franta_2014b#
D. Franta, D. Nečas, L. Zajíčková and I. Ohlídal, “Dispersion model of two-phonon absorption: application to c-Si”, Opt. Mater. Express 4, 1641 (2014)
Source record · Catalogue · Review register
DOI: 10.1364/ome.4.001641
Source kind: fit
Recorded technique: A dispersion model of two-phonon absorption (15 additive and subtractive branch combinations, each a piecewise function with 3D and 2D critical-point singularities between four critical points, Bose-Einstein thermal factors, Gauss-Dawson eps-broadening) fitted to Bruker Vertex 80v transmittance of float-zone c-Si (580 um slab, 450-1000 cm-1; 14 mm slab, 70-450 cm-1) at 300 K, and checked with Woollam IR-VASE ellipsometry at five temperatures, 300-500 K, 300-1000 cm-1 (back side aluminium-coated; only the thermal coefficient nu_T was fitted there) (p. 1650).
Materials in structured entries: Si
Dataset quantities: None recorded.
Contents: 0 datasets; 16 formulas.
Curation notes#
Journal pages 1641-1656 = PDF pages 1-16. Identity checked against the first page. Material: float-zone crystalline silicon (Si); no n or k is tabulated (the optical constants are Fig. 5 only, not digitised).
EQUATION MAP (journal pages). F_2ph = sum over branch pairs of F_(A+B) + F_(A-B) (Eq. (8), p. 1645); Eq. (14): F_(A+-B)(E, T) = f_(A+-B)(E, T) P_(A+-B)(E); critical-point energies Eq. (15), p. 1646: E_(t,A+-B) = h [nu_A(M_t) +- nu_B(M_t)] [1 + nu_T (T - 300)] nu_iso, nu_iso = 0.9982 for natural silicon; the 17 independent frequencies and their degeneracies, Eq. (16); auxiliary variables X, Y of regions I-III, Eqs. (17)-(18), p. 1647; the ten critical-point contributions L_i (Table 1, p. 1647: 0 3D M0 sqrt(X_I); 1 3D M1 1 - sqrt(Y_I); 2 2D M1 -ln Y_I; 3 2D M1 -ln X_II; 4 3D M1 Y_II; 5 3D M2 X_II; 6 2D M1 -ln Y_II; 7 2D M1 -ln X_III; 8 3D M2 1 - sqrt(X_III); 9 3D M3 sqrt(Y_III)), P = H(E) sum_i A_i L_i (Eq. (19), p. 1648), the shape function H(E) with kappa_0..kappa_5, Eq. (20), and its properties Eq. (21); thermal factors Eqs. (22)-(23), f_(A+B) = 1 + f_BE(E_A) + f_BE(E_B), f_(A-B) = f_BE(E_B) - f_BE(E_A), with phonon energies interpolated linearly, Eq. (24); normalisation at 300 K, Eqs. (25)-(26), p. 1649; eps-broadening with the Gauss-Dawson kernel beta^(x) = -(sqrt(2)/(pi B)) D(x/(sqrt(2) B)) + i exp(-x^2/(2 B^2))/(sqrt(2 pi) B), Eq. (27) (franta_2014); B the rms and beta = 2 sqrt(2 ln 2) B/(h c) the FWHM in cm^-1, Eq. (28), used as the fitting parameter; eps^0_(A+-B) = beta^ * F^0/E, Eq. (29); combined TO+TA and LA+TA terms weighted by C, Eq. (30); eps_(A+-B) = alpha_(A+-B) N_Si eps^0_(A+-B), Eq. (31), N_Si the transition strength of the silicon nuclei (franta_2013); the two-phonon effective charge e*_2ph = Z_Si e sqrt(sum alpha), Eq. (32).
