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 `__ · :doc:`Catalogue ` · :doc:`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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 :doc:`review register ` for the checks and decisions.