Spitzer resistivity plasma Coulomb calculator

The Spitzer resistivity calculator determines the electrical resistivity η of a fully ionized plasma dominated by electron-ion Coulomb collisions, per the formulation by Lyman Spitzer Jr. and Richard Härm 1953 Physical Review 89, 977 "Transport Phenomena in a Completely Ionized Gas" Princeton. The practical formula: η = 5.2×10⁻⁵ · Z · ln(Λ) / T_e^(3/2) [Ω·m], with T_e electron temperature in eV, Z ion charge number, ln(Λ) Coulomb logarithm typically 10-20. Exact form: η_Spitzer = (m_e^(1/2) · Z · e² · ln(Λ)) / (3 · ε₀² · (k_B·T_e)^(3/2)) · (1/(4π)), with m_e = 9.109×10⁻³¹ kg electron mass, e = 1.602×10⁻¹⁹ C, ε₀ = 8.854×10⁻¹² F/m, k_B = 1.381×10⁻²³ J/K. Physics: current in a plasma is carried by electrons drifting through the ion fluid; resistivity arises from small-angle Coulomb collisions integrated on impact parameter from b_min (closest approach) to λ_D (Debye length), defining ln Λ = ln(λ_D / b_min). For hydrogen plasma (Z=1) at T_e = 1 keV: η ≈ 10⁻⁸ Ω·m, comparable to copper; at T_e = 10 keV (ITER): η ≈ 3×10⁻¹⁰ Ω·m, ten times better than copper. 4 plasma categories:

• Tokamak fusion plasma T_e = 1-25 keV, η = 10⁻¹⁰-10⁻⁸ Ω·m (ITER, DTT, FTU, JET)
• Solar/stellar plasma corona T_e ≈ 100-200 eV, η ≈ 10⁻⁶ Ω·m (Solar Orbiter, SOHO)
• Astrophysical plasma ICM cluster T_e ≈ keV (XMM-Newton, Athena)
• Lab low-T plasma T_e ≈ 1-10 eV, η ≈ 10⁻⁴ Ω·m. Fundamental physics applications:

• Tokamak current diffusion timescale τ_R = μ₀·a²/η (a = minor radius) → ITER a = 2 m, η = 3×10⁻¹⁰ → τ_R ≈ 1000 s
• Solar corona resistivity in MHD coronal heating models
• Magnetic reconnection Sweet-Parker regime v_rec ∝ η^(1/2)
• Pulsar magnetosphere and accretion disks
• Plasma diagnostics via loop voltage V_loop = η·j·2π·R
• Inertial Confinement Fusion (ICF) NIF Lawrence Livermore + Asterix München. Italian applications:

• ENEA Frascati 1955 + DTT Divertor Tokamak Test 2028 EUROfusion R=2.19 m, T_e=10 keV η=3×10⁻¹⁰ Ω·m, B=6 T, I_p=5.5 MA, liquid tungsten-tin divertor
• FTU Frascati Tokamak Upgrade 1989-2020 ENEA R=0.93 m, T_e up to 13 keV, η Spitzer baseline LH lower hybrid current drive studies
• ITER Cadarache France 2025 first plasma + Ansaldo Nucleare Genoa engineering Q=10 fusion gain, T_e=20 keV η=2×10⁻¹⁰ Ω·m, τ_R≈1000 s
• INAF IAPS Rome + Solar Orbiter ESA 2020 ASI 25% solar corona T_e=10⁶ K=86 eV η=10⁻⁶ Ω·m reconnection
• INFN-LNS Catania cyclotron K800 ECR plasma ion source 14.5 GHz
• Cineca Bologna Marconi100 30 PFlops + Leonardo 250 PFlops 2023 MHD plasma simulations JOREK BOUT++ NIMROD
• AGILE ASI 2007 + INFN Frascati LNF DAFNE 1999 gamma astrophysics + e+e- collider

Italian Spitzer plasma resistivity tables

(Source: Spitzer & Härm 1953 Phys. Rev. 89, 977 + Braginskii 1965 Reviews Plasma Physics + ENEA Frascati 1955 + ITER Organization 2025 + INAF Rome 2002).

