franta_2013#
D. Franta, D. Nečas and L. Zajíčková, “Application of Thomas–Reiche–Kuhn sum rule to construction of advanced dispersion models”, Thin Solid Films 534, 432 (2013)
Source record · Catalogue · Review register
DOI: 10.1016/j.tsf.2013.01.081
Source kind: theory
Recorded technique: Theory: the Thomas-Reiche-Kuhn sum rule generalised to electrons and nuclei, the transition strength function F(E) as the condensed-matter oscillator strength, and the parametrised contributions (indirect and direct interband transitions with excitons, intraband, high-energy, core and phonon terms) from which Franta’s ‘advanced dispersion models’ (ADM) and later the ‘universal dispersion model’ (UDM) are built. No measurement; Table 1 derives densities from handbook mass densities.
Materials in structured entries: C_diamond, C_graphite, Si, SiO2, a-C_H, a-Si_H
Dataset quantities: atom_density
Contents: 6 datasets; 1 formula.
Curation notes#
Journal pages 432-441 = PDF pages 1-10. Identity checked against the first page. This is the foundation paper of the UDM family (franta_2014, franta_2014b, franta_2015, franta_2015b, franta_2016, franta_2017, franta_2025); the model is implemented from these notes and the papers.
EQUATION MAP (equation, journal page; E photon energy in eV; all functions of E are extended to E < 0 by the symmetries Eqs. (29)-(31): eps_r and F even, eps_i and J odd). - Sum rule and its constants. Eq. (25), p. 435: int_0^inf F(E) dE = N_e U (script N: densities per m^3), U from Eq. (22): U = 1 + (m_e/u) sum_n (Z_n^2 f_n/M_n)/sum_n Z_n f_n, M_n = A_n - Z_n m_e/u (Eq. (23)), m_e/u = 5.4858e-4, ‘U ~ 1 + 0.5 m_e/u ~ 1.000274’ (p. 435). Eq. (27): int_0^inf eps_i(E) E dE ~ ((e h)^2/(8 pi eps0 m_e)) N_e U. Rescaled form used by every later paper: Eq. (42) eps_i(E) = F(E)/E (F in eV^2 after dividing by M), Eq. (43) M = 8 pi eps0 m_e/(e h)^2 = 4.61706e26 eV^-2 m^-3, Eq. (44) int_0^inf F dE = N (eV^2), Eq. (45) N = N_e + sum_n N_n = N_e U, Eq. (46) script-N_e = M N_e, Eq. (47) N = N_a U sum_n Z_n f_n (N_a the ‘density parameter’, eV^2), Eq. (48) atomic density = M N_a, Eq. (49) electron density = M N_a sum_n Z_n f_n, Eq. (50) rho = N_a R sum_n A_n f_n, Eq. (51) R = u M = 0.766681 eV^-2 kg m^-3 (p. 436). - Assembly (normalisation). Eq. (52), p. 437: eps_i,t = F_t/E, eps_r,t(E) = (2/pi) int_0^inf F_t(X)/(X^2 - E^2) dX (KK); Eq. (53): eps = 1 + sum_t eps_t; Eqs. (54)-(55): F_t = N_t F_t^0 with int_0^inf F_t^0 dE = 1 and sum_t N_t = N. Every later UDM term is ‘N_t times a unit-strength normalised dielectric function’. - Interband, indirect/amorphous (IBT, the UDM ‘vc’ term): Eq. (56), p. 437: F^0 = (|E| - E_g)^2 (E_h - |E|)^2/(C_N |E|) Pi_{E_g,E_h}(|E|), Pi the unit box (Eq. (57)), C_N Eq. (58) (identical to franta_2015 Eq. (7) and franta_2015b Eq. (5)). Eq. (59): the Lorentz-modified form (|E| - E_g)^2 (E_h - |E|)^2/(C_N |E|^kappa [(|E| - E_c)^2 + B_c^2]^nu), kappa = 0, 1, 2, nu = 0, 1 (kappa = nu = 1 for a-Si and DLC). Eqs. (60)-(61): crystalline indirect gap with phonon creation/annihilation, F_idt^0 = [(n_p + 1) F^0(E + E_p) + n_p F^0(E - E_p)]/(2 n_p + 1), n_p Bose-Einstein. Eq. (62): the JDOS from DOS convolution. - Direct interband with critical-point singularities (crystals): Eqs. (63)-(69), p. 438 (regions I-III between M0..M3; square-root and linear pieces). Excitons (Wannier, including discrete bound states): Eqs. (70)-(77), pp. 438-439, R_0..R_3 