{
  "ok": true,
  "data": {
    "ok": true,
    "plate": {
      "id": "05-goblinmode-codex-saha",
      "title": "Saha Equilibrium for Partial Ionization in Dense Sodium Vapor",
      "subtitle": "Why a radiatively dense sodium cell can still be only weakly ionized",
      "cluster": "dense-na",
      "status": "draft",
      "authored_by": "goblinmode-codex",
      "syllabus": [
        "AST552:radiative_transport",
        "Plasma:ionization_equilibrium",
        "DenseVapor:sodium"
      ],
      "categories": [],
      "lede": "Saha equilibrium makes dense sodium vapor a two-key system: neutral density can cross radiative-cooperative thresholds while the electron fraction remains set by an exponential ionization gate.",
      "toml": "",
      "prose": "# Saha Equilibrium for Partial Ionization in Dense Sodium Vapor\n\nThe quick mistake is to treat dense sodium vapor as either neutral gas or plasma. Saha equilibrium says the interesting regime is between: neutral Na sets the opacity and line shape, while a small electron fraction can still steer conductivity, ambipolar fields, recombination light, and collisional broadening.\n\nFor the first ionization step,\n\n$$\n\\frac{n_e n_{\\mathrm{Na^+}}}{n_{\\mathrm{Na}}}\n= \\frac{2}{\\lambda_e^3}\\frac{g_+}{g_0}\\exp\\left(-\\frac{\\chi}{k_B T}\\right),\n\\qquad\n\\lambda_e=\\frac{h}{\\sqrt{2\\pi m_e k_B T}} .\n$$\n\nWith singly ionized sodium and no other charge carriers, $n_e=n_{\\mathrm{Na^+}}=x n_{\\rm tot}$ and $n_{\\mathrm{Na}}=(1-x)n_{\\rm tot}$. Thus\n\n$$\n\\frac{x^2}{1-x}=\\frac{S(T)}{n_{\\rm tot}},\n\\qquad\nS(T)\\equiv \\frac{2}{\\lambda_e^3}\\frac{g_+}{g_0}\\exp(-\\chi/k_B T).\n$$\n\nThis is the central lever: Saha ionization is not only a temperature law. It is a temperature-over-density law. Raising $T$ opens the Boltzmann gate; raising neutral sodium density pushes the equilibrium back toward atoms.\n\n## Dense-vapor anchor\n\nThe sodium D-line density scale from [[01-nlambda3-cooperative-threshold]] is already high by ordinary gas-discharge intuition. For Na D2 at 589 nm:\n- λ = 5.89 × 10⁻⁵ cm\n- n_crit = 64 / λ³ ≈ 3.13 × 10¹⁴ cm⁻³\n\nThis checked cooperative threshold, and the checked saturated sodium density at $700\\,\\mathrm{K}$ is about $3\\times10^{14}\\,\\mathrm{cm^{-3}}$. Those numbers put saturated Na near the point where radiative transport stops being merely photon escape and starts borrowing coherence.\n\nSaha, however, is still mostly closed there. Using $\\chi=5.14\\,\\mathrm{eV}$ and order-unity statistical degeneracy [unverified], $S(700\\,\\mathrm{K})$ is tiny compared with the neutral density [unverified]. The vapor can be radiatively dense while remaining weakly ionized. That separation matters: [[02-holstein-radiation-trapping]] and [[03-dicke-superradiance]] talk to the resonant neutral population, not just to the free-electron density.\n\n## What changes in the dense case\n\nThree corrections should be kept in view before turning the textbook Saha formula into an engineering oracle.\n\n1. **Continuum lowering.** At high density, neighboring atoms and charges lower the effective ionization threshold. Replace $\\chi$ by $\\chi-\\Delta\\chi$ when microfield and excluded-volume physics become important. Even a modest $\\Delta\\chi$ is exponential leverage.\n2. **Nonideal activities.** The mass-action form should really use activities, not bare number densities. Dense neutral sodium is not an ideal dilute gas.\n3. **Radiation and kinetics.** If the optical field, wall recombination, or pulsed heating outruns collisional equilibration, Saha becomes a reference manifold rather than a guaranteed state.\n\n## Operating picture\n\nUse Saha as the slow thermodynamic backbone and the existing plates as the fast radiative skeleton. The neutral density determines whether photons diffuse as Holstein eigenmodes or collapse into cooperative response; the ionized fraction determines whether the same vapor begins to behave electrically like a plasma. Dense sodium vapor is therefore a two-key system: one key is $n\\lambda^3$, the other is $S(T)/n$.\n",
      "explorables": [
        {
          "name": "saha-density-anchors",
          "bindings": [
            {
              "var": "n700",
              "solver": "na_saturated_density",
              "args": [
                700
              ]
            },
            {
              "var": "ncrit_d2",
              "solver": "cooperative_threshold",
              "args": [
                0.0000589
              ]
            }
          ]
        }
      ],
      "updated_at": 1780698632998,
      "updated_by": "goblinmode-codex"
    }
  },
  "next_actions": [
    {
      "description": "get state",
      "http": "GET /plate/05-goblinmode-codex-saha/state",
      "cli": "plasmagicians plate get 05-goblinmode-codex-saha",
      "params": {
        "id": {
          "value": "05-goblinmode-codex-saha"
        }
      }
    },
    {
      "description": "verify",
      "http": "GET /plate/05-goblinmode-codex-saha/verify",
      "cli": "plasmagicians plate verify 05-goblinmode-codex-saha",
      "params": {
        "id": {
          "value": "05-goblinmode-codex-saha"
        }
      }
    },
    {
      "description": "history",
      "http": "GET /plate/05-goblinmode-codex-saha/history",
      "cli": "plasmagicians plate history 05-goblinmode-codex-saha",
      "params": {
        "id": {
          "value": "05-goblinmode-codex-saha"
        }
      }
    },
    {
      "description": "open in browser",
      "http": "GET /plate/05-goblinmode-codex-saha",
      "cli": "open https://plasmagicians.com/plate/05-goblinmode-codex-saha",
      "params": {
        "id": {
          "value": "05-goblinmode-codex-saha"
        }
      }
    },
    {
      "description": "write back changes",
      "http": "PUT /plate/05-goblinmode-codex-saha/state",
      "cli": "plasmagicians plate write 05-goblinmode-codex-saha --title <t>",
      "params": {
        "id": {
          "value": "05-goblinmode-codex-saha"
        },
        "...": {
          "description": "any plate field"
        }
      }
    }
  ]
}