Italian Spitzer resistivity calculation examples

• ITER Cadarache France 2025 + Ansaldo Nucleare Genoa engineering: deuterium-tritium plasma, T_e = 10 keV = 10⁴ eV, Z = 1, ln Λ ≈ 17. η = 5.2×10⁻⁵ · 1 · 17 / (10⁴)^1.5 = 8.8×10⁻¹⁰ Ω·m. Conductivity σ = 1/η = 1.1×10⁹ S/m (≈18 times copper 5.96×10⁷). Current diffusion timescale τ_R = μ₀·a²/η = 4π×10⁻⁷·(2)²/8.8×10⁻¹⁰ ≈ 5700 s ≈ 1.6 hours
• DTT Frascati ENEA EUROfusion 2028: T_e = 10 keV, Z = 1, ln Λ = 17. η = 8.8×10⁻¹⁰ Ω·m identical to ITER. R = 2.19 m, a = 0.7 m, B_T = 6 T, I_p = 5.5 MA, liquid tungsten-tin divertor
• FTU Frascati Tokamak Upgrade 1989-2020 ENEA: T_e = 2 keV, Z = 1, ln Λ = 15. η = 5.2×10⁻⁵ · 15 / (2000)^1.5 = 8.7×10⁻⁹ Ω·m. R = 0.93 m, B_T = 8 T, I_p = 1.6 MA, lower hybrid current drive 8 GHz studies
• Solar Orbiter ESA 2020 ASI 25% INAF IAPS Rome solar corona: T_e = 10⁶ K = 86 eV, Z = 1, ln Λ = 18. η = 5.2×10⁻⁵ · 18 / (86)^1.5 = 1.2×10⁻⁶ Ω·m. Sweet-Parker reconnection v_rec/v_A = (η/μ₀·v_A·L)^0.5
• Lab arc plasma INFN-LNS Catania ECR ion source 14.5 GHz: T_e = 1 eV (cold non-fully-ionized plasma), Z = 1, ln Λ = 10. η = 5.2×10⁻⁵ · 10 / 1^1.5 = 5.2×10⁻⁴ Ω·m (10⁴× copper)
• ICM Coma cluster XMM-Newton ESA RGS Padua-Bologna: T_e = 8 keV, Z ≈ 1 (H/He plasma), ln Λ ≈ 40 (dilute plasma large Debye λ_D). η = 5.2×10⁻⁵ · 40 / (8000)^1.5 = 2.9×10⁻⁹ Ω·m. Sweet-Parker reconnection time 10⁹-10¹⁰ years > Hubble age
• Pulsar Crab magnetosphere INAF AGILE ASI 2007 + Fermi LAT: ultra-relativistic e+e- plasma T ≈ 1 MeV = 10⁶ eV, η ≈ 3×10⁻¹⁵ Ω·m (Spitzer non-relativistic limit, Spitzer-Lorentz corrected ×5)

Italian plasma physics: ENEA INFN INAF

Italian plasma physics has deep roots: ENEA Frascati Research Center 1955 (Comitato Nazionale Energia Nucleare CNEN, Director Felice Ippolito): FT Frascati Tokamak 1976 R=0.83 m B=10 T first Italian high-field tokamak; FTU Frascati Tokamak Upgrade 1989-2020 R=0.93 m B=8 T I_p=1.6 MA, T_e up to 13 keV, lower hybrid current drive 8 GHz, ECRH 140 GHz 1.5 MW; Frascati Neutron Generator FNG DT 14 MeV source for fusion materials qualification

• DTT Divertor Tokamak Test 2028 EUROfusion Frascati: R=2.19 m, a=0.7 m, B_T=6 T, I_p=5.5 MA, P_aux=45 MW (NBI 10+ECRH 32+ICRH 3), innovative liquid tungsten-tin divertor, plasma exhaust target solved 100% for DEMO 2050
• ITER Cadarache 2025-2035: 8% Italian contribution via Fusion for Energy F4E, Ansaldo Nucleare Genoa engineering vacuum vessel + TF magnets, OCEM Power Modena power supply, ASG Superconductors Genoa NbTi cables, Walter Tosto Chieti vessel components
• INAF Italian National Institute of Astrophysics Rome 2002: IAPS Rome 1965 space plasma physics + Solar Orbiter ESA 2020 ASI 25% (10 instruments, 4 Italian) in-situ solar corona; OAR Rome 1786; OACT Catania-Etna 1880 solar magnetosphere; OAPa Palermo 1790 X-ray INTEGRAL; Trieste 1898 Planck CMB SZ
• INFN Italian National Institute of Nuclear Physics 1951: LNF Frascati 1957 DAFNE 1999 e+e- φ-factory KLOE-2; LNS Catania 1976 superconducting cyclotron K800 + ECR ion source 14.5 GHz plasma sources protontherapy CATANA; LNGS Gran Sasso 1987 XENONnT + DarkSide-50 dark matter underground; LNL Legnaro 1968 XTU tandem + ALPI superconducting linac
• Cineca Casalecchio Bologna 1969: Marconi100 30 PFlops + Leonardo HPC pre-exascale 250 PFlops 2023 5th TOP500 worldwide MHD plasma simulations JOREK BOUT++ NIMROD GENE Eulerian Vlasov-Poisson tokamak edge turbulence ELM mitigation
• Carlo Rubbia 1984 Nobel CERN UA1 + Riccardo Giacconi 2002 Nobel X-ray astronomy founder + Enrico Fermi 1938 Nobel Rome neutrons
• Plasma physics universities: Sapienza Rome 1303 Physics Department + ICRA 1985 Remo Ruffini; Politecnico Milan 1863 Energy Department fusion; Politecnico Turin 1859 NEMO neutrons; Padua 1222 RFX Reversed Field eXperiment 1986 + neutral beam injector NBI ITER prototype; Pisa 1343 Scuola Normale 1810; Trieste SISSA 1978 + ICTP Abdus Salam 1964 plasma astrophysics; IIT Genoa 2003 quantum plasma; INRIM Turin 1923 metrology
• PNRR Mission 2 green hydrogen €3.2B + Mission 4 research €6.9B including plasma fusion initiatives. R&D tax bonuses 2025-2026: R&D Tax Credit L. 160/2019 art. 1 c. 200 up to 25% (max €10M/year); Patent Box L. 178/2020 50% deduction patents software design know-how; Industry/Transition 5.0 Tax Credit DL 39/2024 art. 38 up to 35% green/digital investments; Capital Goods 4.0 L. 178/2020 up to 50%; FIT MIMIT 2024 industrial research fund; ISI INAIL 2024 €100M lab safety. Disclaimer: the calculator provides Spitzer baseline η without neoclassical corrections (Hirshman-Sigmar 0.51 factor for Z=1 trapped particles), or non-Spitzer collisional → collisionless transition regime (parameter ν*)

Updated July 2026