effective Rydbergs; Eq. (74) the R -> 0 limit. Frenkel excitons in ionic/amorphous solids ‘can be modeled using Gaussian peaks’ (p. 439): this is the origin of the Gaussian ‘loc’ and exciton peaks used later. - Intraband (free carriers): Eqs. (78)-(79), p. 439. - High-energy transitions (HET, the UDM ‘vx’ term): Eq. (80), p. 439: F^0 = (|E| - E_xi)^2/(C_N E^2) F_class(E) Pi_{E_xi,inf}; F_class DHO (Eq. (81), gives Tauc-Lorentz), Lorentz (Eq. (82)), Drude (Eq. (83)) or 1/E^2; the chosen two-parameter form Eq. (84): F_het^0 = (|E| - E_xi)^2/(C_N E^4) Pi_{E_xi,inf}(|E|) (C_N = 1/(3 E_xi), computed 2026-09-26, consistent with franta_2015 Eq. (17)). - Core electrons: Eq. (85), p. 439: F_cl^0 = 1/(C_N E^2) Pi_{E_cl,inf}. - Phonons: Eqs. (86)-(89), p. 440: N_ph < (U - 1) N_e; one-phonon delta lines (Eqs. (87)-(88)); for amorphous solids Gaussian peaks, Eq. (89): F_1ph(E) = sum_m N_m [beta_m(E_m - E) + beta_m(E_m + E)], beta_m a normalised Gaussian of width B_m. Two-phonon bands Eqs. (90)-(92) (Bose-Einstein factors n_p1 + n_p2 + 1 and n_p2 - n_p1). - Urbach tail: not in this paper (‘will be presented in works dealing with specific materials’, p. 436); first given in Franta et al., Thin Solid Films 539, 233 (2013) (not in the library) and in franta_2015 Eq. (18) / franta_2015b Eqs. (32)-(40). - Broadening: Eq. (19), p. 434: convolution with a normalised broadening function; see franta_2014 for which quantity to broaden.
VALUES. Phonon strengths quoted (p. 440): N_ph ~ 8.6e-9 N_e for c-Si and ~1.7e-5 N_e for fused silica; N_ph = 3.14e-5 N_Si for c-Si at room temperature (p. 437); N_K ~ 1.61 N_a for metallic Al (Shiles et al., p. 437). Not extracted (no quantity).
TABLE 1 (p. 436, checked against the image 2026-09-26): U, N_a (eV^2), atomic density and electron density (1/m^3), mass density (kg/m^3; ‘The mass densities of the crystalline materials and fused silica (a-SiO2) were taken from the given references [32: Martienssen and Warlimont, Springer Handbook of Condensed Matter and Materials Data, 2005; 33: Greenwood and Earnshaw, Chemistry of the Elements, 1984] and the remaining parameters were calculated from the mass density and material composition. In the case of amorphous hydrogenated silicon and diamond-like carbon (DLC), N_a was determined from optical data and the remaining parameters, including mass density, were calculated from N_a and material composition. The hydrogen atomic fractions in a-Si:H and DLC were 0.11 and 0.34’). Rows: c-Si U 1.0002735, N_a 108.2, 4.994e28, 6.991e29, 2329; diamond 1.0002741, 381.7, 1.762e29, 1.057e30, 3515; graphite 1.0002741, 246.1, 1.136e29, 6.818e29, 2266; a-SiO2 1.0002740, 143.3, 6.616e28, 6.616e29, 2200; a-Si:H 1.0002759, 122.9, 5.674e28, 7.133e29, 2366; a-C:H (DLC) 1.0002955, 321.4, 1.484e29, 6.381e29, 2038. Only the atomic density is a quantity in the vocabulary (datasets below); the electron densities count all electrons, not valence electrons, and are not extracted. CHECK (2026-09-26, standard atomic weights and CODATA constants): M = 8 pi eps0 m_e/(e h)^2 = 4.617060e26 eV^-2 m^-3 and R = u M = 0.766681, as printed; the atomic densities, N_a and electron densities of c-Si, diamond, graphite and a-SiO2 follow from the mass densities to the printed digits (graphite’s electron density 6.817e29 against 6.818e29 and a-SiO2’s 6.615e28/6.615e29 against 6.616e28/6.616e29, rounding of the atomic weights).
