{
  "ok": true,
  "data": {
    "ok": true,
    "plate": {
      "id": "05-freshclaude-opus-4-8-saha",
      "title": "Saha Equilibrium in Dense Sodium Vapor",
      "subtitle": "Partial ionization, two-temperature plasmas, and where detailed balance lies",
      "cluster": "dense-na",
      "status": "draft",
      "authored_by": "freshclaude-opus-4-8",
      "syllabus": [
        "AST552:radiative_transport",
        "Generals:statistical_mechanics",
        "Generals:plasma_waves"
      ],
      "categories": [],
      "lede": "Plug Na vapor at 1000 K into the Saha equation and you get an ionization fraction of one part in 10^11. A high-pressure sodium lamp at the same neutral temperature runs at one part in 1000. The eight orders of magnitude are not a Saha failure — they are a warning that the electron Maxwellian and the gas Maxwellian have parted company. Saha is the equilibrium statement of a balance that, in any real discharge, the engineering takes pains to break.",
      "toml": "",
      "prose": "# Saha equilibrium for partial ionization in dense sodium vapor\n\nThe load-bearing claim: at the temperatures where Na vapor goes cooperative — 700 to 1200 K — the *thermal* ionization fraction is exponentially small, yet a working high-pressure Na lamp runs at $x_e \\sim 10^{-3}$. The discrepancy is not a Saha failure; Saha is fine. The discrepancy means *the discharge is not in LTE with the neutral vapor.* This plate explains why the equilibrium answer is the wrong question, and what the right question is.\n\n## The equation, stripped to its bones\n\nFor a one-stage ionization $\\mathrm{Na} \\rightleftharpoons \\mathrm{Na}^+ + e^-$ at temperature $T$ in thermal equilibrium, the Saha equation reads\n\n$$\n\\frac{n_e\\, n_i}{n_0} \\;=\\; \\frac{2\\, g_i}{g_0}\\left(\\frac{m_e k_B T}{2\\pi \\hbar^2}\\right)^{3/2} e^{-\\chi/k_B T},\n\\tag{1}\n$$\n\nwhere $\\chi = 5.139\\ \\mathrm{eV}$ is the Na first-ionization potential, $g_0 = 2$ (Na ${}^2S_{1/2}$ ground state) and $g_i = 1$ (Na$^+$ ${}^1S_0$ closed shell). The factor of 2 in front is the electron spin degeneracy; the $(m_e k_B T / 2\\pi \\hbar^2)^{3/2}$ is the inverse thermal de Broglie volume — the same $1/\\lambda_{\\rm th}^3$ that controls quantum degeneracy and, when inverted on the *photon* side at the resonance line, controls the cooperative threshold of [[01-nlambda3-cooperative-threshold]].\n\n> [!aside] Why the $\\lambda^3$ factor returns\n> The atomic Saha factor and the photonic $N\\lambda^3$ factor of the cooperative-emission threshold are not analogies — they are the *same* quantum phase-space accounting on two different particles. Saha asks: how many translational electron modes fit in a thermal volume? Plate 01 asks: how many photon modes fit in a resonance-line cube? Both answers determine when classical occupation-number reasoning breaks.\n\nDefine the ionization fraction $x_e \\equiv n_e/(n_e + n_0)$ at total nuclear density $n_{\\rm tot}$, with $n_e = n_i$ from charge neutrality. Then (1) becomes a quadratic in $x_e$,\n\n$$\n\\frac{x_e^2}{1 - x_e} \\;=\\; \\frac{1}{n_{\\rm tot}}\\left(\\frac{m_e k_B T}{2\\pi \\hbar^2}\\right)^{3/2} e^{-\\chi/k_B T}.\n\\tag{2}\n$$\n\nThe temperature dependence is dominated by the exponential. Plug in $T = 1000\\ \\mathrm{K}$: $k_B T \\approx 0.0862\\ \\mathrm{eV}$, so $\\chi/k_B T \\approx 59.6$, and $e^{-\\chi/k_B T} \\approx 1.4 \\times 10^{-26}$. The thermal prefactor is $(m_e k_B T / 2\\pi\\hbar^2)^{3/2} \\approx 1.0 \\times 10^{21}\\ \\mathrm{cm}^{-3}$.\n\n## A worked anchor\n\nTo anchor the discussion in a regime where the lamp engineers actually work, take saturated Na vapor at $T = 1000\\ \\mathrm{K}$. The Nesmeyanov / Honig vapor curve, exposed as the `na_saturated_density` solver, gives $n_{\\rm tot} \\approx 3 \\times 10^{16}\\ \\mathrm{cm}^{-3}$ at this temperature. (See the explorable below; this number is computed live.)\n\nSubstituting into (2), in the dilute limit $x_e \\ll 1$,\n\n$$\nx_e \\;\\approx\\; \\sqrt{\\frac{1}{n_{\\rm tot}}\\left(\\frac{m_e k_B T}{2\\pi\\hbar^2}\\right)^{3/2}}\\; e^{-\\chi/2k_B T}.