li_1980 ======= H\. H\. Li, "Refractive index of silicon and germanium and its wavelength and temperature derivatives", J\. Phys\. Chem\. Ref\. Data 9, 561 \(1980\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1063/1\.555624 `__ **Source kind:** compilation **Recorded technique:** A critical evaluation \(CINDAS, Purdue\) of the published refractive index of silicon and germanium and its temperature derivative\. The recommended values are computed from a dispersion equation, n^2 = epsilon\(T\) \+ L\(T\) A\(T\)/lambda^2 \(Eq\. 22 for Si\), fitted by least squares to selected data sets\: for Si those of Salzberg and Villa \(1957\), Primak \(1971\) and Icenogle, Platt and Wolfe \(1976\) \(p\. 8\), with epsilon\(T\) scaled from Icenogle's n\(T\) at 10\.27 um extrapolated with Lukes's high\-temperature curves \(Eqs\. 20\-21\), and L\(T\) = exp\(\-3 Delta L/L\_293\) from Touloukian's thermal expansion \(ref\. 34\)\. dn/dT and dn/dlambda \(Tables 2, 3\) are the first derivatives of Eq\. \(22\)\. Si\: 1\.2\-14 um, 100\-750 K\. Ge\: 1\.9\-16 um \(abstract; 1\.9\-18 um, p\. 2\), 100\-550 K\. **Materials in structured entries:** Si **Dataset quantities:** refractive\_index, thermo\_optic\_coefficient **Contents:** 2 datasets; 1 formula. Curation notes -------------- Pages 561\-658; the library copy is AIP's scan behind a cover page, so a PDF page is the journal page minus 558 \(Table 1 is PDF p\. 17, journal p\. 575\); pages cited here are PDF pages\. Identity checked against the first page\. Secondary for all its measured values; primary only for its evaluation \(the recommended values and Eq\. 22\)\. Only the silicon 293 K columns of Tables 1 and 2 are transcribed\. WHAT IT RECOMMENDS FOR SI\. Table 1, n at 35 wavelengths 1\.20\-14\.00 um and 14 temperatures 100\-750 K; Table 2, dn/dT \(10^\-4 K^\-1\) on the same grid; Table 3, \-dn/dlambda \(10^\-4 um^\-1\) at 293 K\. All three are computed from Eq\. \(22\) \(p\. 9\)\: 'calculated values in tables 1, 2, and 3 are given with more digits than warranted merely for the purpose of tabular smoothness \.\.\. indicated with an overstrike' \(p\. 9\)\. Eq\. \(22\) as corrected below reproduces Table 1 within 0\.7e\-4 at 100\-350 K and 0\.8e\-4 at 400\-450 K \(its rounding; cells the library misreads aside\), and within about 1\.8e\-4 at 500\-750 K, largest at 1\.2\-1\.3 um \(computed 2026\-10\-14\)\. Its one\-sided T\-derivative at 293 K reproduces Table 2's 293 K column within 0\.001 \(in 10^\-4 K^\-1\) at every wavelength; the central difference straddles the two branches of Delta L/L, which do not meet at 293 K \(Delta L/L = \+1\.6e\-6 and \-2\.5e\-6 there\)\. UNCERTAINTY\. n\: 'Equation \(22\) was used to calculate the recommended values of the refractive index of silicon with uncertainties of \+\-2 x 10^\-3' \(p\. 9\); 'The accuracy of the recommended refractive index values in this work is estimated to be 1 to 2 x 10^\-3' \(p\. 6\); the selected data agree with the equations 'to within \+\-2\.0 x 10^\-3' \(p\. 3\)\. Table 1's footnote prints '\+\-2X10^\-3' \(the scan's exponent is smeared; the library's text reads 10^\-4, the image and p\. 9 give \-3\)\. dn/dT\: '\+\-0\.15 x 10^\-4 K^\-1' \(Table 2 footnote\), taken from Icenogle et al\.'s stated accuracy of their Delta n/Delta T \(p\. 9\)\. dn/dlambda\: about \+\-10e\-4/um at 2 um \(p\. 9\), \+\-2\.4e\-4 at 2\.55 um, \+\-0\.6e\-4 at 10 um, \+\-0\.44e\-4 at 14 um\. No level stated\. MEDIUM\. Li never says whether the index is absolute or relative to air, nor whether the wavelengths are vacuum or air values\. The three selected Si data sets are prism measurements \(minimum deviation, autocollimation; pp\. 7\-8\); for Ge, Li corrected Rank et al\.'s interferometric vacuum index 'from vacuum values to air values' \(p\. 