7. References

[1] Foley, G. M., High-speed optical pyrometer, Rev. Sci. Instr. 41, 827 (1970).

[2] Cezairliyan, A., M. S. Morse, H. A. Berman, and C. W. Beckett, High-speed (subsecond) measurement of heat capacity, electrical resistivity, and thermal radiation_properties of molybdenum in the range 1900 to 2800 K, J. Res. Nat. Bur. Stand. (U.S.) 74A (Phys. and Chem.), 65 (1970).

[3] Cezairliyan, A., J. L. McClure, and C. W. Beckett, High-speed (subsecond) measurement of heat capacity, electrical resistivity, and thermal radiation properties of tantalum in the range 1900 to 3200 K, J. Res. Nat. Bur. Stand. (U.S.) 75A (Phys. and Chem.), 1 (1971).

[4] Tietz, T. E. and J. W. Wilson, Behavior and properties of refractory metals (Stanford University Press, California, 1965). p. 28. [5] International Practical Temperature Scale of 1968, Metrologia 5, 35 (1969).

[6] Hultgren, R., R. L. Orr, P. D. Anderson, and K. K. Kelly, Selected Values of Thermodynamic Properties of Metals and Alloys (John Wiley, New York, 1963).

[7] Schultz, H., “Quenching of vacancies in tungsten," in Lattice Defects in Quenched Metals, R. M. J. Cotterill, M. Doyama, J. J. Jackson and M. Meshi, Eds. (Academic Press, New York, 1965). p. 761.

[8] Gripshover, R. J., M. Khoshnevisan, J. S. Zetts, and J. Bass, A study of vacancies in tungsten wires quenched in superfluid helium, Phil. Mag. 22, 757 (1970).

[9] Meakin, J. D., A. Lawley, and R. C. Koo, "Vacancy loops in quenched molybdenum," in Lattice Defects in Quenched Metals, R. M. J. Cotterill, M. Doyama, J. J. Jackson, and M. Meshi, Eds. (Academic Press, New York, 1965). p. 767. [10] Worthing, A. G., Atomic heats of tungsten and of carbon at incandescent temperatures, Phys. Rev. 12, 199 (1918). [11] Jaeger, F. M. and E. Rosenbohm, Exact measurement of specific heat of solid substances at high temperatures, III: Pd and W, Koninkl. Ned. Akad. Wetenschap., Proc., Ser. B 33, 457 (1930).

[12] Hoch, M. and H. L. Johnston, A high-temperature drop calorimeter, the heat capacities of tantalum and tungsten between 1000 and 3000 K, J. Chem. Phys. 65, 855 (1961).

[13] Kirillin, V. A., A. E. Sheindlin, V. Ya. Chekhovskoi, and V. A. Petrov, Thermodynamic properties of tungsten, J. Phys. Chem. (USSR) 37, 1212 (1963).

[14] Kraftmakher, Ya. A. and P. G. Strelkov, Energy of formation of and concentration of vacancies in tungsten, Solid State Physics (USSR) 4, 1662 (1963).

[15] Lowenthal, G. C., The specific heat of metals between 1200 K and 2400 K, Australian J. Phys. 16, 47 (1963).

[16] Hein, R. A. and P. N. Flagella, Enthalpy measurements of UO2 and tungsten to 3260 K, General Electric Report GEMP-578, 1968.

[17] Leibowitz, L., M. G. Chasanov and L. W. Mishler, The enthalpy of solid tungsten from 2800 K to its melting point, Trans. Met. Soc. AIME 245, 981 (1969).

[18] West, E. D. and S. Ishihara, Enthalpy of tungsten, private communication.

[19] Forsythe, W. E. and A. G. Worthing, The properties of tungsten and the characteristics of tungsten lamps, Astrophys. J. 61, 146 (1925).

[20] Jones, H. A., A temperature scale for tungsten, Phys. Rev. 28, 202 (1926).

[21] Osborn, R. H., Thermal conductivities of tungsten and molyb denum at incandescent temperatures, J. Opt. Soc. Am. 31, 428 (1941).