PARAMETERS (formulas below). Table 2 (p. 1650): phonon frequencies (THz) at the symmetry points at 300 K, fitted (A(Gamma) = 0 fixed); nu_T = -3.4e-5 K^-1 (p. 1654, from the ellipsometry); Table 5 (p. 1653): alpha, beta (cm^-1), C, A_0..A_9, kappa_0..kappa_5 for the 15 terms, daggers = fixed (‘we fixed A_5 = 1’, C_(A+-B(1)) = 1, and kappa_0, kappa_5 = 0 at Gamma-point M0/M3, p. 1650). Tables 3 and 4 (pp. 1651-1652) identify the critical points of each term as sums and differences of Table 2’s frequencies, with the photon wavenumbers; they add no parameters. Checked: the THz sums reproduce Table 2 (e.g. TO+TO M0 = 2 x 14.09 = 28.18); the wavenumber columns are the THz values over c to 0.1 cm^-1 within rounding (28.18 THz = 940.0 cm^-1, printed 939.9: the columns were evidently computed from unrounded frequencies).
LIMITS (p. 1654): good agreement at 300 K over 70-1000 cm^-1; above 300 K the model misses absorption at 750-850 cm^-1 (three-phonon processes, added in franta_2017); broadening independent of temperature over 300-500 K. The paper’s Table 5 needs N_Si (franta_2013: N_n of Eq. (46)) to be used absolutely; franta_2017 supersedes this model for temperatures other than 300 K and gives new phonon frequencies at 0 and 300 K (its Table 4).
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#
franta_2014b/phonon_frequencies#
{
"id": "phonon_frequencies",
"material": "Si",
"equation": "Eqs. (15)-(16), p. 1646; Table 2, p. 1650; nu_T p. 1654",
"printed": "E_{t,A\\pm B} = h[\\nu_{A(M_t)} \\pm \\nu_{B(M_t)}][1 + \\nu_T (T - 300)]\\nu_{iso}",
"variable": "nu: phonon frequencies at the symmetry points (THz) at 300 K; T in K",
"coefficients": {
"nu_A_Gamma": "0",
"nu_O_Gamma": "15.63",
"nu_TA_X": "4.71",
"nu_L_X": "11.98",
"nu_TO_X": "14.09",
"nu_TA_L": "3.14",
"nu_LA_L": "11.05",
"nu_LO_L": "12.93",
"nu_TO_L": "14.99",
"nu_TA_W": "6.39",
"nu_L_W": "10.63",
"nu_TO_W": "14.33",
"nu_TA1_K": "4.82",
"nu_TA2_K": "6.70",
"nu_LA_K": "10.88",
"nu_LO_K": "11.30",
"nu_TO1_K": "13.90",
"nu_TO2_K": "14.91",
"nu_T": "-3.4e-5",
"nu_iso": "0.9982"
},
"check": "Table 2 checked against its image (2026-09-26): the Gamma row's 0 (dagger, fixed) spans the acoustic columns and 15.63 the optical ones; X and W give one value for TA1=TA2, LA=LO and TO1=TO2 (Eq. (16)). Tables 3 and 4 reproduce these frequencies (source notes)."
}
nu_A_Gamma = 0 is fixed (dagger). nu_T in K^-1 (fitted to the 300-500 K ellipsometry, p. 1654); nu_iso for natural silicon (p. 1646). franta_2017 Table 4 gives newer values at 0 and 300 K.
franta_2014b/two_phonon_TO_plus_TO#
{
"id": "two_phonon_TO_plus_TO",
"material": "Si",
"equation": "Eqs. (14)-(31), pp. 1645-1649; Table 5, p. 1653",
"printed": "Two-phonon band TO+TO: F_(A+-B) = f_(A+-B)(E,T) P_(A+-B)(E), P = H(E) sum_i A_i L_i(E), normalised at 300 K, eps-broadened (Gauss-Dawson), eps_(A+-B) = alpha N_Si eps^0",
"variable": "E: photon energy; critical-point energies from the phonon frequencies (formula phonon_frequencies); beta: Gaussian FWHM in cm^-1 (Eq. (28))",
"coefficients": {
"alpha": "1.24e-6",
"beta": "16.12",
"A0": "1.18",
"A1": "0.344",
"A2": "0.0138",
"A3": "0",
"A4": "1.23",
"A5": "1",
"A6": "0.000725",
"A7": "0.144",
"A8": "0.0177",
"A9": "0.842",
"kappa0": "1.08",
"kappa1": "0.747",
"kappa2": "3.67",
"kappa3": "4.22",
"kappa4": "9.46",
"kappa5": "0"
},
"check": "Transcribed from the library JSON of Table 5 by a scratch generator and compared with the table image (2026-09-26). The model cannot be evaluated from the paper alone (N_Si and the critical-point identification of Tables 3-4 are needed); not reproduced numerically."