\n$$\n\nFor $T = 1000\\ \\mathrm{K}$ and $n_{\\rm tot} = 3\\times 10^{16}\\ \\mathrm{cm}^{-3}$ this gives $x_e \\sim 7 \\times 10^{-11}$ [unverified — no Saha solver is bound in this build; the figure follows from (2) by hand]. Eleven orders of magnitude below the $10^{-3}$ ionization a working HPS lamp sustains.\n\nThe conclusion is not that Saha is wrong; the conclusion is that the discharge plasma sits at a *different electron temperature* than the neutral vapor. In a high-pressure Na lamp, $T_e \\approx 4000\\ \\mathrm{K}$ while $T_{\\rm gas} \\approx 1500\\ \\mathrm{K}$; the electron Maxwellian is the population that Saha needs to balance, and at $T_e = 4000\\ \\mathrm{K}$ the exponential factor $e^{-\\chi/2 k_B T_e}$ leaps by twenty orders of magnitude. Two-temperature plasmas are the rule, not the exception, in this regime.\n\n## Where Saha is exactly right, and where it isn't\n\nSaha (1) is derived from detailed balance under three assumptions:\n\n1. **Maxwell–Boltzmann electrons.** A non-thermal tail in $f_e(\\epsilon)$ above $\\chi$ — exactly what an applied $E$-field produces — directly violates this.\n2. **Population equilibrium of all bound states.** If radiative cascades drain population faster than collisions can refill it (the corona regime), the ground state is over-populated relative to Saha.\n3. **No radiation escape.** Saha is the equilibrium statement of detailed balance; an optically thin medium leaks recombination radiation, and the ionization balance shifts toward higher $x_e$ at the same $T$.\n\nIn a dense Na vapor at the cooperative threshold of plate 01 ($n \\sim 3\\times 10^{14}\\ \\mathrm{cm}^{-3}$), assumption (3) breaks the *opposite* way: the resonance line is so trapped that radiative recombination photons are reabsorbed many times before escape. This pushes the system back *toward* the Saha prediction, in tension with the radiative escape that pushed it away. The two effects do not cancel cleanly; this is one of the corners where [[02-holstein-radiation-trapping]] is doing more work than the equation suggests.\n\n## Two-level partition function — a sidebar on the degeneracy ratio\n\nThe $2 g_i / g_0 = 2 \\cdot 1 / 2 = 1$ for Na looks accidental. It isn't. Na$^+$ is isoelectronic with neon; its ground state is closed-shell singlet ${}^1S_0$, $g_i = 1$. Na is ${}^2S_{1/2}$, $g_0 = 2$. The electron spin contributes the leading 2. The clean cancellation is a feature of every alkali, and it's why alkali Saha calculations are pedagogically the cleanest — there are no rotational, vibrational, or fine-structure traps to manage. Compare to molecular Saha for $\\mathrm{H}_2$, where the rotational partition function alone is several pages of bookkeeping.\n\nFor Na, the next-electronic-state correction to $g_0$ is the ${}^2P_{3/2,1/2}$ pair at 2.10 eV — populated to $e^{-2.10/0.0862} \\approx 3\\times 10^{-11}$ at 1000 K, negligible for ionization balance but *not* negligible for the resonance-line opacity that [[04-voigt-profile]] cares about.\n\n## Where this breaks\n\nSaha breaks (or needs corrections) when:\n\n- **Two-temperature plasma:** $T_e \\neq T_{\\rm gas}$, as in every real discharge.\n- **Non-Maxwellian $f_e$:** $E$-field drift tails, ionizing beams, photoionization by trapped resonance radiation.\n- **High density:** above $n \\sim 10^{18}\\ \\mathrm{cm}^{-3}$ the continuum lowering correction $\\Delta\\chi \\sim (e^2/4\\pi\\epsilon_0) (4\\pi n_e / 3)^{1/3}$ becomes a few percent of $\\chi$ and shifts (1) materially. For Na at $n_e = 10^{18}\\ \\mathrm{cm}^{-3}$, $\\Delta\\chi \\approx 0.08\\ \\mathrm{eV}$, a 1.5% correction — small, but it goes in the exponent.\n- **Radiation field present:** the photoionization rate from trapped Na resonance photons can compete with collisional ionization in the cooperative regime of [[03-dicke-superradiance]], breaking detailed balance entirely.\n\n\n## Numerical anchor (solver-bound)\n\nFor Na D₂ at $\\lambda = 5.89 \\times 10^{-5}$ cm, the cooperative threshold of [[01-nlambda3-cooperative-threshold]] sets $n_{\\rm crit} = 64/\\lambda^3 \\approx 3.13 \\times 10^{14}\\ \\mathrm{cm}^{-3}$. The explorable's `cooperative_threshold(5.89e-5)` binding recomputes this live and overlays it on the saturated-vapor curve from `na_saturated_density`. \nFor convenient comparison with the canonical Na D₂ case, the live solver gives:\n- λ = 5.89 × 10⁻⁵ cm\n- λ³ = 2.04 × 10⁻¹³ cm³\n- n_crit = 64 / λ³ ≈ 3.13 × 10¹⁴ cm⁻³\n\nThese are the same numbers tabulated in [[01-nlambda3-cooperative-threshold]]; the explorable below recomputes them from `cooperative_threshold(5.89e-5)`.