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 · Catalogue · Review register
DOI: 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#
{
"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#
{
"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#
{
"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 review register for the checks and decisions.