}
Table 5 (p. 1653), column ‘TO+TO’. Fixed values (dagger): A3, A5, kappa5. Printed ‘-’ for C: no C for a single-term band.
franta_2014b/two_phonon_TO_plus_LO#
{
"id": "two_phonon_TO_plus_LO",
"material": "Si",
"equation": "Eqs. (14)-(31), pp. 1645-1649; Table 5, p. 1653",
"printed": "Two-phonon band TO+LO: F_(A+-B) = f_(A+-B)(E,T) P_(A+-B)(E), P = H(E) sum_i A_i L_i(E), normalised at 300 K, eps-broadened (Gauss-Dawson), eps_(A+-B) = alpha N_Si eps^0",
"variable": "E: photon energy; critical-point energies from the phonon frequencies (formula phonon_frequencies); beta: Gaussian FWHM in cm^-1 (Eq. (28))",
"coefficients": {
"alpha": "4.57e-6",
"beta": "14.96",
"A0": "0.817",
"A1": "0.0516",
"A2": "0.422",
"A3": "0.000367",
"A4": "1.65",
"A5": "1",
"A6": "0.0266",
"A7": "0.234",
"A8": "0.000226",
"A9": "0.626",
"kappa0": "1.76",
"kappa1": "0.315",
"kappa2": "3.8",
"kappa3": "2.06",
"kappa4": "0.723",
"kappa5": "0"
},
"check": "Transcribed from the library JSON of Table 5 by a scratch generator and compared with the table image (2026-09-26). The model cannot be evaluated from the paper alone (N_Si and the critical-point identification of Tables 3-4 are needed); not reproduced numerically."
}
Table 5 (p. 1653), column ‘TO+LO’. Fixed values (dagger): A5, kappa5. Printed ‘-’ for C: no C for a single-term band.
franta_2014b/two_phonon_TO_plus_LA#
{
"id": "two_phonon_TO_plus_LA",
"material": "Si",
"equation": "Eqs. (14)-(31), pp. 1645-1649; Table 5, p. 1653",
"printed": "Two-phonon band TO+LA: F_(A+-B) = f_(A+-B)(E,T) P_(A+-B)(E), P = H(E) sum_i A_i L_i(E), normalised at 300 K, eps-broadened (Gauss-Dawson), eps_(A+-B) = alpha N_Si eps^0",
"variable": "E: photon energy; critical-point energies from the phonon frequencies (formula phonon_frequencies); beta: Gaussian FWHM in cm^-1 (Eq. (28))",
"coefficients": {
"alpha": "3.95e-6",
"beta": "14.59",
"A0": "0.0806",
"A1": "1.45",
"A2": "5.05e-5",
"A3": "0.00127",
"A4": "0.00072",
"A5": "1",
"A6": "0.258",
"A7": "0.000271",
"A8": "0.00174",
"A9": "0.321",
"kappa0": "0",
"kappa1": "0.312",
"kappa2": "1.24",
"kappa3": "4.97",
"kappa4": "3.66",
"kappa5": "0.809"
},
"check": "Transcribed from the library JSON of Table 5 by a scratch generator and compared with the table image (2026-09-26). The model cannot be evaluated from the paper alone (N_Si and the critical-point identification of Tables 3-4 are needed); not reproduced numerically."