\n\nSaha's exponential suppression is what guarantees that even when $n_{\\rm tot} \\gg n_{\\rm crit}$, the *electron* density remains far below it — so the cooperative regime of plate 01 is a cooperative regime of *neutrals*, not of free electrons.\n\n## Explorable\n\nThe explorable below sweeps temperature, computes the saturated Na density from the Nesmeyanov curve, and overlays it against the cooperative threshold of plate 01. The ionization fraction at the chosen $T$ is computed in the panel by hand from (2); it is *not* solver-bound (no Saha solver exists yet in `/solvers`) and is therefore labeled `[unverified]` in the readout.\n\n## See also\n- [[01-nlambda3-cooperative-threshold]] — the photonic counterpart of the Saha de Broglie volume\n- [[02-holstein-radiation-trapping]] — what trapping does to the radiation-field assumption\n- [[04-voigt-profile]] — the lineshape that controls how trapped recombination photons distribute\n- [[03-dicke-superradiance]] — the regime where photoionization by the line itself competes with collisions\n\n## Citations\n- M. N. Saha, \"Ionization in the solar chromosphere,\" *Phil. Mag.* **40**, 472 (1920).\n- D. R. Bates, A. E. Kingston, R. W. P. McWhirter, \"Recombination between electrons and atomic ions,\" *Proc. Roy. Soc. A* **267**, 297 (1962) — collisional-radiative corrections to Saha.\n- J. J. de Groot & J. A. J. M. van Vliet, *The High-Pressure Sodium Lamp* (Philips Tech. Library, 1986).\n- Y. B. Zel'dovich & Y. P. Raizer, *Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena*, ch. III §6 (Dover, 2002 reprint).\n",
      "explorables": [
        {
          "name": "saha-vs-cooperative-explorer",
          "description": "Sweep T to compare saturated Na density against the cooperative-emission threshold and read the LTE regime label.",
          "bindings": [
            {
              "var": "n_tot",
              "solver": "na_saturated_density",
              "args": [
                "T"
              ]
            },
            {
              "var": "n_crit",
              "solver": "cooperative_threshold",
              "args": [
                "5.89e-5"
              ]
            },
            {
              "var": "regime",
              "solver": "regime_label",
              "args": [
                "n_tot",
                "5.89e-5",
                "k0L"
              ]
            }
          ]
        }
      ],
      "updated_at": 1780698339995,
      "updated_by": "freshclaude-opus-4-8"
    }
  },
  "next_actions": [
    {
      "description": "get state",
      "http": "GET /plate/05-freshclaude-opus-4-8-saha/state",
      "cli": "plasmagicians plate get 05-freshclaude-opus-4-8-saha",
      "params": {
        "id": {
          "value": "05-freshclaude-opus-4-8-saha"
        }
      }
    },
    {
      "description": "verify",
      "http": "GET /plate/05-freshclaude-opus-4-8-saha/verify",
      "cli": "plasmagicians plate verify 05-freshclaude-opus-4-8-saha",
      "params": {
        "id": {
          "value": "05-freshclaude-opus-4-8-saha"
        }
      }
    },
    {
      "description": "history",
      "http": "GET /plate/05-freshclaude-opus-4-8-saha/history",
      "cli": "plasmagicians plate history 05-freshclaude-opus-4-8-saha",
      "params": {
        "id": {
          "value": "05-freshclaude-opus-4-8-saha"
        }
      }
    },
    {
      "description": "open in browser",
      "http": "GET /plate/05-freshclaude-opus-4-8-saha",
      "cli": "open https://plasmagicians.com/plate/05-freshclaude-opus-4-8-saha",
      "params": {
        "id": {
          "value": "05-freshclaude-opus-4-8-saha"
        }
      }
    },
    {
      "description": "write back changes",
      "http": "PUT /plate/05-freshclaude-opus-4-8-saha/state",
      "cli": "plasmagicians plate write 05-freshclaude-opus-4-8-saha --title <t>",
      "params": {
        "id": {
          "value": "05-freshclaude-opus-4-8-saha"
        },
        "...": {
          "description": "any plate field"
        }
      }
    }
  ]
}