JOURNAL OF THERMOPHYSICS AND HEAT TRANSFER Vol. 20, No. 1, January-March 2006 Temperature and Wavelength-Dependent Spectral Absorptivities of Metallic Materials in the Infrared Samuel B. Boyden* New Mexico State University, Las Cruces, New Mexico 88003 and Yuwen Zhangt University of Missouri-Columbia, Columbia, Missouri 65211 The optical constants and absorptivity of selected elemental metals and alloys are calculated based on the Drude-type model. The calculations are made primarily at the C02 (10.6-μm) and the Nd:YAG (1.06-μm) laser wavelengths, with consideration for laser material processing. The absorptivities at these wavelengths are calculated for some important metallic materials. For the application of laser material processing, temperature-dependent values are calculated based on available experimental data. The absorptivity values for alloys are calculated by assuming that the sum of contributions of the proportionality of the valence electrons to the effective mass of an electron for each constituent metal is equal to the proportionality of the valence electrons to the effective mass of an electron for the alloy. The absorptivities of the elemental metals at 10.6 μm agreed with experimental data very well, except for transition metals. Agreement of alloy and element absorptivity calculated values and experimental data is good at 10.6 μm but not at 1.06 μm. Overall the calculation by the Drude model gives good estimates of absorptivity at 10.6 μm. e k m m* N n R T a y eo A. p CTo T w Wp = = Nomenclature electron electric charge, C complex refractive index, dimensionless mass of the electron, kg optical mass of the electron, kg number of free electrons per cubic centimeter refractive index, dimensionless reflectivity, dimensionless temperature, K spectral absorptivity, dimensionless damping frequency, s- 1 permittivity of free space, F · m- 1 wavelength of incident radiation, μm resistivity of material, ohm · m conductivity of material, ohm- 1 · m- 1 relaxation time of electrons, s angular frequency of radiation, rad . s- 1 plasma frequency, rad · s 1 Introduction A NY model for laser processing of materials must have a com- plete description of the coupling between the laser source and the material. The coupling is defined by the spectral absorp- tivity of the material at the wavelength of operation; hence, the normal spectral absorptivity is a critical parameter of interest for many applications in this area. 1 Optical properties of bulk metals are typically functions of wavelength, temperature, surface geom- etry (roughness), incident intensity, and physical atomic structural and electrical properties of the material. Because absorptivity is Received 11 January 2005; revision received 28 April 2005; accepted for publication 28 April 2005. Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rose- wood Drive, Danvers, MA 01923; include the code 0887-8722/06 $10.00 in correspondence with the CCC. *Graduate Research Assistant, Department of Mechanical Engineering. t Associate Professor, Department of Mechanical and Aerospace Engi- neering. Senior Member AIAA. 9 dependent on temperature, it is also important to consider the ab- sorptivity change in the modeling of laser heating. 2 Wavelength-dependent absorptivity data may be found by Touloukian, 3 Touloukian and DeWitt, 4 and Touloukian and Ho. 5 Temperature-dependent absorptivity data are few, 6 possibly due to the difficulty in conducting temperature-dependent experiments. The experimental study by Wieting and Schriempf provides the only temperature-dependence absorptivity data available for alloys. Zhang and Modest 8 presented experimental results on temperature- dependent absorptances of ceramics for Nd: YAG and C0 2 laser pro- cessing applications. Temperature-dependent absorptivity of many metallic materials at the temperature near their melting points is not directly available in the existing literature. Although obtaining the absorptivity value from experimental in- vestigation is preferred, calculation of the absorptivity based on a theoretical model is also important for the situation when the ab- sorptivity is not readily available for a particular material of interest. One method to predict the absorptivity of an electromagnetic field that obeys Maxwell's equations is to use the following equation based on the Fresnel reflection relation: (n - 1) 2 + k 2 a=1-R=l------ (n + 1)2 + k2 (1) where n and k are the real and imaginary parts of refractive index, which can be determined using various models. The Drude theory is a modified oscillator-type model developed for reflection and absorption estimation and has been used by many researchers (see Refs. 9-12). This model uses the electrical properties of the material and optical properties of the material. The Hagen-Ruben relation is a model, which provides ease of calculation and can be applied for frequencies much less than the mean collision rate of the electrons in the metal. 13 Sokolov 14 presented a method of determining optical properties of alloys, with a modified damping function. Dausinger and Shen 6 provided temperature-dependent models based on the Drude model. Weaver and Frederikse 15 presented a compilation of data sets for a number of practical metals. The temperature-dependent absorptivity of selected pure metals and alloys at infrared regions of the electromagnetic spectrum were calculated and will be presented and compared with the existing experimental data. The absorptivity values of the selected metals are of necessity for accurate laser processing models.