optmat.gas.lorentz_lorenz#

Convert between the per-particle polarizability and the refractive index.

The package stores gas dispersion as the dynamic dipole polarizability alpha(omega) of one atom or molecule, a property of the particle that does not depend on the state of the gas. The refractive index at number density N follows from the Lorentz-Lorenz (Clausius-Mossotti) relation

\[\frac{n^2 - 1}{n^2 + 2} = \frac{N \alpha}{3 \varepsilon_0} \equiv y, \qquad n^2 - 1 = \chi = \frac{3 y}{1 - y},\]

with alpha in SI (C m^2/V). The dilute limit chi = N alpha / eps0 is the linear density approximation; it differs from Lorentz-Lorenz by the factor 1 / (1 - y) ~ 1 + chi/3, a relative error of about 2e-4 for argon at 1 bar but about 2 % at 100 bar.

Beyond Lorentz-Lorenz, the refractivity virial expansion (n^2 - 1)/(n^2 + 2) = A_R rho + B_R rho^2 + ... (rho the molar density, A_R the molar refractivity) adds the pair term. Every function here takes an optional ratio virial = B_R / A_R (m^3/mol) and uses y = (N alpha / 3 eps0)(1 + virial rho); the ratio is of order 1e-7 to 1e-6 m^3/mol, a correction of 1e-6 to 1e-5 at 1 atm.

Atomic-unit polarizabilities convert with AU_POLARIZABILITY; the molar refractivity A_R = N_A alpha / (3 eps0), used by gas metrology, with molar_refractivity().

Module Attributes

EVALUATION_VERSION

The version of this module's evaluation code, in the identity of every model that evaluates through it; raised whenever the code changes its output (OEP 1, section 4.8).

Functions

`dpolarizability_dsusceptibility`(chi, density)	Return d alpha / d chi of the inverse Lorentz-Lorenz relation.
`molar_refractivity`(polarizability)	Return the molar refractivity `A_R = N_A alpha / (3 eps0)`.
`polarizability_from_refractivity`(...[, virial])	Return the polarizability from `n - 1` at number density `N`.
`polarizability_from_susceptibility`(chi, density)	Return the polarizability from `chi = n^2 - 1` (inverse Lorentz-Lorenz).
`refractive_index`(polarizability, density[, ...])	Return the complex refractive index from a complex polarizability.
`refractivity`(polarizability, density[, virial])	Return `n - 1` from the Lorentz-Lorenz relation, without cancellation.
`susceptibility`(polarizability, density[, virial])	Return `chi = n^2 - 1` from the Lorentz-Lorenz relation.
`susceptibility_dilute`(polarizability, density)	Return the dilute-limit susceptibility `chi = N alpha / eps0`.

optmat.gas.lorentz_lorenz.EVALUATION_VERSION: Final = 1#: The version of this module’s evaluation code, in the identity of every model that evaluates through it; raised whenever the code changes its output (OEP 1, section 4.8).

optmat.gas.lorentz_lorenz.dpolarizability_dsusceptibility(chi, density, virial=0.0)[source]#

Return d alpha / d chi of the inverse Lorentz-Lorenz relation.

With alpha = 3 eps0 chi / [N P (chi + 3)] (polarizability_from_susceptibility(), P the pair factor of the refractivity virial at N), d alpha / d chi = 9 eps0 / [N P (chi + 3)^2]. For a formula in n - 1, d chi / d(n - 1) = 2 n.

Parameters:

chiarray_like or float: Susceptibility n^2 - 1, > -1; complex for an absorbing medium.
densityfloat: Number density in m^-3, > 0.
virialfloat, optional: B_R / A_R in m^3/mol (0 by default: no pair term).

Returns:

numpy.ndarray: d alpha / d chi, C m^2/V.

Examples

>>> h = 1e-9
>>> numeric = (polarizability_from_susceptibility(5e-4 + h, 2.687e25)
...            - polarizability_from_susceptibility(5e-4 - h, 2.687e25)) / (2 * h)
>>> bool(abs(dpolarizability_dsusceptibility(5e-4, 2.687e25) / numeric - 1) < 1e-6)
True

optmat.gas.lorentz_lorenz.molar_refractivity(polarizability)[source]#

Return the molar refractivity A_R = N_A alpha / (3 eps0).

Parameters:

polarizabilityarray_like or float: Per-particle dipole polarizability in C m^2/V.

Returns:

numpy.ndarray: Molar refractivity in m^3/mol (multiply by 1e6 for the cm^3/mol of gas metrology).

Examples

Helium’s static polarizability, 1.383760 a.u., gives the static molar refractivity of gas metrology:

>>> from optmat.constants import AU_POLARIZABILITY
>>> f"{float(molar_refractivity(1.383760 * AU_POLARIZABILITY)) * 1e6:.5f}"
'0.51725'

optmat.gas.lorentz_lorenz.polarizability_from_refractivity(refractivity_value, density, virial=0.0)[source]#

Return the polarizability from n - 1 at number density N.