[22] Platunov, E. S. and V. B. Fedorov, Use of photographic pyrometry in thermal studies, High Temperature 2, 568 (1964). [23] Neimark, B. E. and L. K. Voronin, Thermal conductivity, specific electrical resistivity, and total emissivity of refractory metals at high temperatures, High Temperature 6, 999 (1968). [24] Tye, R. P., Preliminary measurements on the thermal and electrical conductivities of molybdenum, niobium, tantalum, and tungsten, J. Less-Common Metals 3, 13 (1961).

[25] Allen, R. D., L. F. Glasier, and P. L. Jordan, Spectral emissiv. ity, total emissivity, and thermal conductivity of molybdenum, tantalum, and tungsten above 2300 K, J. Appl. Phys. 31, 1382 (1960). [26] Rudkin, R. L., W. J. Parker, and R. J. Jenkins, "Measurement of the thermal properties of metals at elevated temperatures,” in Temperature Its Measurement and Control in Science and Industry, C. M. Herzfeld, Ed., Vol. III, part 2, p. 523, (Reinhold. New York, 1962).

[27] Worthing, A. G., The true temperature scale of tungsten and its emissive power at incandescant temperatures, Phys. Rev. 10,377 (1917).

[28] DeVos, J. C., A new determination of the emissivity of tungsten ribbon, Physica 20,690 (1954).

[29] Larrabee, R. D., Spectral emissivity of tungsten, J. Opt. Soc. Am. 49,619 (1959).

[30] Latyev, L. N., V. Ya. Chekhovskoi, and E. N. Shestakov, Experimental determination of emissive power of tungsten in the visible region of the spectrum in the temperature range 1200 to 2600 K, High Temperature 7,610 (1969). [31] White, G. K. and S. B. Woods, Electrical and thermal resistivity of the transition elements at low temperatures. Phil. Trans. Royal Soc. (London) 251,273 (1959).

[32] Forsythe, W. E. and E. M. Watson, Resistance and radiation of tungsten as a function of temperature, J. Opt. Soc. Am. 24, 114 (1934).

(Paper 75A4-670

1. Introduction

The continuous emission of hydrogen has long been mportance in astrophysics and laboratory plasma sics. Despite this, an accurate calculation of the al emission coefficient has not been accomplished, ce recent calculations [1, 2]1 show discrepancies of much as 20 percent. This could be due to contribus from the continuous emission of H-, H2, and not being taken into account at all, or their emission efficients not being accurately calculated. Newly ailable absorption coefficients for H-, H+, and [3, 4, 5, 6, 7] now permit determination of the total ntinuous hydrogen emission coefficient with an certainty estimated not to exceed 2 percent through

the visible and near-ultraviolet spectral regions. The prime purpose of the calculation is to accurately scribe the spectrum (both continuous and line wing diation) of a hydrogen plasma. This is done for a mperature range from 8000 to 16000 K and for the avelength interval from 400 to 15000 Å.

An accurate calculation of the total continuous nission coefficient will allow the continuous emission ectrum of hydrogen to be used as a spectral radiance libration standard. With the exception of some fairly rrow spectral regions (around hydrogen line central avelengths) this continuum will be usable in the ear ultraviolet down to 1700 A (see section 3f), the sible, and the near infrared portions of the spectrum.

2. Calculation of Particle Densities

The following species were considered: H+, e ̄, and 1. The particle densities are calculated for a LTE2

Figures in brackets indicate the literature references at the end of this paper.

LTE is the abbreviation for local thermodynamic equilibrium which is a condition at the particles of the system obey a Maxwellian velocity distribution and the atomic vels are populated according to Maxwell-Boltzmann statistics. For a more complete scription of LTE see ref. [8].

hydrogen plasma at 1 atm. The usual assumption of charge conservation was made, and the plasma was treated as an ideal gas except for Coulomb interactions between the particles which were treated by the Debye-Hückel [9] approximation. High density corrections [10] were taken into account to obtain partition function cutoffs and modifications to the plasma pressure and Saha-equation. The particle densities obtained agree within 0.1 percent with those calculated by Patch [11].

