length, which shrinks more than tenfold from a light element like silicon to a heavy one like lead, the Moliere radius only shrinks by roughly 50% over the same range. This happens because both X0 and the critical energy Ec fall with atomic number, and their ratio in the Rm formula partly cancels the two effects.
This calculator is useful for particle and detector physics students, calorimeter designers comparing absorber materials, and anyone studying electromagnetic shower development.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Lead
Example 2 - Tungsten
Example 3 - Silicon
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Moliere radius?
The Moliere radius Rm is the transverse distance from an electromagnetic shower's axis that contains about 90% of the shower's deposited energy. Calorimeter designers use it to choose the lateral cell size needed for good shower containment and to separate nearby particle showers.
What is the formula for the Moliere radius?
Rm = X0 x Es / Ec, where X0 is the radiation length, Es = 21.2052 MeV is a fixed scale energy, and Ec = 610/(Z+1.24) MeV is the Rossi critical energy for solids and liquids.
What is the Rossi critical energy?
The Rossi critical energy Ec is the electron energy at which ionization energy loss and bremsstrahlung (radiative) energy loss become equal. Below Ec, ionization dominates; above Ec, radiative loss dominates, which sets where an electromagnetic shower's multiplication slows down.
Why doesn't the Moliere radius shrink as fast as radiation length with atomic number?
Both X0 and Ec fall as Z increases, and their ratio partially cancels in the Rm formula. As a result, Rm only varies by roughly 50% from a light element like aluminum to a heavy one like lead, while X0 alone varies more than tenfold over the same range, a well known effect in calorimeter design.
How much of a shower's energy is contained within one Moliere radius?
About 90% of an electromagnetic shower's total deposited energy lies within a cylinder of radius one Moliere radius around the shower axis, and about 95% within two Moliere radii, standard rules of thumb used to size calorimeter transverse segmentation.
Why is a small Moliere radius desirable in calorimeter design?
A small Moliere radius means the shower stays narrow, so the calorimeter can use finer transverse segmentation to separate two nearby particles (like two photons from a boosted pion decay) without their showers overlapping, improving position and energy resolution.
What is the difference between Rm in g/cm2 and Rm in cm?
Rm in g/cm2 is density-independent, following directly from X0 and Ec. Rm in cm divides that by the material's density to give the actual physical radius, the number a mechanical calorimeter designer would use to size real detector cells.
What is the Moliere radius of lead?
For lead (Z=82, A=207.2), the Moliere radius works out to about 18.26 g/cm2, or about 1.61 cm using lead's density of 11.35 g/cm3, a standard reference value in electromagnetic calorimeter design.
How is the Moliere radius different from the radiation length?
Radiation length X0 describes the longitudinal (depth) development of a shower, how far it takes an electron to lose most of its energy to bremsstrahlung. The Moliere radius Rm describes the transverse (lateral) spread of the shower around its axis, both are needed to fully size a calorimeter.
Which materials are included as presets?
Silicon, aluminum, iron, copper, tungsten, and lead are built in with standard Z, A, and density values, or you can enter a custom material's Z, A, and density directly.