}
Table 5 (p. 1653), column ‘TO+LA’. Fixed values (dagger): A5, kappa0. Printed ‘-’ for C: no C for a single-term band.
franta_2014b/two_phonon_LO_plus_LA#
{
"id": "two_phonon_LO_plus_LA",
"material": "Si",
"equation": "Eqs. (14)-(31), pp. 1645-1649; Table 5, p. 1653",
"printed": "Two-phonon band LO+LA: F_(A+-B) = f_(A+-B)(E,T) P_(A+-B)(E), P = H(E) sum_i A_i L_i(E), normalised at 300 K, eps-broadened (Gauss-Dawson), eps_(A+-B) = alpha N_Si eps^0",
"variable": "E: photon energy; critical-point energies from the phonon frequencies (formula phonon_frequencies); beta: Gaussian FWHM in cm^-1 (Eq. (28))",
"coefficients": {
"alpha": "2.51e-6",
"beta": "4.58",
"A0": "6.94e-5",
"A1": "0.000249",
"A2": "0.0323",
"A3": "0.0107",
"A4": "0.0832",
"A5": "1",
"A6": "0.000179",
"A7": "0.0726",
"A8": "0.00193",
"A9": "0.782",
"kappa0": "0",
"kappa1": "0.1",
"kappa2": "0.811",
"kappa3": "9.97",
"kappa4": "0.679",
"kappa5": "0.895"
},
"check": "Transcribed from the library JSON of Table 5 by a scratch generator and compared with the table image (2026-09-26). The model cannot be evaluated from the paper alone (N_Si and the critical-point identification of Tables 3-4 are needed); not reproduced numerically."
}
Table 5 (p. 1653), column ‘LO+LA’. Fixed values (dagger): A5, kappa0. Printed ‘-’ for C: no C for a single-term band.
franta_2014b/two_phonon_TO_plus_TA1#
{
"id": "two_phonon_TO_plus_TA1",
"material": "Si",
"equation": "Eqs. (14)-(31), pp. 1645-1649; Table 5, p. 1653",
"printed": "Two-phonon band TO+TA(1): F_(A+-B) = f_(A+-B)(E,T) P_(A+-B)(E), P = H(E) sum_i A_i L_i(E), normalised at 300 K, eps-broadened (Gauss-Dawson), eps_(A+-B) = alpha N_Si eps^0",
"variable": "E: photon energy; critical-point energies from the phonon frequencies (formula phonon_frequencies); beta: Gaussian FWHM in cm^-1 (Eq. (28))",
"coefficients": {
"alpha": "7.50e-6",
"beta": "6.58",
"C": "1",
"A0": "0.948",
"A1": "1.49",
"A2": "2.07e-5",
"A3": "6.67e-5",
"A4": "1.66",
"A5": "1",
"A6": "0.515",
"A7": "0.029",
"A8": "0.119",
"A9": "0.264",
"kappa0": "0",
"kappa1": "0.11",
"kappa2": "3.59",
"kappa3": "9.78",
"kappa4": "0.43",
"kappa5": "5.05e-5"
},
"check": "Transcribed from the library JSON of Table 5 by a scratch generator and compared with the table image (2026-09-26). The model cannot be evaluated from the paper alone (N_Si and the critical-point identification of Tables 3-4 are needed); not reproduced numerically."
}
Table 5 (p. 1653), column ‘TO+TA(1)’. Fixed values (dagger): C, A5, kappa0.
