Molière radius

The Molière radius is a characteristic constant of a material giving the scale of the transverse dimension of the fully contained electromagnetic showers initiated by an incident high energy electron or photon. By definition, it is the radius of a cylinder containing on average 90% of the shower's energy deposition. Two Molière radii contain 95% of the shower's energy deposition. It is related to the radiation length $X 0$ by the approximate relation $R M = 0.0265 X 0 (Z + 1.2)$ , where $Z$ is the atomic number.^[1] The Molière radius is useful in experimental particle physics in the design of calorimeters: a smaller Molière radius means better shower position resolution, and better shower separation due to a smaller degree of shower overlaps.

The Molière radius is named after German physicist Paul Friederich Gaspard Gert Molière (1909–64).^[2]

Molière radii for typical materials used in calorimetry

LYSO crystals: 2.07 cm^[3]
Lead tungstate crystals: 2.2 cm^[4]
Caesium iodide: 3.5 cm^[5]
Liquid krypton: 4.7 cm^[6]
Liquid argon: 9.04 cm^[7]
Earth's atmosphere at sea level: 79 m^[8]
Earth's atmosphere above ground: 91 m^[9]

References

^ Molière Radius Archived 2007-10-02 at the Wayback Machine
^ Phillip R. Sloan, Brandon Fogel, "Creating a Physical Biology: The Three-Man Paper and Early Molecular Biology" University of Chicago Press, 2011
^ Collaboration, PIONEER; et al. (2022). "Testing Lepton Flavor Universality and CKM Unitarity with Rare Pion Decays in the PIONEER experiment". arXiv:2203.05505 [hep-ex].
^ The CMS Collaboration (2006). "Chapter 1. Introduction". CMS Physics : Technical Design Report Volume 1: Detector Performance and Software. CERN. p. 14. ISBN 9789290832683. CMS has chosen lead tungstate scintillating crystals for its ECAL. These crystals have short radiation (X0 = 0.89 cm) and Moliere (2.2 cm) lengths, are fast (80% of the light is emitted within 25 ns) and radiation hard (up to 10 Mrad).
^ Atomic and nuclear properties of cesium iodide (CsI)
^ Fanti, V.; Barr, G.D.; Buchholz, P.; Cundy, D.; Doble, N.; Gatignon, L.; Gonidec, A.; Hallgren, B.; Kesseler, G.; Lacourt, A.; Laverrière, G.; Linser, G.; Norton, A.; Schinzel, D.; Seidl, W.; Stemmler, T.; Taureg, H.; Viehhauser, G.; Wahl, H.; Duclos, J.; Gianoli, A.; Martini, M.; Piemontese, L.; Savrié, M.; Kalinin, A.; Kekelidze, J.; Kozhevnikov, Y.; Coward, D.; Leber, F.; Cenci, P.; Lariccia, P.; Lubrano, P.; Pepe, M.; Calafiura, P.; Cerri, C.; Fantechi, R.; Gorini, B.; Laico, F.; Mannelli, I.; Marzulli, V.; Schiuma, D.; Debu, P.; Mazzucato, E.; Migliori, A.; Peyaud, B.; Turlay, R.; Kreutz, A.; Biino, C.; Ceccucci, A.; Palestini, S.; Griesmayer, E.; Markytan, M.; Neuhofer, G.; Pernicka, M.; Taurok, T.; Wulz, C.-E. (May 1994). "Performance of an electromagnetic liquid krypton calorimeter" (PDF). Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. 344 (3): 507–520. doi:10.1016/0168-9002(94)90871-0. Retrieved 1 December 2022.
^ Atomic and nuclear properties of materials: Liquid argon (Ar) (Ar)
^ Greisen, Kenneth (1960). "Cosmic Ray Showers". Annual Review of Nuclear Science. 10. Laboratory of Nuclear Studies, Cornell University, Ithaca, N. Y.: 71. Bibcode:1960ARNPS..10...63G. doi:10.1146/annurev.ns.10.120160.000431.
^ Pierre Auger Collaboration (2009). "Atmospheric effects on extensive air showers observed with the surface detector of the Pierre Auger observatory". Astroparticle Physics. 32 (2): 89–99. arXiv:0906.5497. Bibcode:2009APh....32...89P. doi:10.1016/j.astropartphys.2009.06.004.

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[1] Molière Radius Archived 2007-10-02 at the Wayback Machine

[:0-2] Phillip R. Sloan, Brandon Fogel, "Creating a Physical Biology: The Three-Man Paper and Early Molecular Biology" University of Chicago Press, 2011

[3] Collaboration, PIONEER; et al. (2022). "Testing Lepton Flavor Universality and CKM Unitarity with Rare Pion Decays in the PIONEER experiment". arXiv:2203.05505 [hep-ex].

[4] The CMS Collaboration (2006). "Chapter 1. Introduction". CMS Physics : Technical Design Report Volume 1: Detector Performance and Software. CERN. p. 14. ISBN 9789290832683. CMS has chosen lead tungstate scintillating crystals for its ECAL. These crystals have short radiation (X0 = 0.89 cm) and Moliere (2.2 cm) lengths, are fast (80% of the light is emitted within 25 ns) and radiation hard (up to 10 Mrad).

[5] Atomic and nuclear properties of cesium iodide (CsI)

[6] Fanti, V.; Barr, G.D.; Buchholz, P.; Cundy, D.; Doble, N.; Gatignon, L.; Gonidec, A.; Hallgren, B.; Kesseler, G.; Lacourt, A.; Laverrière, G.; Linser, G.; Norton, A.; Schinzel, D.; Seidl, W.; Stemmler, T.; Taureg, H.; Viehhauser, G.; Wahl, H.; Duclos, J.; Gianoli, A.; Martini, M.; Piemontese, L.; Savrié, M.; Kalinin, A.; Kekelidze, J.; Kozhevnikov, Y.; Coward, D.; Leber, F.; Cenci, P.; Lariccia, P.; Lubrano, P.; Pepe, M.; Calafiura, P.; Cerri, C.; Fantechi, R.; Gorini, B.; Laico, F.; Mannelli, I.; Marzulli, V.; Schiuma, D.; Debu, P.; Mazzucato, E.; Migliori, A.; Peyaud, B.; Turlay, R.; Kreutz, A.; Biino, C.; Ceccucci, A.; Palestini, S.; Griesmayer, E.; Markytan, M.; Neuhofer, G.; Pernicka, M.; Taurok, T.; Wulz, C.-E. (May 1994). "Performance of an electromagnetic liquid krypton calorimeter" (PDF). Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. 344 (3): 507–520. doi:10.1016/0168-9002(94)90871-0. Retrieved 1 December 2022.

[7] Atomic and nuclear properties of materials: Liquid argon (Ar) (Ar)

[8] Greisen, Kenneth (1960). "Cosmic Ray Showers". Annual Review of Nuclear Science. 10. Laboratory of Nuclear Studies, Cornell University, Ithaca, N. Y.: 71. Bibcode:1960ARNPS..10...63G. doi:10.1146/annurev.ns.10.120160.000431.

[9] Pierre Auger Collaboration (2009). "Atmospheric effects on extensive air showers observed with the surface detector of the Pierre Auger observatory". Astroparticle Physics. 32 (2): 89–99. arXiv:0906.5497. Bibcode:2009APh....32...89P. doi:10.1016/j.astropartphys.2009.06.004.

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