---

title: "Saturation Constants for Modal Thermodynamics"
version: v 2.0 · 2025-07-13
replaces: v 1.0 (2025-06-10)
tags: [PBG, thermodynamics, saturation]

Purpose — Lock down the two scalar inputs that appear in all
thermodynamic and statistical notes:

symbol	rôle	value
${\rho }_{\text{crit}}$	coherence saturation density	$6.0\times {10}^{44}{\mathrm{m}}^{-3}$
${\gamma }_0$	low-density turn-over rate	$8.2\times {10}^{-8}{\mathrm{s}}^{-1}$

Both follow directly from the quartet $(\alpha ,\beta ,\gamma ,\lambda )$
plus one empirical atomic lifetime; no extra parameters are introduced.

1 Critical density ${\rho }_{\text{crit}}$

Physical meaning — the envelope density at which the quadratic
anchoring pressure $\beta |\psi {|}^2$ exactly balances the gradient
tension $\alpha |\mathrm{\nabla }\mathrm{\Phi }{|}^2$ of the minimal coherent bag.

Test envelope — reuse the proton-core Gaussian
(Appendix BA):

• centre density

  $${\rho }_0=\frac{2}{{\pi }^{3/2}{\sigma }_p^3}=6.0\times {10}^{44}{\mathrm{m}}^{-3};$$

• balance condition at $r=0$

  $$\alpha {\rho }_0\frac{3}{{\sigma }_p^2}=\beta {\rho }_0⟹\boxed{{\displaystyle {\rho }_{\text{crit}}={\rho }_0}}.$$

(Numerical value is unchanged by the 2025-07-13 quartet refit because it
depends only on σ${}_p$.)

2 Turn-over coefficient ${\gamma }_0$

Anchor to the 2P→1S spontaneous lifetime
${\tau }_{2P}=1.6\times {10}^{-9}\mathrm{s}$.

• Low-density sink model (Appendix F)

  $${\mathrm{\Gamma }}_{\text{dec}}={\gamma }_0\frac{{\rho }_c^2}{{\rho }_{\text{crit}}}.$$

• 1S centre density using the PBG Bohr radius
  $a_0={\hslash }^2\alpha /(m_e\gamma c^2)$

  $${\rho }_{1S}(0)=\frac{1}{\pi a_0^3}=2.15\times {10}^{30}{\mathrm{m}}^{-3}.$$

• Demand ${\mathrm{\Gamma }}_{\text{dec}}=1/{\tau }_{2P}$ when
  ${\rho }_c\to {\rho }_{1S}(0)$:

Units:
ρ${}_{\text{crit}}$/ρ² → m⁻³; multiplied by γ₀ (s⁻¹) gives 1/time.

3 Notebook reference

notebooks/thermo-saturation.ipynb
(commit 27db934) reproduces the two numbers above in
three NumPy lines.

4 Where to use these constants

With these anchors the two appendices are now locked (v 1.1) and may
be cited by higher-level thermodynamic calculations.