franta_2014b/two_phonon_TO_plus_TA2#
{
"id": "two_phonon_TO_plus_TA2",
"material": "Si",
"equation": "Eqs. (14)-(31), pp. 1645-1649; Table 5, p. 1653",
"printed": "Two-phonon band TO+TA(2): F_(A+-B) = f_(A+-B)(E,T) P_(A+-B)(E), P = H(E) sum_i A_i L_i(E), normalised at 300 K, eps-broadened (Gauss-Dawson), eps_(A+-B) = alpha N_Si eps^0",
"variable": "E: photon energy; critical-point energies from the phonon frequencies (formula phonon_frequencies); beta: Gaussian FWHM in cm^-1 (Eq. (28))",
"coefficients": {
"C": "0.0537",
"A0": "0",
"A1": "0",
"A2": "0",
"A3": "0",
"A4": "1.3",
"A5": "1",
"A6": "0",
"A7": "0",
"A8": "2.24",
"A9": "0",
"kappa0": "0",
"kappa1": "1",
"kappa2": "1",
"kappa3": "2.84",
"kappa4": "1",
"kappa5": "1"
},
"check": "Transcribed from the library JSON of Table 5 by a scratch generator and compared with the table image (2026-09-26). The model cannot be evaluated from the paper alone (N_Si and the critical-point identification of Tables 3-4 are needed); not reproduced numerically."
}
Table 5 (p. 1653), column ‘TO+TA(2)’. Fixed values (dagger): A0, A1, A2, A3, A5, A6, A7, A9, kappa0, kappa1, kappa2, kappa4, kappa5. Printed ‘-’ for alpha, beta: this term shares alpha and beta with the (1) term (Eq. (30)).
franta_2014b/two_phonon_TO_plus_TA3#
{
"id": "two_phonon_TO_plus_TA3",
"material": "Si",
"equation": "Eqs. (14)-(31), pp. 1645-1649; Table 5, p. 1653",
"printed": "Two-phonon band TO+TA(3): F_(A+-B) = f_(A+-B)(E,T) P_(A+-B)(E), P = H(E) sum_i A_i L_i(E), normalised at 300 K, eps-broadened (Gauss-Dawson), eps_(A+-B) = alpha N_Si eps^0",
"variable": "E: photon energy; critical-point energies from the phonon frequencies (formula phonon_frequencies); beta: Gaussian FWHM in cm^-1 (Eq. (28))",
"coefficients": {
"C": "0.0846",
"A0": "0",
"A1": "0.876",
"A2": "0",
"A3": "0",
"A4": "0.846",
"A5": "1",
"A6": "0",
"A7": "0",
"A8": "1.43",
"A9": "0",
"kappa0": "0",
"kappa1": "1",
"kappa2": "1",
"kappa3": "1",
"kappa4": "1",
"kappa5": "1"
},
"check": "Transcribed from the library JSON of Table 5 by a scratch generator and compared with the table image (2026-09-26). The model cannot be evaluated from the paper alone (N_Si and the critical-point identification of Tables 3-4 are needed); not reproduced numerically."
}
Table 5 (p. 1653), column ‘TO+TA(3)’. Fixed values (dagger): A0, A2, A3, A5, A6, A7, A9, kappa0, kappa1, kappa2, kappa3, kappa4, kappa5. Printed ‘-’ for alpha, beta: this term shares alpha and beta with the (1) term (Eq. (30)).
franta_2014b/two_phonon_LO_plus_TA#
{
"id": "two_phonon_LO_plus_TA",
"material": "Si",
"equation": "Eqs. (14)-(31), pp. 1645-1649; Table 5, p. 1653",
"printed": "Two-phonon band LO+TA: F_(A+-B) = f_(A+-B)(E,T) P_(A+-B)(E), P = H(E) sum_i A_i L_i(E), normalised at 300 K, eps-broadened (Gauss-Dawson), eps_(A+-B) = alpha N_Si eps^0",
"variable": "E: photon energy; critical-point energies from the phonon frequencies (formula phonon_frequencies); beta: Gaussian FWHM in cm^-1 (Eq. (28))",
"coefficients": {
"alpha": "6.90e-7",
"beta": "7.98",
"A0": "0",
"A1": "0.0345",
"A2": "0.0181",
"A3": "0.000201",
"A4": "29",
"A5": "1",
"A6": "24.7",
"A7": "3",
"A8": "6.84",
"A9": "60.9",
"kappa0": "0",
"kappa1": "7.82",
"kappa2": "4",
"kappa3": "7.53",
"kappa4": "0.191",
"kappa5": "1.23"
},
"check": "Transcribed from the library JSON of Table 5 by a scratch generator and compared with the table image (2026-09-26). The model cannot be evaluated from the paper alone (N_Si and the critical-point identification of Tables 3-4 are needed); not reproduced numerically."
