palastro_2011
=============

J\. P\. Palastro, T\. M\. Antonsen, Jr\. and A\. Pearson, "Models of the delayed nonlinear Raman response in diatomic gases", Phys\. Rev\. A 84, 013829 \(2011\)

`Source record <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/palastro_2011.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1103/PhysRevA\.84\.013829 <https://doi.org/10.1103/PhysRevA.84.013829>`__

**Source kind:** theory

**Recorded technique:** Classical kinetic \(Boltzmann\) theory of the orientational response of a diatomic gas to a polarizing laser field\: the linear Green's function in closed form, a perturbative fully nonlinear impulse response \(second\-order polarizability\), and reduced fluid \(moment\) models with transport coefficients fitted to the kinetic response \(Hammett\-Perkins closures\); compared with full quantum density\-matrix calculations for N2 and O2 \(and H2\) \(secs\. II\-III\)\.

**Materials in structured entries:** N2

**Dataset quantities:** None recorded.

**Contents:** 0 datasets; 3 formulas.

Curation notes
--------------

Journal pages 013829\-1 to \-11 = PDF pages 1\-11\. Identity checked against the first page\. N2, O2, H2 in the comparisons; no gas parameter \(Delta alpha, B, I\) is tabulated or quoted as a number\: NO DATASET\. The models and their fitted dimensionless parameters are recorded as \[\[formulas\]\] below \(gas N2\: the fits are to nitrogen at 294 K\)\.

DEFINITIONS \(Gaussian, p\. 1\)\: V\_L\(theta, t\) = \-\(1/4\)\[alpha\_perp \+ Delta alpha cos^2 theta\] \|E\(t\)\|^2 \(E the field amplitude, cycle averaged\), Delta alpha = alpha\_par \- alpha\_perp; polarization P = rho \[alpha\_perp \+ Delta alpha \<cos^2 theta\>\] E; chi = 4 pi rho alpha\_perp, Gamma = rho Delta alpha \<cos^2 theta\>, wave equation Eq\. \(2\) with \(4 pi / c^2\) d^2\(Gamma E\)/dt^2\. Classical correspondence\: l\_th \>\> 1 \(N2 at room temperature\: l\_th ~ 10\) and times shorter than the Heisenberg time t\_H ~ 2 pi / \(2 l\_th Omega\_0\) \(0\.84 ps for N2; p\. 2\)\. omega\_T = \(2T/I\)^\(1/2\) \(T in energy units, I the moment of inertia; p\. 4\)\. Adiabatic part, Eq\. \(15b\)\: Gamma\_A = \(2/5\) pi rho \(Delta alpha\)^2 \|E\(t\)\|^2 / \(I omega\_T^2\)\. Gamma\_T\(t\) = int G\(t \- t'\) \|E\(t'\)\|^2 dt'\. Second\-order steady alignment, Eq\. \(35\), p\. 9\: Gamma\_2\(t \-\> inf\) = pi rho Delta alpha \(2/225\)\(sigma Delta alpha E0^2 / I omega\_T\)^2 \(sigma the pulse duration\)\.

COMPARISONS \(figures only\)\: N2 and O2 at 294 K, I = 1e12 W/cm2, sigma = 20 fs \(Fig\. 2\); N2 at 20 and 100 K \(Fig\. 3\); H2 at 294 K, where the classical theory fails \(Fig\. 4\); the second\-order theory against the nonlinear quantum calculation for N2 at I = 1e14 W/cm2 \(Fig\. 5\)\: 'negligible up to the intensity threshold for ionization' \(p\. 10\)\.

**Recorded bibliographic identity status:** VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets
--------

No datasets are recorded.

Formulas
--------

palastro\_2011/kinetic\_green\_function
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "kinetic_green_function",
     "material": "N2",
     "equation": "Eqs. (16a), (16b), (17), p. 013829-4",
     "printed": "G(tau) = (g0/4)[tau + pi^(1/2)(1 - tau^2/2) exp(-tau^2/4) erfi(tau/2)], tau = 2 omega_T t, g0 = 8 pi rho (Delta alpha)^2 / (15 I omega_T^2), omega_T = (2T/I)^(1/2); G(Omega) = -g0 int s_R [Omega_R^3 / (Omega^2 - 4 Omega_R^2)] exp(-Omega_R^2) dOmega_R, Omega = omega/omega_T",
     "variable": "t: time after an impulse, s; T: temperature in energy units; I: moment of inertia; rho: number density; Delta alpha: polarizability anisotropy (Gaussian)",
     "state": "any (rho explicit); classical: high rotational temperature (l_th >> 1), times < t_H",
     "density": "rho",
     "range": "N2 and O2 at atmospheric temperatures, before the quarter revival (not H2 at 294 K)",
     "coefficients": {},
     "check": "Not checked numerically here (figures only): 'in excellent agreement' with the density-matrix response for N2 and O2 at 294 K (Fig. 2)."
   }

