AnalyticaldevelopmentsintheWong–Fung–Tam–Gao radiationmodelofthermaldiffusivity U.Lucia a,1 ,G.Maino b,c, * a INFM, Istituto Nazionale per la Fisica della Materia, Corso Perrone 24, I-16152 Genova, Italy b ENEA, Divisione Fisica Applicata, via don G. Fiammelli 2, I - 40129 Bologna, Italy c Facolt  a di Conservazione dei Beni Culturali, University of Bologna, via Mariani 5, I-48100 Ravenna, Italy Abstract When the thermal diffusivity, v, of a thin film on a substrate is measured by means of the mirage method, the photothermaldeflectionoftheprobebeamisdeterminedbytheheatradiationfieldcontributedbythefilmandthe substrate,heatedbythepumpbeam.Atwo-dimensionalalgorithmisherepresentedinordertodeducethemeasureof thediffusivitiesofthefilmandthesubstrateinonesetofmiragedetectionfromtheexperimentaldata. Ó 2003ElsevierB.V.Allrightsreserved. PACS: 44.40.+a;44.50.+f;65.50.+m;65.90.+1 Keywords: Thermal diffusivity; Photothermal deflection; Radiation field; Thermophysical properties; Mirage effect; Film; Charac- terizationofmaterials 1. Introduction Thermal diffusivity is one of the physical quantities providing reliable information on the occurrence of phase transitions [1,2]. Therefore, analysis of thermal diffusivity is one of the basic means and a crucial clue in the study and char- acterizationofmaterials.Ontheapplicationside, it is necessary to know the thermal diffusivity of the materials used in making microelectronic componentsinordertooptimizetherelevantop- erationsinmanycasesofphysicalandengineering interests. Non-destructive methods have been in- troduced to measure the diffusivity of crystals [1] and, within this context, the relevance of the mi- rageeffecthasbeenpointedout[3–5]. As for this experimental technique, the sample is represented by a film-substrate composite and, whenthefilmisverythin,thenasignificantpor- tion of the pump power is absorbed by the sub- strate and the thermal wave can reach the field position where the probe is passing (see, Fig. 1). Up to recent years, theoretical analyses did not takeintoaccounttheradiationfieldgeneratedby the substrate. Wong et al. [1] have firstly derived an explicit expression for the thermal diffusivity forthefilm-substratecompositesamples,including the contribution to the temperature field at the probebeamposition,fromthefilmaswellasthe * Correspondingauthor.Address:AppliedPhysicsDivision, ENEA, Via Don G. Fiammelli 2, 40129 Bologna, Italy. Tel.: +39-051-6098206/6098359. E-mail address: maino@bologna.enea.it (G.Maino). 1 Permanent address: via S. G. Bosco 43, I-15100 Alessan- dria,Italy. 0168-583X/$-seefrontmatter Ó 2003ElsevierB.V.Allrightsreserved. doi:10.1016/S0168-583X(03)01548-9 NuclearInstrumentsandMethodsinPhysicsResearchB213(2004)139–143 www.elsevier.com/locate/nimb