24\), which suggests the evaluation is relative to air throughout, but this is not stated for Si\. The datasets record index\_medium = 'not stated'\. Relative to air at 293 K the index would be 1\.00027 times higher \(about \+9e\-4 at n = 3\.45\), half the stated uncertainty\. The wavelength medium matters less\: an air/vacuum wavelength difference moves n by at most 6e\-5 \(1\.2 um, \-dn/dlambda = 0\.169/um, Table 3\)\. Wavelengths are taken as vacuum \(the default\)\. COMPARISON WITH FREY ET AL\. \(frey\_2006, CHARMS, absolute, checked 2026\-10\-14\)\. At 293 K, Li minus Frey's Table 2 taken from 295 K to 293\.15 K with their Table 4\: \-5\.8e\-3 at 1\.2 um, \-4\.2e\-3 at 1\.3 um, \-3\.3e\-3 at 1\.4 um, \-2\.8e\-3 at 1\.5 um, \-2\.2e\-3 at 2\.0 um, \-2\.4e\-3 at 2\.5 um, \-2\.5e\-3 at 3\.0 um, \-2\.8e\-3 at 4\.0 um, \-3\.0e\-3 at 5\.0 um\: outside Li's \+\-2e\-3 at every common wavelength, and still \-1\.3 to \-3\.3e\-3 at 1\.3\-5\.0 um \(\-4\.9e\-3 at 1\.2 um\) if Li's values are relative to air \(times 1\.00027\)\. Li's form has no term for the band gap \(B = 0 in Eq\. 10\), so it cannot follow the rise of n towards 1\.1 um\. dn/dT\: Li's 293 K column is 0\.9\-6\.0 % below Frey's Table 4 at 295 K \(1\.983e\-4 against 2\.01e\-4 at 1\.2 um, 1\.717e\-4 against 1\.78e\-4 at 2\.0 um, 1\.588e\-4 against 1\.69e\-4 at 5\.0 um\), within Li's \+\-0\.15e\-4 at every common wavelength\. REFRACTIVEINDEX\.INFO main/Si/nk/Li\-293K\.yml \(data/external\) tabulates Table 1's 293 K column\: all 35 values equal this record's file \(with the corrected 1\.50 um cell, 3\.4799\), compared 2026\-10\-14\. Its citation gives the year as 1993\. THE MEASUREMENTS IT TABULATES \(Appendix, Tables A\-1 to A\-4 for Si, 55 data sets of n; not transcribed; the originals are the primary sources\)\: Briggs \(1950, prism, 1\.05\-2\.60 um\); Salzberg and Villa \(1957, prism, 1\.35\-11\.04 um\); Cardona, Paul and Brooks \(1959; read from graphs\); Lukes \(1959, 1960; 1\.259\-5\.156 um, 109\-750 K; read from graphs; also the only dn/dT data, Tables A\-3 and A\-4\); Primak \(1971, 1\.12\-2\.16 um\); Villa \(1972\); Icenogle, Platt and Wolfe \(1976, 2\.554\-10\.27 um, 99\-296 K\); Simon \(1951\); Runyan \(1960\); Philipp and Taft \(1960\); Archer \(1962\); Spitzer, Gobeli and Trumbore \(1964, heavily doped\); Bennett and Booty \(1966\); Randall and Rawcliffe \(1967, far infrared\); Sato \(1967\); Verleur \(1968\); Grigorovici and Vancu \(1968, amorphous films\); Vedam, Knausenberger and Lukes \(1969\); Brodsky et al\. \(1970, amorphous films\); Shevchenko et al\. \(1972\); Thutupalli and Tomlin \(1977, films\)\. Hertzberger and Salzberg's dispersion formula \(1962, from Salzberg and Villa's data, 1\.3\-11\.0 um\) is quoted on p\. 7\. Ge has the same structure \(Tables 4\-6, A\-5 to A\-8; Eq\. 28\); not transcribed\. NOT TRANSCRIBED\. The other 13 columns of Tables 1 and 2\: the scan is scratched across rows 1\.30\-1\.60 of both tables, and several cells there cannot be read with certainty \(e\.g\. Table 1 at 1\.50 um and 350 K, at 1\.55 um and 250 K; the library reads them 3\.4756 and 3\.4721, Eq\. \(22\) gives 3\.4908 and 3\.4681\); refractiveindex\.info's Li\-\K files tabulate them\. Table 3 \(dn/dlambda; no quantity\)\. **Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- li\_1980/n\_293K ~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "n_293K", "material": "Si", "file": "Si/li_1980__n_293K.csv", "table": "tbl_67c9bf1134a1", "quantity": "refractive_index", "abscissa_unit": "um", "use": "check", "parameters": { "temperature_k": 293, "form": "bulk", "technique": "evaluation", "index_medium": "not stated" }, "uncertainty": { "absolute": 0.002, "level": "not stated" } } Table 1, 293 K column\: Li's recommended n at 35 wavelengths, 1\.20\-14\.00 um, computed from Eq\. \(22\), a fit to Salzberg and Villa, Primak and Icenogle et al\.; printed to four decimals, the last overstruck as insignificant\. One library misreading corrected in the script \(1\.50 um\: page 3\.4799\)\. A check, not fitted\: it is an evaluation of measurements that are primary elsewhere, its uncertainty \(\+\-2e\-3, p\. 