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 <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/franta_2013.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1016/j\.tsf\.2013\.01\.081 <https://doi.org/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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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 :doc:`review register </register>` for the checks and decisions.