}
Table 5 (p. 1653), column ‘LO+TA’. Fixed values (dagger): A0, A5, kappa0. Printed ‘-’ for C: no C for a single-term band.
franta_2014b/two_phonon_LA_plus_TA1#
{
"id": "two_phonon_LA_plus_TA1",
"material": "Si",
"equation": "Eqs. (14)-(31), pp. 1645-1649; Table 5, p. 1653",
"printed": "Two-phonon band LA+TA(1): F_(A+-B) = f_(A+-B)(E,T) P_(A+-B)(E), P = H(E) sum_i A_i L_i(E), normalised at 300 K, eps-broadened (Gauss-Dawson), eps_(A+-B) = alpha N_Si eps^0",
"variable": "E: photon energy; critical-point energies from the phonon frequencies (formula phonon_frequencies); beta: Gaussian FWHM in cm^-1 (Eq. (28))",
"coefficients": {
"alpha": "5.06e-6",
"beta": "8.98",
"C": "1",
"A0": "0.0154",
"A1": "0.214",
"A2": "0.0481",
"A3": "2.39e-5",
"A4": "0.42",
"A5": "1",
"A6": "0.0725",
"A7": "0.024",
"A8": "0.000503",
"A9": "0.533",
"kappa0": "0",
"kappa1": "0.256",
"kappa2": "0.248",
"kappa3": "10",
"kappa4": "2.08",
"kappa5": "0.953"
},
"check": "Transcribed from the library JSON of Table 5 by a scratch generator and compared with the table image (2026-09-26). The model cannot be evaluated from the paper alone (N_Si and the critical-point identification of Tables 3-4 are needed); not reproduced numerically."
}
Table 5 (p. 1653), column ‘LA+TA(1)’. Fixed values (dagger): C, A5, kappa0.
franta_2014b/two_phonon_LA_plus_TA2#
{
"id": "two_phonon_LA_plus_TA2",
"material": "Si",
"equation": "Eqs. (14)-(31), pp. 1645-1649; Table 5, p. 1653",
"printed": "Two-phonon band LA+TA(2): F_(A+-B) = f_(A+-B)(E,T) P_(A+-B)(E), P = H(E) sum_i A_i L_i(E), normalised at 300 K, eps-broadened (Gauss-Dawson), eps_(A+-B) = alpha N_Si eps^0",
"variable": "E: photon energy; critical-point energies from the phonon frequencies (formula phonon_frequencies); beta: Gaussian FWHM in cm^-1 (Eq. (28))",
"coefficients": {
"C": "0.0755",
"A0": "0",
"A1": "0",
"A2": "0",
"A3": "0",
"A4": "0",
"A5": "1",
"A6": "0",
"A7": "0",
"A8": "1",
"A9": "0",
"kappa0": "0",
"kappa1": "1",
"kappa2": "1",
"kappa3": "1",
"kappa4": "1",
"kappa5": "1"
},
"check": "Transcribed from the library JSON of Table 5 by a scratch generator and compared with the table image (2026-09-26). The model cannot be evaluated from the paper alone (N_Si and the critical-point identification of Tables 3-4 are needed); not reproduced numerically."
