BH Physics Explainer

This page summarizes the physical model used in bh_molecule.physics.BHModel. It is organized by method; each section states the mathematical definition and key assumptions.

energy

Rovibronic term value \(E(v,N)\) (in cm⁻¹) for a given electronic state, using a Dunham-like expansion truncated to cubic vibrational and quartic (centrifugal distortion) rotational terms:

\[ E(v,N)=T_e + G(v) + F_v(N), \]

with

\[ \begin{aligned} G(v) &= \omega_e\,(v+\tfrac12) \;-\; \omega_e x_e\,(v+\tfrac12)^2 \;+\; \omega_e y_e\,(v+\tfrac12)^3, \\ B_v &= B_e \;-\; \alpha_e\,(v+\tfrac12), \\ D_v &= D_e \;-\; \beta_e\,(v+\tfrac12), \\ F_v(N) &= B_v\,N(N+1)\;-\; D_v\,[N(N+1)]^2 . \end{aligned} \]

Notes. \(T_e\) is the electronic term origin; \(\omega_e,\omega_e x_e,\omega_e y_e\) are vibrational constants; \(B_e,\alpha_e,D_e,\beta_e\) are rotational and centrifugal-distortion constants, all state-specific.

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line_profile

Lines are modeled as Gaussian in wavelength with Doppler and instrumental widths added in quadrature (FWHM):

\[ \Delta\lambda = \sqrt{\Delta\lambda_D^2 + \Delta\lambda_{\rm inst}^2}. \]

The Doppler FWHM at temperature \(T\) (for emitter mass \(m\)) follows the standard expression

\[ \Delta\lambda_D = \lambda \sqrt{\frac{8\ln 2\,k_B T}{m c^2}} . \]

The corresponding standard deviation is

\[ \sigma=\frac{\Delta\lambda}{2\sqrt{2\ln 2}} , \]

and the normalized profile at wavelength \(x\) is

\[ g(x)=\frac{1}{\sqrt{2\pi}\,\sigma}\exp\!\left[-\frac{(x-\lambda)^2}{2\sigma^2}\right]. \]

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A_coeff

Einstein \(A_{ul}\) for a rovibronic line.

For the BH \(A\,^1\Pi \rightarrow X\,^1\Sigma^+\) system, line Einstein coefficients are formed from band \(A_{\rm vib}(v')\) (per upper vibrational level) and Hönl–London rotational factors:

\[ A_{ul}(v', N_2 \to N_1)= \frac{A_{\rm vib}(v')\, H_{\rm HL}(N_2,\Delta N)}{2N_2+1} \]

\[ \Delta N = N_2 - N_1 \in \{-1,0,+1\}, \quad H_{\mathrm{HL}} = \begin{cases} N_2/2, & \Delta N = -1 \quad (\text{P}) \\ (2N_2+1)/2, & \Delta N = 0 \quad (\text{Q}) \\ (N_2+1)/2, & \Delta N = +1 \quad (\text{R}) \end{cases} \]

Notes. \(H_{\rm HL}\) are the case-(a) factors appropriate to a \(^{1}\Pi \to {}^{1}\Sigma^+\) transition and partition intensity among P/Q/R branches (\(\Delta N=-1,0,+1\)). Electronic degeneracies, Λ-doubling, parity, and nuclear-spin substructure are neglected here and can be incorporated via additional weights if needed.

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spectrum

For a chosen rotational branch (P/Q/R) on a wavelength grid \(x\), the model sums lines whose centers \(\lambda_{v'N_2\to v''N_1}\) come from tabulated wavelengths (X-state only fixes positions) while A-state level energies and populations set intensities.

Per-line contribution near \(\lambda_0\) is

\[ I_\ell(x)=\frac{h\nu_0}{4\pi}\; n'(v',N_2)\; A_{ul}(v',N_2\!\to\!N_1)\; g_\lambda(x), \]

where

\(n'(v',N_2)\) : upper-state populations (Boltzmann at \(T_{\rm rot}\), scaled by an overall factor \(C\))

\(A_{ul}\) : as above

\(g_\lambda\) : Gaussian line profile with total FWHM \(\Delta\lambda\) (Doppler + instrumental).

The total branch spectrum is the sum over included \((v',N_2\to v'',N_1)\) within specified bounds.

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full_fit_model

Composite model for the 433 nm window.

The forward model used around 433 nm is the sum of:

1. the BH Q-branch spectrum (as in spectrum, with branch fixed to Q), evaluated on a shifted grid \(x+\delta x\) and rescaled numerically by \(10^8\) for conditioning, plus
2. two auxiliary Gaussian lines at fixed wavelengths \(\lambda_{R7}=433.64776244\,\mathrm{nm}\) and \(\lambda_{R8}=433.33500584\,\mathrm{nm}\) with amplitudes \(I_{R7}, I_{R8}\) and the same instrumental width as the BH part (Doppler set to zero for these), and
3. a constant baseline \(b\).

Putting it together:

\[ \begin{aligned} y(x) ={}& 10^{8}\,S_{\mathrm{Q}}\!\left(x+\delta x;\, C, T_{\mathrm{rot}}, w_{\mathrm{inst}}\right) \\ & {}+ I_{R7}\, g\!\left(x; \lambda_{R7}, w_{\mathrm{inst}}\right) \\ & {}+ I_{R8}\, g\!\left(x; \lambda_{R8}, w_{\mathrm{inst}}\right) + b. \end{aligned} \]

Notes. The X-state affects only line centers (via tables). If Doppler broadening is also required for the auxiliary features, replace their \(g\) with the full Doppler + instrument \(\Delta\lambda\).