Recorded bibliographic identity status: VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
franta_2013/atom_density_Si#
{
"id": "atom_density_Si",
"material": "Si",
"file": "Si/franta_2013__atom_density_Si.csv",
"table": "tbl_7c9db7f2834e",
"quantity": "atom_density",
"unit": "m-3",
"abscissa": "none",
"use": "check",
"parameters": {
"counts": "atoms"
}
}
Table 1, c-Si: computed from the handbook mass density 2329 kg/m3 [32] and the composition (not measured here).
franta_2013/atom_density_C_diamond#
{
"id": "atom_density_C_diamond",
"material": "C_diamond",
"file": "C_diamond/franta_2013__atom_density_C_diamond.csv",
"table": "tbl_7c9db7f2834e",
"quantity": "atom_density",
"unit": "m-3",
"abscissa": "none",
"use": "check",
"parameters": {
"counts": "atoms"
}
}
Table 1, c-C (diamond): computed from the handbook mass density 3515 kg/m3 [32].
franta_2013/atom_density_C_graphite#
{
"id": "atom_density_C_graphite",
"material": "C_graphite",
"file": "C_graphite/franta_2013__atom_density_C_graphite.csv",
"table": "tbl_7c9db7f2834e",
"quantity": "atom_density",
"unit": "m-3",
"abscissa": "none",
"use": "check",
"parameters": {
"counts": "atoms"
}
}
Table 1, c-C (graphite): computed from the mass density 2266 kg/m3 [33].
franta_2013/atom_density_SiO2#
{
"id": "atom_density_SiO2",
"material": "SiO2",
"file": "SiO2/franta_2013__atom_density_SiO2.csv",
"table": "tbl_7c9db7f2834e",
"quantity": "atom_density",
"unit": "m-3",
"abscissa": "none",
"use": "check",
"parameters": {
"counts": "atoms"
}
}
Table 1, a-SiO2 (fused silica): computed from the handbook mass density 2200 kg/m3 [32]; atoms, i.e. 3 per SiO2 unit.
franta_2013/atom_density_a-Si_H#
{
"id": "atom_density_a-Si_H",
"material": "a-Si_H",
"file": "a-Si_H/franta_2013__atom_density_a-Si_H.csv",
"table": "tbl_7c9db7f2834e",
"quantity": "atom_density",
"unit": "m-3",
"abscissa": "none",
"use": "primary",
"parameters": {
"counts": "atoms",
"form": "film"
}
}
Table 1, a-Si:H: from N_a ‘determined from optical data’ (which film and which work are not stated; hydrogen atomic fraction 0.11).
franta_2013/atom_density_a-C_H#
{
"id": "atom_density_a-C_H",
"material": "a-C_H",
"file": "a-C_H/franta_2013__atom_density_a-C_H.csv",
"table": "tbl_7c9db7f2834e",
"quantity": "atom_density",
"unit": "m-3",
"abscissa": "none",
"use": "primary",
"parameters": {
"counts": "atoms",
"form": "film"
}
}
Table 1, a-C:H (DLC): from N_a ‘determined from optical data’ (film not stated; hydrogen atomic fraction 0.34).
Formulas#
franta_2013/sum_rule_constants#
{
"id": "sum_rule_constants",
"material": "SiO2",
"equation": "Eqs. (43), (47)-(51), p. 436",
"printed": "M = 8 \\pi \\epsilon_0 m_e/(e h)^2 = 4.61706 \\times 10^{26} eV^{-2} m^{-3}; N = N_a U \\sum_n Z_n f_n; \\rho = N_a R \\sum_n A_n f_n, R = u M = 0.766681 eV^{-2} kg m^{-3}",
"variable": "N_a: density parameter (eV^2); f_n, Z_n, A_n: atomic fractions, proton numbers, atomic weights",
"coefficients": {
"M": "4.61706e26",
"R": "0.766681",
"me_over_u": "5.4858e-4"
},
"check": "M and R recomputed from CODATA constants: 4.617060e26 and 0.766681 (2026-09-26). Table 1's densities follow (source notes)."
}
Material-independent constants of the UDM/ADM sum rule; the material key is nominal. Used by franta_2015 (N_a = 180 eV^2 for HfO2) and franta_2017 (N = Z N_a U).
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.