}
Table 5 (p. 1653), column ‘LA+TA(2)’. Fixed values (dagger): A0, A1, A2, A3, A4, A5, A6, A7, A8, A9, kappa0, kappa1, kappa2, kappa3, kappa4, kappa5. Printed ‘-’ for alpha, beta: this term shares alpha and beta with the (1) term (Eq. (30)).
franta_2014b/two_phonon_TA_plus_TA#
{
"id": "two_phonon_TA_plus_TA",
"material": "Si",
"equation": "Eqs. (14)-(31), pp. 1645-1649; Table 5, p. 1653",
"printed": "Two-phonon band TA+TA: F_(A+-B) = f_(A+-B)(E,T) P_(A+-B)(E), P = H(E) sum_i A_i L_i(E), normalised at 300 K, eps-broadened (Gauss-Dawson), eps_(A+-B) = alpha N_Si eps^0",
"variable": "E: photon energy; critical-point energies from the phonon frequencies (formula phonon_frequencies); beta: Gaussian FWHM in cm^-1 (Eq. (28))",
"coefficients": {
"alpha": "1.96e-7",
"beta": "8.63",
"A0": "0",
"A1": "0",
"A2": "0",
"A3": "0",
"A4": "0",
"A5": "1",
"A6": "0.00901",
"A7": "1.9e-5",
"A8": "0.187",
"A9": "0",
"kappa0": "0",
"kappa1": "1",
"kappa2": "1",
"kappa3": "0.183",
"kappa4": "9.98",
"kappa5": "0"
},
"check": "Transcribed from the library JSON of Table 5 by a scratch generator and compared with the table image (2026-09-26). The model cannot be evaluated from the paper alone (N_Si and the critical-point identification of Tables 3-4 are needed); not reproduced numerically."
}
Table 5 (p. 1653), column ‘TA+TA’. Fixed values (dagger): A0, A1, A2, A3, A4, A5, A9, kappa0, kappa1, kappa2, kappa5. Printed ‘-’ for C: no C for a single-term band.
franta_2014b/two_phonon_TO_minus_TA#
{
"id": "two_phonon_TO_minus_TA",
"material": "Si",
"equation": "Eqs. (14)-(31), pp. 1645-1649; Table 5, p. 1653",
"printed": "Two-phonon band TO-TA: F_(A+-B) = f_(A+-B)(E,T) P_(A+-B)(E), P = H(E) sum_i A_i L_i(E), normalised at 300 K, eps-broadened (Gauss-Dawson), eps_(A+-B) = alpha N_Si eps^0",
"variable": "E: photon energy; critical-point energies from the phonon frequencies (formula phonon_frequencies); beta: Gaussian FWHM in cm^-1 (Eq. (28))",
"coefficients": {
"alpha": "1.59e-7",
"beta": "6.40",
"A0": "4.26",
"A1": "20.1",
"A2": "0.511",
"A3": "6.41",
"A4": "1.9",
"A5": "1",
"A6": "0.329",
"A7": "0.00061",
"A8": "1.36",
"A9": "0.00811",
"kappa0": "1.35",
"kappa1": "0.785",
"kappa2": "1.07",
"kappa3": "1.8",
"kappa4": "1.1",
"kappa5": "0"
},
"check": "Transcribed from the library JSON of Table 5 by a scratch generator and compared with the table image (2026-09-26). The model cannot be evaluated from the paper alone (N_Si and the critical-point identification of Tables 3-4 are needed); not reproduced numerically."
}
Table 5 (p. 1653), column ‘TO-TA’. Fixed values (dagger): A5, kappa5. Printed ‘-’ for C: no C for a single-term band.
franta_2014b/two_phonon_LO_minus_TA#
{
"id": "two_phonon_LO_minus_TA",
"material": "Si",
"equation": "Eqs. (14)-(31), pp. 1645-1649; Table 5, p. 1653",
"printed": "Two-phonon band LO-TA: F_(A+-B) = f_(A+-B)(E,T) P_(A+-B)(E), P = H(E) sum_i A_i L_i(E), normalised at 300 K, eps-broadened (Gauss-Dawson), eps_(A+-B) = alpha N_Si eps^0",
"variable": "E: photon energy; critical-point energies from the phonon frequencies (formula phonon_frequencies); beta: Gaussian FWHM in cm^-1 (Eq. (28))",
"coefficients": {
"alpha": "3.94e-7",
"beta": "7.30",
"A0": "3.64",
"A1": "17.5",
"A2": "26.2",
"A3": "8.74",
"A4": "27.3",
"A5": "1",
"A6": "0.171",
"A7": "5.84e-5",
"A8": "3.77",
"A9": "0.00146",
"kappa0": "0.186",
"kappa1": "0.933",
"kappa2": "2.89",
"kappa3": "9.99",
"kappa4": "0.483",
"kappa5": "0"
},
"check": "Transcribed from the library JSON of Table 5 by a scratch generator and compared with the table image (2026-09-26). The model cannot be evaluated from the paper alone (N_Si and the critical-point identification of Tables 3-4 are needed); not reproduced numerically."