The linear \(first\-order\) classical response function of a thermal diatomic gas; its shape depends on omega\_T only\.

palastro\_2011/two\_moment\_fluid
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "two_moment_fluid",
     "material": "N2",
     "equation": "Eq. (26c), p. 013829-6; fitted nu hat, p. 013829-7",
     "printed": "G(tau) = g0 sin((1/2) omega_hat_Tnu tau) / omega_hat_Tnu exp(-(1/2) nu_hat tau), nu_hat = nu / omega_T, omega_hat_Tnu = (3 - nu_hat^2)^(1/2); 'With the least-squares fit we find that nu_hat = 0.95'",
     "variable": "tau = 2 omega_T t",
     "state": "N2 at T = 294 K (the fit); I = 1e12 W/cm2, sigma = 20 fs in the comparison",
     "density": "rho (through g0)",
     "range": "fitted to the kinetic response in the time domain; 'reasonable' but not as accurate as desired (p. 7)",
     "coefficients": {
       "nu_hat": "0.95"
     },
     "check": "The fit's quality is shown in Fig. 6 only; not checked here."
   }

Continuity and momentum moments with a viscosity closure \(rate nu of momentum loss\); the zero\-frequency limit of the kinetic Green's function is not preserved\.

palastro\_2011/three\_moment\_fluid
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "three_moment_fluid",
     "material": "N2",
     "equation": "Eqs. (30), (31a), (31b), p. 013829-8",
     "printed": "G(tau) = -g0 [nu2_hat (1 - omega1_hat^2/3) / (nu2_hat^2 + omega1_hat^2 - 2 nu1_hat nu2_hat)] [((Psi - nu1_hat)/omega1_hat) sin(omega1_hat tau/2) exp(-nu1_hat tau/2) + cos(omega1_hat tau/2) exp(-nu1_hat tau/2) - exp(-nu2_hat tau/2)], Psi = omega1_hat^2 (1 + nu2_hat^2/3 - 2 nu1_hat nu2_hat/3) / [nu2_hat (1 - omega1_hat^2/3)]; 'omega1_hat = 2.37, nu1_hat = 1.27, and nu2_hat = 3.28 ... we find mu_hat = 6.16, varpi = -6.45 and nu_hat = -0.17'",
     "variable": "tau = 2 omega_T t; hats: divided by omega_T",
     "state": "N2 at T = 294 K (the fit), same laser parameters as the two-moment fit",
     "density": "rho (through g0)",
     "range": "fitted in the time domain; 'matches the quantum response function quite well' (Fig. 7)",
     "coefficients": {
       "omega1_hat": "2.37",
       "nu1_hat": "1.27",
       "nu2_hat": "3.28",
       "mu_hat": "6.16",
       "varpi": "-6.45",
       "nu_hat": "-0.17"
     },
     "check": "Not checked numerically here. The relations mu_hat = omega1_hat^2 nu2_hat / 3 (2.37^2 x 3.28 / 3 = 6.14, printed 6.16: rounding) and 2 nu_hat + mu_hat = 2 nu1_hat + nu2_hat (2(-0.17) + 6.16 = 5.82 = 2(1.27) + 3.28) and 2 nu_hat mu_hat + 8 - (5/4) varpi = 2 nu1_hat nu2_hat + omega1_hat^2 (13.97 against 13.95) hold (checked 2026-09-26)."
   }

Continuity, momentum and pressure \(energy\) moments; varsigma set to zero \(unphysical equilibria otherwise\)\. A negative viscosity, 'linearly stable'\.

Corrections
-----------

No dataset errata are recorded in this source record.

Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the :doc:`review register </register>` for the checks and decisions.