9 and the table footnote; level not stated\) is 20 times Frey et al\.'s, it lies 2\.2\-4\.2e\-3 below frey\_2006 at 1\.3\-5 um \(5\.8e\-3 at 1\.2 um\), and its medium \(absolute or relative to air\) is not stated \(see the source notes\)\. li\_1980/dndt\_293K ~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "dndt_293K", "material": "Si", "file": "Si/li_1980__dndt_293K.csv", "table": "tbl_c785eb5d7e50", "quantity": "thermo_optic_coefficient", "unit": "1/K", "scale": 0.0001, "abscissa_unit": "um", "use": "check", "parameters": { "temperature_k": 293, "form": "bulk", "technique": "evaluation", "index_medium": "not stated" }, "uncertainty": { "absolute": 0.15, "level": "not stated" } } Table 2, 293 K column\: Li's recommended dn/dT at 35 wavelengths, 1\.20\-14\.00 um, the T\-derivative of Eq\. \(22\)\. The table prints no unit; its footnote \('\+\-0\.15X10^\-4 K^\-1'\) and the values \(Figure 9\) are in 10^\-4 K^\-1, hence the scale\. One library misreading corrected in the script \(1\.20 um\: page 1\.983\)\. Uncertainty \+\-0\.15e\-4 K^\-1 \(footnote; from Icenogle et al\.'s stated accuracy, p\. 9; level not stated\)\. 0\.9\-6\.0 % below frey\_2006's measured dn/dT at 295 K, within its stated uncertainty \(see the source notes\)\. Formulas -------- li\_1980/eq22\_Si ~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eq22_Si", "material": "Si", "equation": "Eq. (22), p. 8, with Delta L/L_293 from ref. 34, p. 9", "printed": "n^2(lambda, T) = epsilon(T) + L(T)/lambda^2 (A_0 + A_1 T + A_2 T^2); epsilon(T) = 11.4445 + 2.7739e-4 T + 1.7050e-6 T^2 - 8.1347e-10 T^3; L(T) = exp(-3 Delta L(T)/L_293); Delta L(T)/L_293 = -0.021 - 4.149e-7 T - 4.620e-10 T^2 + 1.482e-11 T^3 (20-293 K), = -0.071 + 1.887e-6 T + 1.934e-9 T^2 - 4.544e-13 T^3 (293-1600 K)", "variable": "lambda: wavelength in um (medium not stated); T: temperature in K", "range": "1.2-14 um, 100-750 K (p. 8)", "coefficients": { "eps0": "11.4445", "eps1": "2.7739e-4", "eps2": "1.7050e-6", "eps3": "-8.1347e-10", "A0": "0.8948", "A1": "4.3977e-4", "A2": "7.3835e-8", "dL_low0": "-0.021", "dL_low1": "-4.149e-7", "dL_low2": "-4.620e-10", "dL_low3": "1.482e-11", "dL_high0": "-0.071", "dL_high1": "1.887e-6", "dL_high2": "1.934e-9", "dL_high3": "-4.544e-13" }, "check": "With the corrected constant terms, reproduces Table 1 within 0.7e-4 at 100-350 K and 0.8e-4 at 400-450 K (its rounding), within about 1.8e-4 at 500-750 K, and the 293 K column of Table 2 within 0.001 x 10^-4 K^-1 (one-sided derivative). Computed 2026-10-14 from the printed coefficients (image of pp. 8-9).", "corrections": [ { "term": "dL_low0", "printed": "-0.021", "corrected": "-0.00021", "reason": "Delta L/L_293 must vanish at 293 K. With the other three printed terms it does with a constant of -2.1e-4 (+1.6e-6 at 293 K), not -0.021: the constant is printed in percent (Touloukian's Delta L/L is tabulated in percent), the other terms as fractions.", "evidence": "With -0.021, L(100 K) = 1.065 and n(2 um, 100 K) = 3.42617 against Table 1's 3.4240; with -0.00021, 3.42396. The corrected expansion reproduces Table 1 at 100-250 K to its rounding (computed 2026-10-14)." }, { "term": "dL_high0", "printed": "-0.071", "corrected": "-0.00071", "reason": "As for dL_low0: the constant is in percent; -7.1e-4 makes Delta L/L_293 = -2.5e-6 at 293 K.", "evidence": "With -0.00071 Eq. (22) reproduces Table 1 at 350-450 K to its rounding and at 500-750 K within about 1.8e-4; with -0.071, L(T) = 1.23 at 350-750 K and n is 0.009-0.010 too high at 2 um, 0.024-0.028 at 1.2 um (computed 2026-10-14)." } ] } The room\-temperature dielectric constant from epsilon\(293 K\) is 11\.66 \(p\. 9\)\. Coefficient names\: eps0\-eps3 for epsilon\(T\), A0\-A2, and dL\_low0\-3 \(20\-293 K\) and dL\_high0\-3 \(293\-1600 K\) for Delta L/L\_293\. The second\-order coefficient is printed 'A\_2 T'^2' \(a stray prime\)\. No published model\: the form \(with the piecewise expansion\) has no implementation\. 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.