}
Table 5 (p. 1653), column ‘LO-TA’. Fixed values (dagger): A5, kappa5. Printed ‘-’ for C: no C for a single-term band.
franta_2014b/two_phonon_LA_minus_TA#
{
"id": "two_phonon_LA_minus_TA",
"material": "Si",
"equation": "Eqs. (14)-(31), pp. 1645-1649; Table 5, p. 1653",
"printed": "Two-phonon band LA-TA: F_(A+-B) = f_(A+-B)(E,T) P_(A+-B)(E), P = H(E) sum_i A_i L_i(E), normalised at 300 K, eps-broadened (Gauss-Dawson), eps_(A+-B) = alpha N_Si eps^0",
"variable": "E: photon energy; critical-point energies from the phonon frequencies (formula phonon_frequencies); beta: Gaussian FWHM in cm^-1 (Eq. (28))",
"coefficients": {
"alpha": "1.85e-7",
"beta": "7.45",
"A0": "34.6",
"A1": "139",
"A2": "40.1",
"A3": "77.8",
"A4": "218",
"A5": "1",
"A6": "0.513",
"A7": "21.5",
"A8": "0.0417",
"A9": "0.245",
"kappa0": "0",
"kappa1": "1.74",
"kappa2": "9.89",
"kappa3": "9.88",
"kappa4": "0.1",
"kappa5": "0.00079"
},
"check": "Transcribed from the library JSON of Table 5 by a scratch generator and compared with the table image (2026-09-26). The model cannot be evaluated from the paper alone (N_Si and the critical-point identification of Tables 3-4 are needed); not reproduced numerically."
}
Table 5 (p. 1653), column ‘LA-TA’. Fixed values (dagger): A5, kappa0. Printed ‘-’ for C: no C for a single-term band.
franta_2014b/two_phonon_TO_minus_LA#
{
"id": "two_phonon_TO_minus_LA",
"material": "Si",
"equation": "Eqs. (14)-(31), pp. 1645-1649; Table 5, p. 1653",
"printed": "Two-phonon band TO-LA: F_(A+-B) = f_(A+-B)(E,T) P_(A+-B)(E), P = H(E) sum_i A_i L_i(E), normalised at 300 K, eps-broadened (Gauss-Dawson), eps_(A+-B) = alpha N_Si eps^0",
"variable": "E: photon energy; critical-point energies from the phonon frequencies (formula phonon_frequencies); beta: Gaussian FWHM in cm^-1 (Eq. (28))",
"coefficients": {
"alpha": "3.16e-7",
"beta": "11.18",
"A0": "8.42",
"A1": "15.7",
"A2": "17.1",
"A3": "16.3",
"A4": "38.2",
"A5": "1",
"A6": "25.1",
"A7": "7.28",
"A8": "7.24",
"A9": "0.00539",
"kappa0": "0.876",
"kappa1": "0.195",
"kappa2": "0.259",
"kappa3": "0.152",
"kappa4": "0.378",
"kappa5": "0"
},
"check": "Transcribed from the library JSON of Table 5 by a scratch generator and compared with the table image (2026-09-26). The model cannot be evaluated from the paper alone (N_Si and the critical-point identification of Tables 3-4 are needed); not reproduced numerically."
}
Table 5 (p. 1653), column ‘TO-LA’. Fixed values (dagger): A5, kappa5. Printed ‘-’ for C: no C for a single-term band.
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.