3. Calculation of Emission Coefficients

Four major contributions were included in the calculation of the total continuous emission coefficient. They were: free-free and free-bound transitions of electrons in (a) the field of protons (H continuum), as well as (b) hydrogen atoms (H- continuum), (c) free-free and free-bound transitions of protons in the field of hydrogen atoms (H2 continuum), and (d) the continuous emission coefficient of quasi-molecular hydrogen. For an optically thin homogeneous emitting layer the emission coefficient in watts cm-3sr-1cm-1 is given by [12]:

(a) H continuum: The first term in the square brackets represents the contribution from the H continuum [13] with high density corrections according to reference [10]. Ne is the electron density (cm-3), T is the temperature (K), λ is the wavelength (cm), c2 is the second radiation constant (=1.43879 cm K), E; is the ionization potential of hydrogen, AE, is the lowering of the ionization potential given by AE1=e2/pp [10], where PD=[kТ/8πе2 Ne]1/2, Yƒƒ and Y are the free-free and free-bound Gaunt factors respectively [14] averaged over a Maxwellian velocity distribution, and n is the upper state principal quantum number. The summation is taken from a value of n defined by

nmin≥ [Ei/(hc/λ+AE;)]1/2

to n = 15 unless high density corrections require a lower cutoff. When the high density correction allows n 16 all free-bound transitions are approximately taken into account by integrating 1/n3 exp (Ei/n2kT) from n = 16 to n=nmax, where nmax is the quantum number of the last energy level below E¿- AE¿, i.e., max (E/AE) 1/2 and replacing so=Yso (n=15). If high density corrections require nmax < 15 then the term multiplied by y is omitted.

The averaged free-free and free-bound Gaunt factors obtained from reference [14] are tabulated in tables 1 and 2 respectively. This tabulation is done so the Gaunt factors can be easily obtained for a given wavelength and temperature.

(b) H- continuum: The term containing G(λ, T) represents the contribution from the H- continuum, where NH is the density of neutral hydrogen. G(λ, T) is given in terms of the free-bound and free-free Habsorption coefficients [3, 4, 5] as

G(λ, T) = (a(H−)μ+α(H ̄)ƒ) [1-exp

(— c2/XT)]B(λ, T)kT (2)

where B(X, T) is the Planck function and 1-exp (-c2/AT) corrects for induced emission. The absorption coefficient for free-free H- was interpolated for wavelengths below 3038 Å and extrapolated for temperatures above 10080 K. The parametric function used for the extension was found to fit the original data [4] over all wavelengths and temperatures to within 0.1 percent in the standard deviation. Although the interpolation below 3038 Å may be questionable around 1130 Å [15] for the conditions considered here, the H-free-free contribution to the total emission continuum is 20 percent or less below 3038 Å and above 10080 K. Therefore, a possible 5 percent error in this contribution would reflect an uncertainty of 1 percent or less in the total.

(c) He continuum: The term containing F(X, T) represents the contribution from the H continuum where Np is the proton density (Np≈Ne). F(X, T) is represented in terms of the free-bound and free-free H absorption

This transition is the only important contribution to the quasi-molecular hydrogen continuum for wavelengths larger than 1700 A. Below this wavelength, quasi-molecular continuum contributions arise from other molecular states [16]. These are not considered here since the available data are incomplete and of uncertain reliability.

(e) Other possible continuum contributions: Contributions from other species are estimated to be insignificant for the stated conditions, e.g., H†1⁄2 (11). A forbidden continuum resulting from H- has also been hypothesized [17], and should occur principally in the near-infrared. A more recent paper [18] states that this continuum is significant only for electron densities of roughly 1020cm-3 which is three orders of magnitude above the highest density considered in this calculation.

(f) Contributions from lines superimposed on the continuum: The intensity from the wings of Starkbroadened hydrogen lines makes a non-negligible contribution at higher densities in certain spectral regions [19]. This contribution must be included as part of the continuous emission from hydrogen in these spectral regions. Extensive calculations of complete hydrogen line profiles including the wings for the first four lines of the Lyman and Balmer spectral series have been carried out [20, 21]. The far wings of other hydrogen lines can also be estimated [22] to within 10 percent. In the vacuum ultraviolet, molecular line emission from the strong Lyman and Werner bands of H2 is superimposed on the continuum and should be significant up to the highest temperature considered here. Because of the very complicated spectrum of H2 and the incompleteness of the available f-value data, the H2 molecular line emission could not be included in these calculations.

3 The absorption coefficients as stated in eq (3) are those of ref. [6] divided by 4, ie.. per steradian.