Parameters:

refractivity_valuearray_like or float: n - 1.
densityfloat: Number density in m^-3, > 0.
virialfloat, optional: B_R / A_R in m^3/mol (0 by default: no pair term).

Returns:

numpy.ndarray: The polarizability in C m^2/V, the shape of refractivity_value.

Examples

>>> from optmat.constants import AU_POLARIZABILITY
>>> alpha = polarizability_from_refractivity(2.772112e-4, 2.687e25)
>>> f"{float(alpha / AU_POLARIZABILITY):.5f}"
'11.08000'

optmat.gas.lorentz_lorenz.polarizability_from_susceptibility(chi, density, virial=0.0)[source]#

Return the polarizability from chi = n^2 - 1 (inverse Lorentz-Lorenz).

Parameters:

chiarray_like or float: Susceptibility n^2 - 1, > -1.
densityfloat: Number density in m^-3, > 0.
virialfloat, optional: B_R / A_R in m^3/mol (0 by default: no pair term).

Returns:

numpy.ndarray: alpha = 3 eps0 y / N with y = chi / (chi + 3), in C m^2/V.

Examples

>>> alpha = polarizability_from_susceptibility(5.544992e-4, 2.687e25)
>>> from optmat.constants import AU_POLARIZABILITY
>>> f"{float(alpha / AU_POLARIZABILITY):.4f}"
'11.0800'

optmat.gas.lorentz_lorenz.refractive_index(polarizability, density, virial=0.0)[source]#

Return the complex refractive index from a complex polarizability.

Parameters:

polarizabilityarray_like or complex: Per-particle dipole polarizability in C m^2/V; Im alpha > 0 is absorption.
densityfloat: Number density in m^-3.
virialfloat, optional: B_R / A_R in m^3/mol (0 by default: no pair term).

Returns:

numpy.ndarray: n = sqrt(1 + chi) with chi = 3 y / (1 - y) and y = (N alpha / 3 eps0)(1 + b rho) (complex; Im n > 0 is absorption), the shape of polarizability.

Examples

>>> from optmat.constants import AU_POLARIZABILITY
>>> n = refractive_index((11.08 + 0.01j) * AU_POLARIZABILITY, 2.687e25)
>>> f"{float(n.real - 1):.6e} {float(n.imag):.3e}"
'2.772112e-04 2.502e-07'

optmat.gas.lorentz_lorenz.refractivity(polarizability, density, virial=0.0)[source]#

Return n - 1 from the Lorentz-Lorenz relation, without cancellation.

Parameters:

polarizabilityarray_like or float: Per-particle dipole polarizability in C m^2/V.
densityfloat: Number density in m^-3.
virialfloat, optional: B_R / A_R in m^3/mol (0 by default: no pair term).

Returns:

numpy.ndarray: n - 1 = chi / (1 + sqrt(1 + chi)), the shape of polarizability.

Examples

>>> from optmat.constants import AU_POLARIZABILITY
>>> f"{float(refractivity(11.08 * AU_POLARIZABILITY, 2.687e25)):.6e}"
'2.772112e-04'

optmat.gas.lorentz_lorenz.susceptibility(polarizability, density, virial=0.0)[source]#

Return chi = n^2 - 1 from the Lorentz-Lorenz relation.

Parameters:

polarizabilityarray_like or float: Per-particle dipole polarizability alpha in C m^2/V (real).
densityfloat: Number density N in m^-3, >= 0.
virialfloat, optional: B_R / A_R in m^3/mol (0 by default: no pair term).

Returns:

numpy.ndarray: chi = 3 y / (1 - y) with y = (N alpha / 3 eps0)(1 + b rho), the shape of polarizability.

Examples

>>> from optmat.constants import AU_POLARIZABILITY
>>> f"{float(susceptibility(11.08 * AU_POLARIZABILITY, 2.687e25)):.6e}"
'5.544992e-04'

optmat.gas.lorentz_lorenz.susceptibility_dilute(polarizability, density)[source]#

Return the dilute-limit susceptibility chi = N alpha / eps0.

This is linear density scaling, exact only as N -> 0; it is provided to measure the error of that approximation against susceptibility().

Parameters:

polarizabilityarray_like or float: Per-particle dipole polarizability in C m^2/V.
densityfloat: Number density in m^-3.

Returns:

numpy.ndarray: N alpha / eps0, the shape of polarizability.

Examples

>>> from optmat.constants import AU_POLARIZABILITY
>>> f"{float(susceptibility_dilute(11.08 * AU_POLARIZABILITY, 2.687e25)):.6e}"
'5.543968e-04'